Barreira_pspm Lectures on Lyapunov Exponents Pesin

Proceedings of Symposia in Pure Mathematics

Lectures on Lyapunov Exponents and

Smooth Ergodic Theory

L. Barreira and Ya. Pesin

Contents

Introduction

1. Lyapunov Exponents for Differential Equations

2. Abstract Theory of Lyapunov Exponents

3. Regularity of Lyapunov Exponents Associated with Differential Equations

4. Lyapunov Stability Theory

5. The Oseledets Decomposition

6. Dynamical Systems with Nonzero Lyapunov Exponents. Multiplicative Ergodic

Theorem

7. Nonuniform Hyperbolicity. Regular Sets

8. Examples of Nonuniformly Hyperbolic Systems

9. Existence of Local Stable Manifolds

10. Basic Properties of Local Stable and Unstable Manifolds

11. Absolute Continuity. Holonomy Map

12. Absolute Continuity and Smooth Invariant Measures

13. Ergodicity of Nonuniformly Hyperbolic Systems Preserving Smooth Measures

14. Local Ergodicity

15. The Entropy Formula

16. Ergodic Properties of Geodesic Flows on Compact Surfaces of Nonpositive Curvature

Appendix A. Holder Continuity of Invariant Distributions, by M. Brin

Appendix B. An Example of a Smooth Hyperbolic Measure with Countably Many Ergodic

Components, by D. Dolgopyat, H. Hu and Ya. Pesin

2000 Mathematics Subject Classification. Primary: 37D25, 37C40.Key words and phrases. Lyapunov exponents, nonuniformly hyperbolic dynamical

systems, smooth ergodic theory.L. Barreira was partially supported by FCT’s Funding Program and the NATO grant

CRG 970161. Ya. Pesin was partially supported by the National Science Foundation grant#DMS-9704564 and the NATO grant CRG 970161.

c©2000 American Mathematical Society

1

2 L. BARREIRA AND YA. PESIN

Introduction

This manuscript is based on lectures given by Ya. Pesin at the AMSSummer Research Institute (Seattle, Washington, 1999). It presents the coreof the nonuniform hyperbolicity theory of smooth dynamical systems. Thistheory was originated in [26, 27, 28, 29] and has since become a mathemat-ical foundation for the paradigm which is widely-known as “deterministicchaos” — the appearance of irregular “chaotic” motions in pure determinis-tic dynamical systems. We follow the original approach by Ya. Pesin makingsome improvements and necessary modifications.

The nonuniform hyperbolicity theory is based on the theory of Lyapunovexponents which was originated in the works of Lyapunov [19] and Perron[25] and was developed further in [7]. We provide an extended excursioninto this theory. This includes the abstract theory of Lyapunov exponents— that allows one to introduce and study the crucial concept of Lyapunov-Perron regularity (see Section 2) — as well as the advanced stability theoryof differential equations (see Sections 1, 3, and 4).

Using the language of the theory of Lyapunov exponents one can viewnonuniformly hyperbolic dynamical systems as those where the set of pointswhose Lyapunov exponents are all nonzero is “large”, for example, has fullmeasure with respect to an invariant Borel measure (see Sections 6 and 7).In this case the fundamental Multiplicative Ergodic theorem of Oseledets[24] implies that almost every point is Lyapunov–Perron regular. Thus, thepowerful theory of Lyapunov exponents applies and allows one to carry outa thorough analysis of the local stability of trajectories.

The crucial difference between the classical uniform hyperbolicity andits weakened version of nonuniform hyperbolicity is that the hyperbolicityconditions can get worse when one moves along the trajectory of a nonuni-formly hyperbolic point. However, if this point is Lyapunov–Perron regularthen the worsening occurs with subexponential rate and the contraction andexpansion along stable and unstable directions prevail.

One of the crucial manifestations of this fact is the fundamental StableManifold theorem that was established in [27] and is a generalization of theclassical Hadamard–Perron theorem. In Section 9 we present the proof ofthe Stable Manifold theorem following the original approach in [27] whichis essentially an elaboration of the Perron method. In Section 10 we sketchthe proof of a slightly more general version of the Stable Manifold theorem(known as the Graph Transform Property) which is due to Hadamard. Wealso describe several main properties of local stable manifolds of which oneof the most important is that their sizes may decrease along trajectories onlywith subexponential rate (and thus the contraction prevails).

There are several methods for establishing nonuniform hyperbolicity.One of them, which we consider in these lectures, is to show that the Lya-punov exponents of the system are nonzero. One of the first examples was

LECTURES ON LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY 3

constructed in [26]. It is a three dimensional flow on a compact smooth Rie-mannian manifold which is a reconstruction of an Anosov flow. It revealssome mechanisms for the appearance of zero Lyapunov exponents (see thediscussion in Section 8).

Another example are geodesic flows on compact smooth Riemannianmanifolds of nonpositive curvature. Let us stress that geodesic flows havealways been a good source of examples and have provided inspiration fordeveloping the hyperbolicity theory. In these lectures we consider only ge-odesic flows on surfaces of nonpositive curvature and show that they arenonuniformly hyperbolic on an open and dense set (see Section 8).

One of the main goals of the nonuniform hyperbolicity theory is to de-scribe the ergodic properties of a smooth dynamical system preserving asmooth invariant measure. This is done in Sections 11, 12, 13, and 15 wherewe show that such a system has ergodic components of positive measure andalso establish the entropy formula that expresses the entropy of the systemvia its positive Lyapunov exponents. Finally, in Section 16 we apply theseresults to the geodesic flows on compact surfaces of nonpositive curvature.

Acknowledgment. We want to express our deep gratitude to M. Brin,D. Dolgopyat, C. Pugh, and J. Schmeling for their valuable comments anddiscussions.

The lectures were used as a handout for a graduate course on DynamicalSystems with Nonzero Lyapunov Exponents that Ya. Pesin taught during theFall 2000 semester at The Pennsylvania State University. Ya. Pesin wouldlike to thank students for their patience during the course and numerousfruitful remarks which helped improve the text. The authors are speciallygrateful to students C. Carter, T. Fisher, R. Gunesch, I. Ugarcovici, andA. Windsor for many valuable comments and corrections.

1. Lyapunov Exponents for Differential Equations

Consider a linear differential equation

v = A(t)v, (1.1)

where v(t) ∈ Cn and A(t) is a n×n matrix with complex entries dependingcontinuously on t ∈ R. We assume that the matrix function A(t) is bounded,i.e.,

sup‖A(t)‖ : t ∈ R <∞. (1.2)

It follows that for every v0 ∈ Cn there exists a unique solution v(t) = v(t, v0)of Equation (1.1) which is defined for every t ∈ R and satisfies the initialcondition v(0, v0) = v0.

Consider the trivial solution v(t) = 0 for t ≥ 0. If the matrix functionA(t) is constant, i.e., A(t) = A for all t ≥ 0, then the trivial solution isasymptotically (and indeed, exponentially) stable if and only if the real partof every eigenvalue of the matrix A is negative. A similar result holds in thecase when the matrix function A(t) is periodic.


In order to characterize the stability of the trivial solution in the gen-eral case we introduce the Lyapunov exponent χ+ : Cn → R ∪ −∞ ofEquation (1.1) by the formula

χ+(v) = lim supt→+∞

1

tlog‖v(t)‖, (1.3)

for each v ∈ Cn, where v(t) is the unique solution of (1.1) satisfying the ini-tial condition v(0) = v. It follows immediately from (1.3) that the Lyapunovexponent χ+ satisfies:

1. χ+(αv) = χ+(v) for each v ∈ Cn and α 6= 0;2. χ+(v + w) ≤ maxχ+(v), χ+(w) for each v, w ∈ Cn;3. χ+(0) = −∞.

The function χ+ can take on only finitely many distinct values χ+1 < · · · <

χ+s on Cn \ 0, where s ≤ n (see Section 2 below). Each number χ+

ioccurs with some multiplicity ki so that

∑si=1 ki = n. Note that for every

ε > 0 there exists a constant Cε > 0 such that for every solution v(t) ofEquation (1.1) and any t ≥ 0 we have

‖v(t)‖ ≤ Cεe(χ+

s +ε)t‖v(0)‖. (1.4)

It follows from (1.4) that if

χ+s < 0 (1.5)

then for any sufficiently small ε > 0, every solution v(t) → 0 as t → +∞with an exponential rate. In other words the trivial solution v(t) = 0 isasymptotically (and indeed, exponentially) stable.

We now consider a nonlinear differential equation

u = A(t)u+ f(t, u), (1.6)

which is a perturbation of (1.1). We assume that f(t, 0) = 0 and hence,u(t) = 0 is a solution of (1.6). We also assume that there exists a neigh-borhood H of 0 in Cn such that f is continuous on [0,∞)×H and that forevery u1, u2 ∈ H and t ≥ 0, we have

‖f(t, u1) − f(t, u2)‖ ≤ K‖u1 − u2‖q (1.7)

for some constants K > 0 and q > 1. This means that the perturbationf(t, u) is small in H. The number q is called the order of the perturbation.

One of the main problems in the Lyapunov stability theory is whetherCondition (1.5) implies that the solution u(t) = 0 of the perturbed equa-tion (1.6) is asymptotically (and exponentially) stable.

Perron showed that in general the answer is negative (see [25]).

Example 1.1. Consider the following nonlinear system of differentialequations in R2:

u1 = [−ω − a(sin log t+ cos log t)]u1,

u2 = [−ω + a(sin log t+ cos log t)]u2 + |u1|λ+1,(1.8)


for some positive constants ω, a, and λ. It is a perturbation of the followinglinear system of differential equations

v1 = [−ω − a(sin log t+ cos log t)]v1,

v2 = [−ω + a(sin log t+ cos log t)]v2.(1.9)

We assume that

a < ω < (2e−π + 1)a and 0 < λ <2a

ω − a− eπ. (1.10)

Notice that the function f(t, (u1, u2)) = (0, |u1|λ+1) is autonomous, i.e., itdoes not depend on t, and that it satisfies Condition (1.7) with q = λ+1 > 1.

The general solution of (1.8) is given by

u1(t) = c1e−ωt−at sin log t,

u2(t) = c2e−ωt+at sin log t + c1

2e−ωt+at sin log t

∫ t

t0

e−(2+λ)aτ sin log τ−ωλτ dτ,

(1.11)

while the general solution of (1.9) is given by

v1(t) = d1e−ωt−at sin log t, v2(t) = d2e

−ωt+at sin log t.

Here c1, c2, d1, d2, and t0 are arbitrary numbers.It is easy to check that the values of the Lyapunov exponent associated

with Equation (1.9) are χ+1 = χ+

2 = −ω + a < 0.Let u(t) = (u1(t), u2(t)) be a solution of the nonlinear system of differ-

ential equations (1.8). In view of (1.11) it is also a solution of the linearsystem of differential equations

u1 = [−ω − a(sin log t+ cos log t)]u1,

u2 = [−ω + a(sin log t+ cos log t)]u2 + δ(t)u1,(1.12)

whereδ(t) = sgn c1|c1|λe−ωλt−aλt sin log t.

Note that|δ(t)| ≤ |c1|λe(−ω+a)λt

and thus by (1.10), Condition (1.2) holds for Equation (1.12). Fix 0 < ε <π/4 and for each k ∈ N set

tk = e2kπ− 12π, t′k = e2kπ− 1

2π−ε.

Clearly, tk → ∞ and t′k → ∞ as k → ∞. One can also see that∫ tk

t0

e−(2+λ)aτ sin log τ+ωλτ dτ >

∫ tk

t′k

e−(2+λ)aτ sin log τ+ωλτ dτ.

For every τ ∈ [t′k, tk] we have

2kπ − π

2− ε ≤ log τ ≤ 2kπ − π

2,

(2 + λ)aτ cos ε ≤ −(2 + λ)aτ sin log τ.


This implies that∫ tk

t′k

e−(2+λ)aτ sin log τ+ωλτ dτ ≥∫ tk

t′k

e(2+λ)aτ cos ε−ωλτ dτ.

Set r = (2 + λ)a cos ε− ωλ. It follows that if k ∈ N is sufficiently large then∫ tk

t0

e−(2+λ)aτ sin log τ+ωλτ dτ >

∫ tk

t′k

erτ dτ > certk ,

where c = (1 − e−ε)/r. Set

t∗k = tkeπ = e2kπ+ 1

2π.

We obtain

eat∗k

sin log t∗k

∫ t∗k

t0

e−(2+λ)aτ sin log τ+ωλτ dτ > eat∗k

∫ tk

t0

e−(2+λ)aτ sin log τ+ωλτ dτ

> ceat∗k+rtk = ce(a+re−π)t∗

k .

It follows from (1.10) that if c1 6= 0 and ε is sufficiently small, then theLyapunov exponent of any solution u(t) of (1.8) (which is also a solution of(1.12)) satisfies

χ+(u) ≥ −ω + a+ re−π = −ω + a+ [(2 + λ)a cos ε− ωλ]e−π > 0.

Therefore, the solution u(t) is not asymptotically stable. This completes theconstruction of the example.

Lyapunov introduced regularity conditions which guarantee asymptotic(and indeed, exponential) stability of the solution u(t) = 0 of the perturbedequation (1.6). Although there are many different ways to state the regu-larity conditions (which we discuss below), for a given differential equationthe regularity is often difficult to verify.

We now state the Lyapunov Stability Theorem (see [7]). It claims thatunder an additional assumption known as forward regularity, Condition (1.5)indeed implies the stability of the trivial solution u(t) = 0 of (1.6).

Theorem 1.2. Assume that the Lyapunov exponent χ+ of Equation(1.1), with the matrix function A(t) satisfying (1.2), is forward regular andsatisfies Condition (1.5). Then the solution u(t) = 0 of the perturbed equa-tion (1.6) is asymptotically and exponentially stable.

The notion of forward regularity will be introduced and discussed inSections 2 and 3.

2. Abstract Theory of Lyapunov Exponents

Before we proceed with the proof of the Lyapunov Stability Theorem 1.2we discuss the notions of Lyapunov exponent and regularity. It is moreconvenient to do this in a formal axiomatic setting using the basic properties1, 2, and 3 of the Lyapunov exponent χ+ described in Section 1. This allows


one to study Lyapunov exponents in other situations such as sequences ofmatrices, cocycles, dynamical systems with continuous and discrete time,etc.

Let V be an n-dimensional real vector space. A function χ : V → R ∪−∞ is called a Lyapunov characteristic exponent or simply a Lyapunovexponent on V if:

1. χ(αv) = χ(v) for each v ∈ V and α ∈ R \ 0;2. χ(v + w) ≤ maxχ(v), χ(w) for each v, w ∈ V ;3. χ(0) = −∞ (normalization property).

We describe some basic properties of Lyapunov exponents.

Theorem 2.1. If χ is a Lyapunov exponent, then the following proper-ties hold:

1. if v, w ∈ V are such that χ(v) 6= χ(w) then

χ(v + w) = maxχ(v), χ(w);

2. if v1, . . ., vm ∈ V and α1, . . ., αm ∈ R \ 0 then

χ(α1v1 + · · · + αmvm) ≤ maxχ(vi) : 1 ≤ i ≤ m;

if, in addition, there exists i such that χ(vi) > χ(vj) for all j 6= ithen

χ(α1v1 + · · · + αmvm) = χ(vi);

3. if for some v1, . . ., vm ∈ V \ 0 the numbers χ(v1), . . ., χ(vm) aredistinct, then the vectors v1, · · · , vm are linearly independent; if, inaddition, m = n, then the vectors v1, . . ., vn form a basis of V ;

4. the function χ can take no more than n distinct finite values.

Proof. Suppose that χ(v) < χ(w). We have

χ(v + w) ≤ χ(w) = χ(v + w − v) ≤ maxχ(v + w), χ(v).

It follows that if χ(v + w) < χ(v) then χ(w) ≤ χ(v) which contradicts ourassumption. Hence, χ(v + w) ≥ χ(v), and thus, χ(v + w) = χ(w). State-ment 1 follows. Statement 2 is an immediate consequence of Statement 1and Properties 1 and 2 in the definition of Lyapunov exponent.

In order to prove Statement 3 assume on the contrary that the vectorsv1, . . ., vm are linearly dependent, i.e., α1v1 + · · · + αmvm = 0 with notall constants αi equal to zero, while χ(v1), . . ., χ(vm) are distinct. ByStatement 2 and Property 3 in the definition of Lyapunov exponent, weobtain

−∞ = χ(α1v1 + · · · + αmvm) = maxχ(vi) : 1 ≤ i ≤ m and αi 6= 0 6= −∞.

This contradiction implies Statement 3. Statement 4 follows from State-ment 3.


By Theorem 2.1, the Lyapunov exponent χ can take on only finitelymany distinct values on V \ 0. We denote them by

χ1 < · · · < χs

for some s ≤ n. In general, χ1 may be −∞. For each 1 ≤ i ≤ s, define

Vi = v ∈ V : χ(v) ≤ χi. (2.1)

Put V0 = 0. It follows from Theorem 2.1 that Vi is a linear subspace of Vfor each i, and

0 = V0 $ V1 $ · · · $ Vs = V. (2.2)

We call a collection V = Vi : i = 0, . . ., s of linear subspaces of V satisfy-ing (2.2) a linear filtration or simply a filtration of V .

The following result gives an equivalent characterization of Lyapunovexponents in terms of filtrations.

Theorem 2.2. A function χ : V → R ∪ −∞ is a Lyapunov exponentif and only if there exist numbers χ1 < · · · < χs for some 1 ≤ s ≤ n, and afiltration V = Vi : i = 0, . . ., s of V such that:

1. χ(v) ≤ χi for every v ∈ Vi;2. χ(v) = χi for every v ∈ Vi \ Vi−1 and 1 ≤ i ≤ s;3. χ(0) = −∞.

Proof. If χ is a Lyapunov exponent then the filtration

V = Vi : i = 0, . . ., sdefined by (2.1) satisfies Conditions 1 and 3 of Theorem 2.2. Moreover, forany v ∈ Vi \ Vi−1 we have χi−1 < χ(v) ≤ χi. Since χ takes no value strictlybetween χi−1 and χi, we obtain χ(v) = χi and Condition 2 follows.

Now suppose that a function χ and a filtration V satisfy the conditionsof the theorem. Observe that v ∈ Vi\Vi−1 if and only if αv ∈ Vi\Vi−1 for anyα ∈ R \ 0. Therefore, by Condition 2, χ(αv) = χ(v). Choose now vectorsv1, v2 ∈ V . Let χ(vj) = χij for j = 1, 2. It follows from Conditions 1 and 2that (2.1) holds. Therefore, vj ∈ Vij for j = 1, 2. Without loss of generalitywe may assume that i1 < i2. This implies that v1 + v2 ∈ Vi1 ∪ Vi2 = Vi2 .Hence, by Condition 1, we have

χ(v1 + v2) ≤ χi2 = maxχ(v1), χ(v2),and thus, χ is a Lyapunov exponent.

We refer to the filtration V = Vi : i = 0, . . ., s of V , defined by (2.1),as the filtration of V associated to χ and denote it by Vχ. We call thenumber

ki = dimVi − dimVi−1

the multiplicity of the value χi, and the collection of pairs

Spχ = (χi, ki) : 1 ≤ i ≤ sthe Lyapunov spectrum of χ.


Given a filtration V = Vi : i = 0, . . ., s of V and numbers χ1 < · · · <χs define a function χ : V → R∪−∞ by χ(v) = χi for every v ∈ Vi \Vi−1,1 ≤ i ≤ s, and χ(0) = −∞. It is easy to see that the function χ andthe filtration V satisfy the conditions of Theorem 2.2, and thus determine aLyapunov exponent on V .

We now consider a special class of bases in Rn which are well-adaptedto filtrations in Rn.

A basis v = (v1, . . ., vn) of V is said to be normal with respect to thefiltration V = Vi : i = 0, . . ., s if for every 1 ≤ i ≤ s there exists a basisof Vi composed of ni vectors from v1, . . ., vn. A normal basis v is calledordered if for every 1 ≤ i ≤ s, the vectors v1, . . ., vni

form a basis of Vi.Note that for every filtration V there exists a basis which is normal with

respect to V. Moreover, two filtrations coincide if and only if each normalbasis with respect to one of them is also normal with respect to the otherone. Furthermore, if χ is a Lyapunov exponent and v is a normal basis withrespect to the filtration Vχ, then among the numbers χ(v1), . . ., χ(vn) thevalue χi occurs exactly ki times, for each i = 1, . . ., s. Hence,

n∑

j=1

χ(vj) =s∑

i=1

kiχi. (2.3)

We use this observation to prove the following result.

Theorem 2.3. A basis v of V is normal with respect to the filtration Vχ

associated to the Lyapunov exponent χ if and only if

inf

n∑

j=1

χ(wj) : w is a basis of V

=

n∑

j=1

χ(vj). (2.4)

Proof. One can easily verify that there exists a normal ordered basis v

adapted to Vχ.

Lemma 2.4. If w is a basis of V for which χ(w1) ≤ · · · ≤ χ(wn), then:

1. χ(wj) ≥ χ(vj) for every 1 ≤ j ≤ n, and χ(wn) = χ(vn);2.∑n

j=1 χ(wj) ≥∑n

j=1 χ(vj);

3. w is normal if and only if χ(wj) = χ(vj) for every 1 ≤ j ≤ n;4. w is normal if and only if

∑nj=1 χ(wj) =

∑nj=1 χ(vj).

Proof of the lemma. We notice that since χ1 is the minimal valueof χ on V \0 we have χ(wj) ≥ χ(vj) = χ1 for every j = 1, . . ., n1. Assumethat χ(wn1+1) = χ1. Then χ(w1) = · · · = χ(wn1+1) = χ1 and

n1 ≥ dim spanw1, . . ., wn1+1 = n1 + 1,

where spanZ denotes the linear space generated by the set of vectors Z. Thiscontradiction implies that χ(wn1+1) ≥ χ2 and hence χ(wj) ≥ χ(vj) = χ2 forevery j = n1 + 1, . . ., n2.

Repeating the same argument finitely many times we obtain χ(wj) ≥χ(vj) for every 1 ≤ j ≤ n. In particular, χ(wn) = χ(vn) since χ(vn) is the


maximum value of χ. Statement 1 follows. Statement 2 is an immediateconsequence of Statement 1.

By (2.1), χ(wj) = χ(vj) for every 1 ≤ j ≤ n if and only if w1, . . ., wni

is a basis of Vi for every 1 ≤ i ≤ s and hence, if and only if the basis w isnormal. This implies Statement 3. The last statement is a consequence ofStatements 2 and 3.

By the lemma, the infimum in (2.4) is equal to

inf

n∑

j=1

χ(wj) : w is a normal basis of V

=

s∑

i=1

kiχi.

In view of Statement 4 in the lemma, the basis v is normal if and only if(2.3) holds, and hence, if and only if (2.4) holds.

There is a very useful Lyapunov’s construction of normal bases whichwe now describe. This construction defines a sequence of bases

wi = (wi1, . . ., win) = (w1, . . ., wi, vi+1, . . ., vn),

which makes the sum∑n

j=1 χ(wij) decrease as i increases. Since χ takes ononly finitely many values, this process ends up after a finite number of steps.

Set wn = vn and let i < n. If

w ∈ spanvi, . . ., vn \ spanvi+1, . . ., vn, (2.5)

then w = α(vi + w′i) for some real number α 6= 0 and some vector

w′i ∈ spanvi+1, . . ., vn.

Since χ takes on only finitely many values one can choose a vector w′i such

that if wi = vi + w′i then χ(wi) is the minimum of χ(w) for w satisfying

(2.5). Observe that there exists a linear transformation A : V → V whichhas an upper triangular form with respect to the basis v such that wi = Avi

for each i = 1, . . ., n. Moreover, since

wi − vi ∈ spanvi+1, . . ., vn,all entries on the diagonal of A are equal to 1, and hence detA 6= 0 andw = (w1, . . ., wn) is a basis.

We now show that w is a normal basis. Assuming the contrary and

using Lemma 2.4 we can find a vector v such that v =∑k

j=1 αjwij for some1 ≤ k ≤ n, i1 < i2 < · · · < ik, and αj 6= 0 for j = 1, . . ., k, which satisfies

χ(v) < maxχ(wi1), . . ., χ(wik). (2.6)

Since v ∈ spanwi1 , . . ., win, we have χ(v) ≥ χ(wi1) and hence,

χ(v − α1wi1) ≤ maxχ(v), χ(wi1) = χ(v). (2.7)

By (2.6) and (2.7),

χ(v − α1wi1) < maxχ(wi2), . . ., χ(wik).


Iterating this procedure we obtain

χ(wik) = χ(v −k−1∑

j=1

αjwij ) < χ(wik).

This contradiction implies that the basis w is normal.We now establish the behavior of normal bases under linear transforma-

tions.Let v be a normal ordered basis with respect to a filtration V = Vi :

i = 0, . . ., s of a vector space V , and A : V → V an invertible linear trans-formation.

Theorem 2.5. The following properties are equivalent:

1. (Av1, . . ., Avn) is a normal ordered basis;2. the transformation A preserves the filtration V, i.e, AVi = Vi for

every 1 ≤ i ≤ s;3. the transformation A, with respect to a basis which is a reordering

of the basis v, has the lower block-triangular form

A1 0 · · · 0

A2. . .

.... . . 0

As

,

where each Ai is a ki × ki matrix with detAi 6= 0.

Proof. The proof is elementary.

Let W be another filtration of V . As a consequence of the above state-ment one can show that there exists a basis which is normal with respect toboth filtrations. Indeed, Let v be a normal basis with respect to W. Thereexists a lower triangular n × n matrix such that w = (Av1, . . ., Avn) is anormal basis with respect to V. One can easily show that the basis w is alsonormal with respect to W.

Let V be a vector space and W a dual vector space to V . Let alsov = (v1, . . ., vn) be a basis in V and w = (w1, . . ., wn) a basis in W . Wesay that v is dual to w and write v ∼ w if 〈vi, wj〉 = δij for each i and j.

Let χ be a Lyapunov exponent on V and χ a Lyapunov exponent on W .We say that the exponents χ and χ are dual and write χ ∼ χ if for any dualbases v and w, and every 1 ≤ i ≤ n, we have

χ(vi) + χ(wi) ≥ 0.

We denote by χ′1 ≤ · · · ≤ χ′

n the values of χ counted with their multiplicities,i.e., χ′

i = χ(vi) for some normal ordered basis v of V . Similarly, we denoteby χ′

1 ≥ · · · ≥ χ′n the values of χ counted with their multiplicities.

We define the regularity coefficient of the dual Lyapunov exponents χand χ by

γ(χ, χ) = minmaxχ(vi) + χ(wi) : 1 ≤ i ≤ n,


where the minimum is taken over all dual bases v and w of V and W . ThePerron coefficient of χ and χ is defined by

π(χ, χ) = maxχ′i + χ′

i : 1 ≤ i ≤ n.Theorem 2.6. The following statements hold:

1. π(χ, χ) ≤ γ(χ, χ);2. if χ ∼ χ, then 0 ≤ π(χ, χ) ≤ γ(χ, χ) ≤ nπ(χ, χ).

Proof. We begin with the following lemma.

Lemma 2.7. Given numbers λ1 ≤ · · · ≤ λn and µ1 ≥ · · · ≥ µn, and apermutation σ of 1, . . ., n, we have

minλi + µσ(i) : 1 ≤ i ≤ n ≤ minλi + µi : 1 ≤ i ≤ n,maxλi + µσ(i) : 1 ≤ i ≤ n ≥ maxλi + µi : 1 ≤ i ≤ n.

Proof of the lemma. Notice that the second inequality follows fromthe first one in view of the following relations:

maxλi + µσ(i) : 1 ≤ i ≤ n = −min−µσ(i) − λi : 1 ≤ i ≤ n= −min−µi − λσ−1(i) : 1 ≤ i ≤ n≥ −min−µi − λi : 1 ≤ i ≤ n,= maxλi + µi : 1 ≤ i ≤ n.

We now prove the first inequality. We may assume that σ is not the identitypermutation (otherwise the result is trivial). Fix an integer i such that1 ≤ i ≤ n. If i ≤ σ(i), then µσ(i) ≤ µi and

minλi + µσ(i) : 1 ≤ i ≤ n ≤ λi + µσ(i) ≤ λi + µi.

If i > σ(i), then there exists k < i such that i ≤ σ(k). Otherwise, we wouldhave σ(1), . . ., σ(i− 1) ≤ i− 1 and hence, σ(i) ≥ i. It follows that

minλi + µσ(i) : 1 ≤ i ≤ n ≤ λk + µσ(k) ≤ λi + µi.

The desired result now follows.

We proceed with the proof of the theorem. Consider dual bases v andw. Without loss of generality, we may assume that χ(v1) ≤ · · · ≤ χ(vn).Let σ be a permutation of 1, . . ., n such that the numbers µσ(i) = χ(wi)satisfy µ1 ≥ · · · ≥ µn. We have χ(vi) ≥ χ′

i and µi ≥ χ′i. By Lemma 2.7 we

obtain

maxχ(vi) + χ(wi) : 1 ≤ i ≤ n ≥ maxχ(vi) + µi : 1 ≤ i ≤ n≥ maxχ′

i + χ′i : 1 ≤ i ≤ n

= π(χ, χ).

Therefore, γ(χ, χ) ≥ π(χ, χ) and Statement 1 is proven.We assume now that χ ∼ χ. It is not difficult to show that one can

choose bases v and w, which are dual and both normal; furthermore, we


assume that v is ordered. It follows that χ(vi) = χ′i and µi = χ′

i for each i.Thus,

γ(χ, χ) ≤ maxχ(vi) + χ(wi) : 1 ≤ i ≤ n

≤n∑

i=1

(χ(vi) + χ(wi)) =

n∑

i=1

(χ′i + χ′

i)

≤ nmaxχ′i + χ′

i : 1 ≤ i ≤ n = nπ(χ, χ).

Finally, since χ ∼ χ we have γ(χ, χ) ≥ 0. This implies that π(χ, χ) ≥ 0,and Statement 2 follows.

We now introduce the crucial concept of regularity of a pair of Lyapunovexponents χ and χ in dual vector spaces V and W . Roughly speaking reg-ularity means that the filtrations Vχ and Vχ are well-adapted to each other(essentially they are orthogonal; see Theorem 2.8 below), and thus impliessome special properties which determine its role in the stability theory. Ona first glance the regularity requirements seem quite strong and even a bitartificial. However, they hold in “typical” situations.

The pair of Lyapunov exponents (χ, χ) is called regular if χ ∼ χ andγ(χ, χ) = 0. By Theorem 2.6, this holds if and only if π(χ, χ) = 0, and alsoif and only if χ′

i = −χ′i.

Theorem 2.8. If the pair (χ, χ) is regular, then the filtrations Vχ = Vi :i = 0, . . ., s and Vχ = Wi : i = 0, . . ., r are orthogonal, that is, s = r,dimVi + dimWs−i = n, and 〈v, w〉 = 0 for every v ∈ Vi and w ∈Ws−i.

Proof. Set mi = n− dimWs−i + 1. Then

Ws−i = w ∈W : χ(w) ≤ χ′mi

and χ′mi

= −χ′mi

in view of Theorem 2.6. Let v be a normal ordered basisof V and w the basis of W dual to v. Since χ ∼ χ we obtain

Ws−i = w ∈W : χ′mi

+ χ(w) < 0= w ∈W : χ(vi) + χ(w) < 0 if and only if i < mi= spanwmi

, . . ., wn.

This implies that w is normal and ordered. Now for each i and any normalordered basis v = (v1, . . ., vni

, vni+1, . . ., vn) of V and the dual basis w

of W the last n − ni components of w have to coincide with those of w.This implies that r = s, and

Ws−i = spanwni+1, . . ., wn = V ⊥i .

The desired result now follows.

With slight changes one can also consider Lyapunov exponents on com-plex vector spaces.


3. Regularity of Lyapunov Exponents Associated with

Differential Equations

We now discuss the regularity properties of the Lyapunov exponent χ+

defined by (1.3) for Equation (1.1) provided the matrix function A(t) satisfies(1.2). Consider the linear differential equation which is dual to (1.1),

w = −A(t)∗w, (3.1)

where w(t) ∈ Cn and A(t)∗ denotes the complex-conjugated transpose ofA(t). Let w(t) be a unique solution of this equation such that w(0) = w.The function χ+ : Cn → R ∪ −∞ given by

χ+(w) = lim supt→+∞

1

tlog‖w(t)‖

defines a Lyapunov exponent associated with Equation (3.1). We note thatthe exponents χ+ and χ+ are dual. To see that, let v(t) be a solution of theequation (1.1) and w(t) a solution of the dual equation (3.1). Observe thatfor every t ∈ R,

d

dt〈v(t), w(t)〉 = 〈A(t)v(t), w(t)〉 + 〈v(t),−A(t)∗w(t)〉

= 〈A(t)v(t), w(t)〉 − 〈A(t)v(t), w(t)〉 = 0,

where 〈·, ·〉 denote the standard inner product in Cn. Hence

〈v(t), w(t)〉 = 〈v(0), w(0)〉for any t ∈ R. Choose now dual bases (v1, . . ., vn) and (w1, . . ., wn) of Cn.Let vi(t) be the unique solution of (1.1) such that vi(0) = vi, and wi(t) theunique solution of (3.1) such that wi(0) = wi, for each i. We obtain

‖vi(t)‖ · ‖wi(t)‖ ≥ 1

for every t ∈ R, and hence, χ+(vi) + χ+(wi) ≥ 0 for every i. It follows thatthe exponents χ+ and χ+ are dual.

We will study the regularity of the pair of exponents (χ+, χ+). Letv = (v1, . . ., vn) be a basis of Cn. We denote by Γm(t) = Γv

m(t) the m-volume of the parallelepiped defined by the vectors vi(t), for i = 1, . . ., m,that are solutions of (1.1) satisfying the initial conditions vi(0) = vi. LetVm(t) be the m×m matrix whose entries are the scalar products 〈vi(t), vj(t)〉for i, j = 1, . . ., m. Then

Γm(t) = Γv

m(t) = |detVm(t)|1/2.

In particular, Γ1(t) = |v1(t)| and Γn(t) = Γn(0)|det A(t)|, where

A(t) = exp

(∫ t

0A(τ) dτ

).

Note that

det A(t) = exp

(∫ t

0trA(τ) dτ

). (3.2)


The following theorem provides some crucial criteria for the pair (χ+, χ+)to be regular.

Theorem 3.1. Assume that the matrix function A(t) satisfies (1.2). Thefollowing statements are equivalent:

1. the pair (χ+, χ+) is regular;2.

limt→+∞

1

tlog|detA(t)| =

s∑

i=1

kiχ+i . (3.3)

3. for any normal ordered basis v of Cn and any 1 ≤ m ≤ n thefollowing limit exists:

limt→+∞

1

tlog Γv

m(t).

In addition, if the pair (χ+, χ+) is regular then for any normal ordered basisv of Cn and any 1 ≤ m ≤ n we have

limt→+∞

1

tlog Γv

m(t) =m∑

i=1

χ+(vi). (3.4)

Proof. We adopt the following notations in the proof. Given a functionf : (0,∞) → R we set

χ(f) = lim supt→+∞

1

tlog |f(t)| and χ(f) = lim inf

t→+∞

1

tlog |f(t)|.

If, in addition, f is integrable we shall also write

f = lim supt→+∞

1

t

∫ t

0f(τ) dτ and f = lim inf

t→+∞

1

t

∫ t

0f(τ) dτ.

We first show that Statement 1 implies Statement 2. We start with anauxiliary result.

Lemma 3.2. The following statements hold:

1. χ(det A) = Re trA and χ(det A) = Re trA;2. if (v1, . . ., vn) is a basis of Cn, then

−n∑

i=1

χ+(vi) ≤ χ(det A) ≤ χ(detA) ≤n∑

i=1

χ+(vi).

Proof of the lemma. It follows from (3.2) that

χ(det A) = lim inft→+∞

1

tRe

∫ t

0trA(τ) dτ = Re trA

and

χ(det A) = lim supt→+∞

1

tRe

∫ t

0trA(τ) dτ = Re trA.

This establishes the first statement.


Since Γn(0)|det A(t)| gives the volume of the parallelepiped determinedby the vectors v1(t), . . ., vn(t), we have |detA(t)| ≤∏n

i=1‖vi(t)‖, and hence,

χ(det A) ≤n∑

i=1

χ+(vi).

In a similar way,

−χ(det A) = −Re trA = Re tr(−A∗) ≤n∑

i=1

χ+(vi).

The lemma follows.

Let χ′i and χ′

i be the values of the Lyapunov exponents χ+ and χ+,counted with their multiplicities. Choosing a normal basis (v1, . . ., vn) ofCn, it follows from Lemma 3.2 that

−n∑

i=1

χ′i ≤ χ(det A) ≤ χ(detA) ≤

n∑

i=1

χ′i.

Therefore,

χ(det A) − χ(det A) ≤n∑

i=1

(χ′i + χ′

i) ≤ nπ(χ+, χ+),

This shows that if the pair (χ+, χ+) is regular then (3.3) holds.We now show that Statement 3 implies Statement 1. We split the proof

into two steps.Step 1. For every t ≥ 0 consider a linear coordinate change in Cn given

by a matrix U(t). We assume that the matrix function U(t) is differentiable.Setting z(t) = U(t)−1v(t) we obtain

v(t) = U(t)z(t) + U(t)z(t) = A(t)v(t) = A(t)U(t)z(t).

It follows that z = B(t)z, where the matrix B(t) = (bij(t)) is defined by

B(t) = U(t)−1A(t)U(t) − U(t)−1U(t). (3.5)

We need the following lemma of Perron. Its main manifestation is to showhow to reduce Equation (1.1) with a general matrix function A(t) to a lineardifferential equation with a triangular matrix function.

Lemma 3.3. There exists a differentiable matrix function U(t) such that:

1. U(t) is unitary for each t ≥ 0;2. the matrix B(t) is upper triangular for each t;3. sup|bij(t)| : t ≥ 0, i 6= j <∞;4. if k = 1, . . ., n then

Re bkk(t) =d

dtlog

Γv

k(t)

Γv

k−1(t).


Proof of Lemma 3.3. Given a basis v = (v1, . . ., vn) we constructthe desired matrix function U(t) by applying the Gram–Schmidt orthogo-nalization procedure to the basis vi(t), with i = 1, . . ., n, where vi(t) is thesolution of (1.1) satisfying the initial condition vi(0) = vi. Thus we obtain acollection of functions u1(t), . . ., un(t) such that 〈ui(t), uj(t)〉 = δij where δijis the Kronecker symbol. Let V (t) and U(t) be the matrices with columnsv1(t), . . ., vn(t) and u1(t), . . ., un(t), respectively. The matrix U(t) is unitaryfor each t. Moreover, the Gram–Schmidt procedure can be effected in sucha way that each function uk(t) is a linear combination of functions v1(t), . . .,vk(t). It follows that the matrix Z(t) = U(t)−1V (t) is upper triangular foreach t.

The columns z1(t) = U(t)−1v1(t), . . ., zn(t) = U(t)−1vn(t) of the matrixZ(t) form a basis of the space of solutions of the linear differential equation

z = B(t)z. Furthermore, B(t) = Z(t)Z(t)−1, and as Z(t) is upper triangularso is the matrix B(t).

Since U(t) is unitary, using (3.5) we obtain

B(t) +B(t)∗ = U(t)∗(A(t) +A(t)∗)U(t) − (U(t)∗U(t) + U(t)∗U(t))

= U(t)∗(A(t) +A(t)∗)U(t) − d

dt(U(t)∗U(t))

= U(t)∗(A(t) +A(t)∗)U(t).

Since B(t) is triangular we conclude that |bij(t)| ≤ 2‖A(t)‖ < ∞ uniformlyover t ≥ 0 and i 6= j, thus establishing the third statement.

In order to prove the last statement of the lemma assume first that allentries of the matrix Z(t) = (zij(t)) are real. Then the entries of the matrixB(t) are real too and

bkk(t) =zkk(t)

zkk(t)=

d

dtlog zkk(t).

Observe that

vi(t) =∑

1≤ℓ≤i

uℓ(t)zℓi(t).

Therefore,

〈vi(t), vj(t)〉 =∑

1≤ℓ≤i, 1≤m≤j

δℓmzℓi(t)zmj(t)

=∑

1≤ℓ≤mini,j

zℓi(t)zℓj(t) = 〈zi(t), zj(t)〉.

Set z = (z1(0), . . ., zn(0)). This implies that Γv

k(t) = Γz

k(t) for each k andthus,

Γv

k(t)

Γv

k−1(t)=

Γz

k(t)

Γz

k−1(t)= zkk(t).


In the general case (when the entries of Z(t) are not necessarily real) we canwrite Γv

k(t)/Γv

k−1(t) = |zkk(t)| and

d

dtlog |zkk(t)| =

1

2

d

dtlog(zkk(t)zkk(t))

=1

2

(zkk(t)

zkk(t)+zkk(t)

zkk(t)

)

=1

2(bkk(t) + bkk(t)) = Re bkk(t).

This completes the proof of the lemma.

We define the n×n matrix function Z(t) = (zij(t)) as follows: zij(t) = 0if j < i,

zij(t) = e∫ t

0bii(τ) dτ

if j = i, and

zij(t) =

∫ t

aij

j∑

k=i+1

bik(s)zkj(s)e∫ t

sbii(τ) dτds

if j > i.

Lemma 3.4. For any constants aij, with 1 ≤ i < j ≤ n, the columns ofthe matrix Z(t) form a basis of solutions of the equation z = B(t)z.

Proof of Lemma 3.4. For each i we have zii(t) = bii(t)zii(t), and

zij(t) =

j∑

k=i+1

bik(t)zkj(t) + bii(t)zij(t) =

j∑

k=i

bik(t)zkj(t)

for each j > i. This shows that Z(t) = B(t)Z(t) and hence the columnsof Z(t) (i.e., the vectors zi(t) = (z1i(t), . . ., zni(t))) are solutions of theequation z = B(t)z. Since Z(t) is upper triangular, we have

detZ(t) = exp

(n∑

i=1

∫ t

0bii(τ) dτ

)6= 0,

and hence the vectors zi(t) form a basis.

Step 2. Assume that χ(Γv

m) = χ(Γv

m) for any normal ordered basis

v and any 1 ≤ m ≤ n. We show that the pair (χ+, χ+) is regular. ByLemma 3.3, it is sufficient to consider the equation z = B(t)z, where B(t)is a n× n upper triangular matrix for every t.

Lemma 3.5. If Re bii = Re biidef= Bi for each i = 1, . . ., n then:

1. the pair of Lyapunov exponents corresponding to the equations z =B(t)z and w = −B(t)∗w is regular;

2. the numbers B1, . . ., Bn are the values of the Lyapunov exponent χ+;3. the numbers −B1, . . ., −Bn are the values of the Lyapunov expo-

nent χ+.


Proof of Lemma 3.5. We consider the solutions of the equation z =B(t)z described in Lemma 3.4 and show that each column

zi(t) = (z1i(t), . . ., zni(t))

of Z(t) satisfies

χ+(zi) = lim supt→+∞

1

tlog‖zi(t)‖ = Bi

for some choice of the constants aij . Since Re bii = Bi, we clearly haveχ+(zii) = Bi. Assume now that χ+(zkj) ≤ Bj for each i + 1 ≤ k ≤ j. Weshow that χ+(zij) ≤ Bj . Observe that for each ε > 0, we have

χ+(zij) ≤ lim supt→+∞

1

t

(log∣∣∣e∫ t

0bii(τ) dτ

∣∣∣

+ log

∣∣∣∣∣

∫ t

aij

j∑

k=i+1

bik(s)zkj(s)e−∫ s

0bii(τ) dτds

∣∣∣∣∣

)

≤ Bi + lim supt→+∞

1

tlog

∣∣∣∣∣

∫ t

aij

Kne(Bj−Bi+ε)s ds

∣∣∣∣∣ .

We exploit here the fact that, by Lemma 3.3, |bij(t)| ≤ K for some con-stant K independent of i, j, and t. For each j > i set aij = 0 if Bj −Bi ≥ 0and aij = +∞ if Bj − Bi < 0. Then for every sufficiently small ε > 0, wehave

χ+(zij) ≤ Bi + lim supt→+∞

1

tlog

Kn(e(Bj−Bi+ε)t − 1)

Bj −Bi + ε

if Bj −Bi ≥ 0 and

χ+(zij) ≤ Bi + lim supt→+∞

1

tlog

Kne(Bj−Bi+ε)t

Bj −Bi + ε

if Bj −Bi < 0. Therefore,

χ+(zij) ≤ Bi +Bj −Bi + ε = Bj + ε.

Since ε is arbitrary we obtain χ+(zij) ≤ Bj . This shows that χ+(zi) = Bi

for each 1 ≤ i ≤ n.In a similar way, one can show that there exists a lower triangular matrix

W (t) such that W (t) = −B(t)∗W (t). The entries of the matrix W (t) aredefined by wij(t) = 0 if j > i,

wij(t) = e−∫ t

0bjj(τ) dτ

if j = i, and

wij(t) = −∫ t

aji

i−1∑

k=j

bki(s)wkj(s)e−∫ t

sbii(τ) dτ ds


if j < i where the constants aji are chosen as above. Since Re bii = Bi, thecolumns w1(t), . . ., wn(t) of W (t) satisfy

χ+(wi) = lim supt→+∞

1

tlog‖wi(t)‖ = −Bi = −χ+(zi).

Note that χ+(zi) + χ+(wi) = 0 for each i. In order to prove that the pair(χ+, χ+) is regular, it remains to show that the bases z and w are dual.Clearly, 〈zi(0), wj(0)〉 = 0 for every i < j. Moreover, 〈zi(0), wj(0)〉 = 1 foreach 1 ≤ i ≤ n. Fix i > j and t > 0. We have that

〈zi(t), wj(t)〉 =i∑

k=j

zki(t)wkj(t). (3.6)

Since χ+(zij) ≤ Bj and χ+(wij) ≤ −Bj for every i, j, and ε > 0, we obtain

χ(〈zi(0), wj(0)〉) ≤ maxj+1≤k≤i−1

χ(zkiwkj)

≤ maxj+1≤k≤i−1

lim supt→+∞

1

t

(log

∣∣∣∣∫ t

aki

Kne(Bi−Bk+ε)s ds

∣∣∣∣

+ log

∣∣∣∣∣

∫ t

ajk

Kne(−Bj+Bk+ε)s ds

∣∣∣∣∣

)

≤ maxj+1≤k≤i−1

(Bi −Bk −Bj +Bk + 2ε) = Bi −Bj + 2ε.

Since ε is arbitrary, if Bi −Bj < 0 we obtain χ(〈zi(0), wj(0)〉) < 0 and

〈zi(0), wj(0)〉 = limt→+∞

〈zi(t), wj(t)〉 = 0.

If Bi − Bj ≥ 0, then aji = 0. Moreover, for each k we have Bi − Bk ≥ 0 orBk −Bj ≥ 0, and hence aki = 0 or ajk = 0. Letting t→ 0 in (3.6) we obtain

〈zi(0), wj(0)〉 = zji(0)wjj(0) + zii(0)wij(0) +i−1∑

k=j+1

zki(0)wkj(0).

Since i > j and aji = 0, we obtain zji(0) = wij(0) = 0. Moreover, for eachk such that j + 1 ≤ k ≤ i − 1 we have aki = 0 or ajk = 0, and hence,zki(0) = 0 or wkj(0) = 0. Therefore each term in the above sum is zero.Thus 〈zi(0), wj(0)〉 = 0, and the lemma follows.

By Lemma 3.3 and Statement 3, we have

1

t

∫ t

0Re bii(τ) dτ =

1

t

∫ t

0

d

dτlog

Γv

i (t)

Γv

i−1(t)dτ

=1

tlog

Γv

i (t)/Γv

i−1(t)

Γv

i (0)/Γv

i−1(0)→ χ(Γv

i ) − χ(Γv

i−1)

as t → +∞. By Lemma 3.5 we conclude that the Lyapunov exponentcorresponding to the equation z = B(t)z is regular.


We now show that Statement 2 implies Statement 3. By Lemma 3.3 wehave

Γv

m(t) = Γz

m(t) =m∏

k=1

zkk(t). (3.7)

By Lyapunov’s construction of normal bases (see the description after The-orem 2.3) there exists a normal basis (v1(0), . . ., vn(0)) of Cn such thatzk(0) = ek + fk for some fk ∈ spanek+1, . . . , en, where (e1, . . ., en) isthe canonical basis of Cn. Since the matrix solution Z(t) is upper tri-angular for each t, the vectors ek and Z(t)fk are orthogonal. Therefore,‖Z(t)fk‖ ≥ |zkk(t)|, and

χ+(zk) ≥ lim supt→∞

1

tlog|zkk(t)| def

= λk. (3.8)

Without loss of generality we are considering the norm in Cn given by‖(w1, . . . , wn)‖ = |w1| + · · · + |wn|. By (3.7) we obtain

limt→∞

1

tlog Γv

n(t) =s∑

i=1

kiχ+i ≥

n∑

k=1

χ+(zk) ≥n∑

k=1

λk. (3.9)

Furthermore, by (3.2) and Lemma 3.4 we have

Γz

n(t)/Γz

n(0) = exp

(∫ t

0trB(τ) dτ

)=

n∏

k=1

zkk(t),

and hence, by (3.7),

limt→∞

1

tlog Γv

n(t) ≤n∑

k=1

λk. (3.10)

It follows from (3.8), (3.9), and (3.10) that

χ+(vk) = χ+(zk) = limt→+∞

1

tlog‖zk(t)‖ = λk

for each k = 1, . . ., n. Thus, again by (3.7), for each 1 ≤ m ≤ n we concludethat

χ(Γv

m) = χ(Γv

m) =

m∑

k=1

χ+(vk).

This completes the proof of the theorem.

The following example illustrates that Statement 2 of Theorem 3.1 can-not be weakened. Consider the system of differential equations

v1 = −p(t)v1, v2 = p(t)v2

for t > 0, where p(t) = cos log t − sin log t − 1. The general solution of thesystem can be written in the form

v1(t) = C1q(t)−1, v2(t) = C2q(t),


for some constants C1 and C2, where

q(t) = exp

(∫ t

1p(τ) dτ

)= exp(t(cos log t− 1)).

We have

lim inft→+∞

1

t

∫ t

1p(τ) dτ = −2 and lim sup

t→+∞

1

t

∫ t

1p(τ) dτ = 0. (3.11)

Observe that detA(t) = 1 for every t, and hence, the limit in (3.3) exists.On the other hand, it follows from (3.11) that the the limit of 1-volumesmay not exist, and hence the pair of Lyapunov exponents (χ+, χ+) is notregular. In this case the Lyapunov exponent is equal to χ+(v) = 2 for everyv = (v1, v2) 6= 0 with C1 6= 0, and χ+(v) = 0 otherwise. Hence,

0 = limt→+∞

1

tlog|detA(t)| < χ+

1 + χ+2 = 2.

We also illustrate that Statement 2 of Theorem 3.1 cannot be weakenedby replacing the limit in (3.3) by the upper limit. Consider a system ofdifferential equations

v1 = v2, v2 = p(t)v2for t > 0. Observe that the general solution of the system can be written inthe form

v1(t) = C1 + C2

∫ t

1q(τ) dτ, v2(t) = C2q(t),

for some constants C1 and C2. One can easily show that for every vectorv = (v1, v2) 6= 0, we have

χ+(v) = limt→+∞

1

tlog‖v(t)‖ = 0.

On the other hand, it follows from (3.11) that the limit in (3.3) does notexist, and hence, the pair of Lyapunov exponents (χ+, χ+) is not regular.

We say that the Lyapunov exponent χ+ is forward regular if the pairof Lyapunov exponents (χ+, χ+) is regular. We also say that the Lyapunovexponent χ+ is exact if χ(Γv

m) = χ(Γv

m) for any 1 ≤ m ≤ n and any basisv of Cn. We emphasize that exactness does not require the identity in(3.4), but only the existence of the limit in (3.4). By Theorem 3.1, if thefunction t 7→ ‖A(t)‖ is bounded (i.e., if (1.2) holds) then forward regularity isequivalent to exactness. In the case of unbounded matrix functions exactnessmay turn out to be weaker then forward regularity as the following exampleillustrates.

Consider the system of differential equations

v1 = av1 + ebtv2, v2 = −av2for t ∈ R, where a and b are positive constants such that 2a < b. The generalsolution of the system can be written in the form

v1(t) = C1eat +

C2

b− 2ae(b−a)t, v2(t) = C2e

−at,


for some constants C1 and C2. One can easily show that the Lyapunovexponent is exact. In particular, for every vector v = (v1, v2) 6= 0 we haveχ+(v) = a whenever C1 6= 0, and χ+(v) = b − a otherwise. By (3.2) weobtain detA(t) = 1 and

0 = limt→+∞

1

tlog|detA(t)| < χ+

1 + χ+2 = b.

Therefore, the identity in (3.4) does not hold, and the Lyapunov exponentis not forward regular.

One can also show that forward regularity is equivalent to exactnessprovided the function t 7→ ‖A(t)‖ is integrable.

4. Lyapunov Stability Theory

We now proceed with the proof of the Lyapunov Stability Theorem 1.2.Consider the perturbed differential equation (1.6) where the function f(t, u)satisfies (1.7). Let v1(t), . . ., vn(t) be n linearly independent solutions of(1.1) and V (t) the corresponding monodromy matrix. We assume thatV (0) = Id. Denote by V(t, s) the Cauchy matrix of (1.1) defined by V(t, s) =V (t)V (s)−1.

Theorem 4.1. Assume that there are two continuous functions R andr on [0,∞) such that for every t ≥ 0,

∫ t

0R(τ) dτ ≤ D1 < +∞, (4.1)

∫ ∞

0e(q−1)

∫ s

0(R+r)(τ) dτ ds = D2 < +∞, (4.2)

and for every 0 ≤ s ≤ t,

‖V(t, s)‖ ≤ D3e∫ t

sR(τ) dτ+(q−1)

∫ s

0r(τ) dτ . (4.3)

Then there exist D > 0 and δ > 0 such that for every u0 ∈ Cn with ‖u0‖ < δ,there is a unique solution u of Equation (1.6) which satisfies:

1. u is well-defined on the interval [0,∞), and u(0) = u0;2. for every t,

‖u(t)‖ ≤ ‖u0‖De∫ t

0R(τ) dτ . (4.4)

Proof. Equation (1.6) is equivalent to the integral equation

u(t) = V(t, 0)u0 +

∫ t

0V(t, s)f(s, u(s)) ds. (4.5)

Moreover, by (1.7) for every u0 ∈ Cn with u0 ∈ H there exists a uniquesolution u(t) satisfying the initial condition u(0) = u0.

We consider the linear space BC of Cn valued continuous functions x onthe interval [0,∞) such that

‖x(t)‖ ≤ Ce∫ t

0R(τ) dτ


for every t ≥ 0, where C > 0 is a constant. We endow this space with thenorm

‖x‖R = sup‖x(t)‖e−∫ t

0R(τ) dτ : t ≥ 0.

It follows from (4.1) that ‖x(t)‖ ≤ ‖x‖ReD1 . Therefore, if ‖x‖R is sufficiently

small, then x(t) ∈ H for every t ≥ 0. This means that the operator

(Jx)(t) =

∫ t

0V(t, s)f(s, x(s)) ds

is well-defined on BC . By (1.7), (4.2), and (4.3) we have

‖(Jx1)(t)−(Jx2)(t)‖

≤∫ t

0D3e

∫ t

sR(τ) dτ+(q−1)

∫ s

0r(τ) dτK(‖x1 − x2‖R)qeq

∫ s

0R(τ) dτ ds

=D3K(‖x1 − x2‖R)qe∫ t

0R(τ) dτ

∫ t

0e(q−1)

∫ s

0(R+r)(τ) dτ ds

≤D3KD2(‖x1 − x2‖R)qe∫ t

0R(τ) dτ .

It follows that

‖Jx1 − Jx2‖R ≤ D3KD2(‖x1 − x2‖R)q

= D3KD2(‖x1 − x2‖R)q−1‖x1 − x2‖R.

Choose ε > 0 such that

θdef= D3KD2(2ε)

q−1 < 1. (4.6)

Whenever x1, x2 ∈ Bε we have

‖Jx1 − Jx2‖R ≤ θ‖x1 − x2‖R. (4.7)

Choose a point u0 ∈ Cn and set ξ(t) = V(t, 0)u0. It follows from (4.3) that

‖ξ(t)‖ ≤ ‖u0‖D3e∫ t

0R(τ) dτ

and hence, ‖ξ‖R ≤ ‖u0‖D3. Consider the operator J defined by

(Jx)(t) = ξ(t) + (Jx)(t)

on the space Bε. Note that Bε is a complete metric space with the distanced(x, y) = ‖x− y‖R. We have

‖Jx‖R ≤ D3‖u0‖ + θ‖x‖R < D3δ + θε < ε

provided that δ is sufficiently small. Therefore J(Bε) ⊂ Bε. By (4.6) and

(4.7), the operator J is a contraction, and hence, there exists a unique func-tion u ∈ Bε which is a solution of (4.5). Furthermore, u can be obtained by

u(t) = limn→∞

(Jn0)(t) =∞∑

n=0

(Jnξ)(t)


and hence, by (4.7),

‖u‖R ≤∞∑

n=0

‖Jnξ‖R ≤∞∑

n=0

θn‖ξ‖R =‖ξ‖R

1 − θ≤ D3‖u0‖

1 − θ.

Therefore, the function u(t) satisfies (4.4).

As an immediate consequence of this theorem we obtain the followingstatement due to Malkin (see [20]).

Theorem 4.2. Assume that the Cauchy matrix of (1.1) admits the es-timate

‖V(t, s)‖ ≤ Deα(t−s)+βs, (4.8)

for any 0 ≤ s ≤ t, and some constants α, β, and D, such that

(q − 1)α+ β < 0. (4.9)

Then the trivial solution of Equation (1.6) is asymptotically and exponen-tially stable.

Proof. Set

R(t) = α and r(t) =β

q − 1.

It follows from (4.8) and (4.9) that Conditions (4.2) and (4.3) hold. In orderto verify Condition (4.1) note that by setting t = s in (4.3) we obtain

1 = ‖V(t, t)‖ ≤ D3e(q−1)

∫ t

0r(τ) dτ .

It follows that for every t,

βt

q − 1=

∫ t

0r(τ) dτ ≥ −C, (4.10)

where C ≥ 0 is a constant. Therefore, by (4.9),

αt =

∫ t

0R(τ) dτ ≤ C +

∫ t

0(R(τ) + r(τ)) dτ ≤ C (4.11)

for all t. This implies Condition (4.1). Therefore, by (4.4), any solution uof (1.6) satisfies

‖u(t)‖ ≤ ‖u0‖Deαt ≤ ‖u0‖DeC . (4.12)

This shows that the trivial solution of (1.6) is asymptotically stable.If every solution of (1.6) is defined for all t then by (4.10) and (4.11),

β ≥ 0 and α ≤ 0. In view of (4.9), α < 0, and in view of (4.12), the solutionu(t) = 0 is exponentially stable. The desired result follows.

Another important consequence of Theorem 4.1 is the following state-ment due to Lyapunov (see [19]).


Theorem 4.3. Assume that the maximal value χmax of the Lyapunovexponent of (1.1) is strictly negative (see (1.5)), and that

(q − 1)χmax + γ < 0,

where γ is the regularity coefficient. Then the trivial solution of Equation(1.6) is asymptotically and exponentially stable.

Proof. Consider a normal basis (v1(t), . . ., vn(t)) for the space of so-lutions of (1.1), such that the numbers

χ′1 ≤ · · · ≤ χ′

n = χmax

are the values of the Lyapunov exponent χ+ for the vectors v1(t), . . ., vn(t).We may assume that the matrix V (t) whose columns are v1(t), . . ., vn(t) issuch that the columns w1(t), . . ., wn(t) of the matrix W (t) = [V (t)∗]−1 forma normal basis for the space of solutions of the dual equation w = −A(t)∗w.

Let µ1, . . ., µn be the values of the Lyapunov exponent χ+ for the vectorsw1(t), . . ., wn(t). For every ε > 0 there exists a constant Dε > 0 such that

‖vj(t)‖ ≤ Dεe(χ′

j+ε)t and ‖wj(t)‖ ≤ Dεe(µj+ε)t

for every t. We also have

γ = maxχ′j + µj : j = 1, . . ., n.

It follows that χ′j + µj ≤ γ for every j = 1, . . ., n. Consider the Cauchy

matrixV(t, s) = V (t)V (s)−1 = V (t)W (s)∗.

Its entries are

vik(t, s) =n∑

j=1

vij(t)wkj(s),

where vij(t) is the i-th coordinate of the vector vj(t) and wkj(s) is the k-thcoordinate of the vector wj(s). It follows that

|vik(t, s)| ≤n∑

j=1

|vij(t)| · |wkj(s)| ≤n∑

j=1

‖vj(t)‖ · ‖wj(s)‖

≤ nDε2e(χ

′j+ε)t+(µj+ε)s = nDε

2e(χ′j+ε)(t−s)+(χ′

j+µj+2ε)s.

Therefore, there exists a constant D > 0 such that

‖V(t, s)‖ ≤ De(χmax+ε)(t−s)+(γ+2ε)s. (4.13)

Since ε can be chosen arbitrarily small the desired result follows from The-orem 4.2 if we set α = χmax + ε and β = γ + 2ε.

We conclude by observing that the Lyapunov Stability Theorem 1.2 isan immediate corollary of Theorem 4.3. Moreover, if χ+ is forward regular,setting γ = 0 in (4.13) we obtain that for every solution u(t) of (1.6) andevery s ∈ R,

‖u(t)‖ ≤ De(χmax+ε)(t−s)+2εs‖u(s)‖ = De2εse(χmax+ε)(t−s)‖u(s)‖. (4.14)


In other words the contraction constant may get worse along the orbit of asolution of (1.6). This implies that the “size” of the neighborhood at times where the asymptotic (and exponential) stability of the trivial solutionis guaranteed may decay with subexponential rate. In Section 6 we willestablish an analog of (4.14) for dynamical systems with discrete time (seeStatement (L7) of Theorem 6.3) and illustrate the crucial role that it playsin the nonuniform hyperbolicity theory.

We now describe a generalization of the Lyapunov Stability Theoremin the case when not all but only some Lyapunov exponents are negative.Namely, let χ+

1 < · · · < χ+s be the distinct values of the Lyapunov exponent

χ+ for Equation (1.1). Assume that

χ+k < 0, (4.15)

where 1 ≤ k < s and the number χ+k+1 can be negative, positive, or equal

to 0 (compare to (1.5)).

Theorem 4.4 (see [7]). Assume that the Lyapunov exponent χ+ forEquation (1.1) with the matrix function A(t) satisfying (1.2), is forwardregular and satisfies Condition (4.15). Then there exists a local smoothsubmanifold V s ⊂ Cn which passes through 0, is tangent at 0 to the linearsubspace Vk (defined by (2.1)), and is such that the trivial solution of (1.6)is asymptotically and exponentially stable along V s.

This statement is a particular case of a more general and strong resultthat we discuss in Section 9.

5. The Oseledets Decomposition

Using results of Section 2, we conclude that the Lyapunov exponent χ+

associated with Equation (1.1), as defined by (1.3), takes on only finitelymany values on Cn \ 0. To stress that we deal with positive time we shallnow denote its values by

χ+1 < · · · < χ+

s+ ,

where s+ ≤ n. We also denote by V+ the filtration of Cn associated to χ+.We have

0 = V +0 $ V +

1 $ · · · $ V +s+ = Cn,

where V +i = v ∈ Cn : χ+(v) ≤ χ+

i . Finally we denote by k+i = dimV +

i −dimV +

i−1 the multiplicity of the value χ+i . Note that

∑s+

i=1 k+i = n.

An important advantage of Theorem 3.1 is that one can verify the regu-larity property of the pair (χ+, χ+) dealing only with the Lyapunov exponentχ+. In this case we say that the Lyapunov exponent χ+ is forward regularto stress that we allow only positive time.

Reversing the time we introduce the Lyapunov exponent χ− : Cn →R ∪ −∞ by

χ−(v) = lim supt→−∞

1

|t| log ‖v(t)‖,


where v(t) is the solution of (1.1) satisfying the initial condition v(0) = v.The function χ− takes on only finitely many values on Cn \ 0, which wedenote by

χ−1 > · · · > χ−

s−,

where s− ≤ n. We denote by V− the filtration of Cn associated to χ−. Wehave

Cn = V −1 % V −

1 % · · · % V −s−

% V −s−+1

= 0,where V −

i = v ∈ Cn : χ−(v) ≤ χ−i . We also denote by k−i = dimV −

i −dimV −

i+1 the multiplicity of the value χ−i , and we have

∑s−

i=1 k−i = n.

We also consider the Lyapunov exponent χ− : Cn → R∪−∞ given by

χ−(w) = lim supt→−∞

1

|t| log ‖w(t)‖,

where w(t) is the solution of (3.1) satisfying the initial condition w(0) = w.We say that the Lyapunov exponent χ− is backward regular if the pair

(χ−, χ−) is regular. We say that the filtrations V+ and V− comply if thefollowing properties hold:

1. s+ = s−def= s;

2. there exists a decomposition

Cn =

s⊕

i=1

Ei (5.1)

into subspaces Ei such that if i = 1, . . ., s then

V +i =

i⊕

j=1

Ej and V −i =

s⊕

j=i

Ej ;

3. χ+i = −χ−

idef= χi for i = 1, . . ., s;

4. if i = 1, . . ., s and v ∈ Ei \ 0 then

limt→±∞

1

tlog ‖v(t)‖ = χi

with uniform convergence on v ∈ Ei : ‖v‖ = 1 (recall that v(t) isthe solution of Equation (1.1) with initial condition v(0) = v).

The decomposition (5.1) is called the Oseledets decomposition associatedwith the Lyapunov exponent χ+ (or with the pair of Lyapunov exponents(χ+, χ−)).

We say that the Lyapunov exponent χ+ (or the pair of Lyapunov expo-nents (χ+, χ−)) is Lyapunov regular if the exponent χ+ is forward regular,the exponent χ− is backward regular, and the filtrations V+ and V− comply.

Remark 5.1. We stress that simultaneous forward and backward regu-larity of the Lyapunov exponents does not imply the Lyapunov regularity.Roughly speaking, whether the Lyapunov exponent is forward (respectively,


backward) regular does not depend only on the backward (respectively, for-ward) behavior of the system. On the other hand Lyapunov regularity re-quires some compatibility between the forward and backward behavior whichis expressed in terms of the filtrations V+ and V−.

The regularity property is quite strong and requires a quite rigid behav-ior of the system.

Theorem 5.2. If the Lyapunov exponent χ+ is Lyapunov regular thenthe following statements hold:

1. the exponents χ+ and χ− are exact;

2. dimEi = k+i = k−i

def= ki;

3. if v = (v1, . . ., vki) is a basis of Ei, then

limt→±∞

1

tlog Γv

ki(t) = χiki;

4. if vi(t) and vj(t) are solutions of Equation (1.1) such that vi(0) ∈Ei \ 0 and vj(0) ∈ Ej \ 0 with i 6= j then

limt→±∞

1

tlog ∠(vi(t), vj(t)) = 0.

Proof. The first three statements follow from Theorem 3.1. For the laststatement, let Γ(t) denote the area of the rectangle formed by the vectorsvi(t) and vj(t). We have

Γ(t) = ‖vi(t)‖ · ‖vj(t)‖ sin ∠(vi(t), vj(t)).

Since sin∠(vi(t), vj(t)) ≤ 1, and the exponent χ+ is exact we obtain

χ+i + χ+

j = limt→+∞

1

tlog Γ(t)

≤χ+(vi) + χ+(vj) + lim inft→+∞

1

tlog sin ∠(vi(t), vj(t))

≤χ+i + χ+

j + lim supt→+∞

1

tlog sin ∠(vi(t), vj(t)) ≤ χ+

i + χ+j .

A similar argument applies to the exponent χ− and we obtain the desiredstatement.

One can construct examples of nonstationary linear differential equa-tions whose Lyapunov exponent is forward and backward regular but is notLyapunov regular. In dimension 1, an example is given by the equationv = a(t)v, where a : R → R is a bounded continuous function such thata(t) → a+ as t → +∞ and a(t) → a− as t → −∞, for some constantsa+ 6= a−. The general solution of this equation is

v(t) = exp

(∫ t

0a(s) ds

)v(0),


and hence the forward and backward exponents are respectively equal to a+

and −a−. In dimension 2, an example is given by

v1 = a(t)v1, v2 = a(−t)v2,with a as above. In this case the values of the exponents χ+ and χ− co-incide (up to the change of sign required for Lyapunov regularity), but thefiltrations V+ and V− do not comply.

6. Dynamical Systems with Nonzero Lyapunov Exponents.

Multiplicative Ergodic Theorem

Let ϕt be a smooth flow on a smooth compact Riemannian p-manifoldM . It is generated by the vector field X on M given by

X(x) =dϕt(x)

dt|t=0.

For every x0 ∈ M the trajectory x(x0, t) = ϕt(x0) : t ∈ R represents asolution of the nonlinear differential equation

v = X(v) (6.1)

on the manifold M . This solution is uniquely defined by the initial conditionx(x0, 0) = x0.

Given a point x ∈M and the trajectory ϕt(x) : t ∈ R passing throughx we introduce the variational differential equation

w(t) = A(x, t)w(t), (6.2)

where

A(x, t) = dϕt(x)X.

This is a nonstationary linear differential equation along the trajectoryϕt(x) : t ∈ R known also as the equation of the first linear approximation.

With the flow ϕt one can associate a certain collection of single lineardifferential equations (6.1) “along” each trajectory of the flow. The stabilityof a given trajectory can be described by studying small perturbations ofthe variational differential equation (6.2). The perturbation term is of type(1.6) and satisfies (1.7). The results of the previous section can be applied tostudy the stability of trajectories via the Lyapunov exponents. Although it isstill a very difficult problem to verify whether a given trajectory is Lyapunovregular it turns out that “most” trajectories (in the sense of measure theory)have this property.

A similar approach is used to establish the stability of trajectories ofdynamical systems with discrete time. Since it is technically simpler wedescribe such an approach here.

Let f : M → M be a diffeomorphism of a compact smooth Riemannianp-manifoldM . Given x ∈M , consider the trajectory fmxm∈Z. The familyof maps

dfmxfm∈Z


can be viewed as an analog of the variational differential equation (6.2) inthe continuous time case. Given x ∈M and v ∈ TxM the formula

χ(x, v) = lim supm→+∞

1

mlog ‖dxf

mv‖

defines the Lyapunov exponent specified by the diffeomorphism f at thepoint x. The function χ(x, ·) takes on only finitely many values on TxM\0,which we denote by

χ1(x) < · · · < χs(x)(x),

where s(x) ≤ p. We denote by Vx the filtration of TxM associated to χ(x, ·).We have

0 = V0(x) $ V1(x) $ · · · $ Vs(x)(x) = TxM,

where Vi(x) = v ∈ TxM : χ(x, v) ≤ χi(x). We also introduce ki(x) =dimVi(x) − dimVi−1(x), which is the multiplicity of the value χi(x). We

have∑s(x)

i=1 ki(x) = p. Finally, the collection of pairs

Spχ(x) = (χi(x), ki(x)) : 1 ≤ i ≤ s(x)is called the Lyapunov spectrum of the exponent χ(x, ·).

We note that the functions χi(x), s(x), and ki(x) are invariant under f ,and (Borel) measurable (but not necessarily continuous).

We say that a point x ∈M is forward, or backward, or Lyapunov regularif so is the Lyapunov exponent χ(x, ·). Note that if x is regular then so isthe point f(x) and thus one can speak on the whole trajectory fm(x) asbeing forward, backward, or Lyapunov regular.

Summarizing our discussion in the previous section we come to the fol-lowing result.

Theorem 6.1. If a point x ∈ M is Lyapunov regular, then there existsthe Oseledets decomposition

TxM =

s(x)⊕

i=1

Ei(x)

into subspaces Ei(x) such that:

1. the subspaces Ei(x) are invariant under the differential dxf , i.e.,dxfEi(x) = Ei(f(x)), and depend (Borel) measurably on x;

2. if v ∈ Ei(x) \ 0 then

limm→±∞

1

mlog ‖dxf

mv‖ = χi(x)

with uniform convergence on v ∈ Ei(x) : ‖v‖ = 1;3. if v = (v1, . . ., vki(x)) is a basis of Ei(x), then

limn→±∞

1

nlog Γv

ki(x)(n) = χi(x)ki(x);

in particular, the Lyapunov exponent χ(x, ·) is exact;


4. there exists a decomposition of the co-tangent bundle

T ∗xM =

s(x)⊕

i=1

E∗i (x)

into subspaces E∗i (x) associated with the Lyapunov exponent χ+; the

subspaces E∗i (x) are invariant under the co-differential d∗xf , that is

d∗xfE∗i (x) = E∗

i (f(x)), and depend (Borel) measurably on x; more-over, if vi(x) : i = 1, . . ., p is a normal basis with vi(x) ∈ Ej(x)for nj−1(x) < i ≤ nj(x), and v∗i (x) : i = 1, . . ., p is a dual basis,then v∗i (x) ∈ E∗

j (x) for nj−1(x) < i ≤ nj(x).

The following theorem due to Oseledets (see [24]) is a key result instudying the regularity of trajectories of dynamical systems. It shows thatregularity is “typical” from the measure-theoretical point of view.

Theorem 6.2 (Multiplicative Ergodic Theorem). If f is a C1 diffeomor-phism of a compact smooth Riemannian manifold, then the set of Lyapunovregular points has full measure with respect to any f-invariant Borel proba-bility measure on M .

We stress that there may exist trajectories which are both forward andbackward regular but not Lyapunov regular. However, such trajectoriesform a negligible set with respect to any f -invariant Borel probability mea-sure.

We also emphasize that the notion of Lyapunov regularity does not re-quire any invariant measure. Consider the set L ⊂ M of points that areLyapunov regular. Due to Theorem 6.2 this set is nonempty and indeedhas full measure with respect to any f -invariant Borel probability measureon M .

We note that only in some exceptional cases every point in M is Lya-punov regular. To illustrate this consider a Smale horseshoe, i.e., a diffeo-morphism f : R2 → R2 which maps the unit square S onto a horseshoe-likeset (see for example [14]). The map f has the locally maximal invariant

set Adef=⋂

n∈ZfnS ⊂ S, and f |A is topologically conjugate to the full

shift on two symbols. Moreover, the map f is uniformly hyperbolic on Aand hence, TxR2 = E1(x) ⊕ E2(x) at every point x ∈ A, where E1(x)and E2(x) are the one-dimensional stable and unstable subspaces at x. Setλ−(x) = log‖dxf |E1(x)‖ < 0 and λ+(x) = log‖dxf |E2(x)‖ > 0. For everyx ∈ A and v ∈ E1(x) we have

χ(x, v) = lim supn→∞

1

n

n−1∑

k=0

λ−(fkx),

and for every x ∈ A and v ∈ E2(x) we have

χ(x, v) = lim supn→∞

1

n

n−1∑

k=0

λ+(fkx).


For a Smale horseshoe the stable and unstable subspaces are uniformlytransverse, and hence a point x is regular if and only if the Lyapunovexponent at x is exact (see Statement 3 of Theorem 6.1). Furthermore,Theorem 6.2 is a consequence of the Birkhoff ergodic theorem applied to theHolder continuous functions λ− and λ+. Therefore, in the case of a linearhorseshoe (when the functions λ− and λ+ are constant on A), or more gen-erally when the functions λ− and λ+ are cohomologous to constants (i.e.,λ− = a− + ϕ− f − ϕ− and λ+ = a+ + ϕ+ f − ϕ+ for some constantsa−, a+ ∈ R and some continuous functions ϕ− and ϕ+ on A), every pointx ∈ A is regular. On the other hand, when at least one of the functions λ−

and λ+ is noncohomologous to a constant (and thus for a generic C1 surfacediffeomorphism with a Smale horseshoe), it follows from work of Barreiraand Schmeling [3] that the set of nonregular points is nonempty and hasHausdorff dimension equal to the Hausdorff dimension of the set A.

Recall that an f -invariant measure ν is ergodic if for each f -invariantmeasurable set A ⊂M either A or M \A has zero measure. One can showthat ν is ergodic if and only if every f -invariant (mod 0) measurable function(i.e., a measurable function ϕ such that ϕ f = ϕ almost everywhere) isconstant almost everywhere.

Let ν be a Borel f -invariant ergodic measure. There exist numbersχi = χν

i and ki = kνi for i = 1, . . ., s (where s = sν) such that

χi(x) = χi, ki(x) = ki, s(x) = s (6.3)

for ν-almost every x. The collection of pairs

Spχ(ν) = (χi, ki) : 1 ≤ i ≤ s

is called the Lyapunov spectrum of the measure ν.We now consider dynamical systems whose spectrum of the Lyapunov

exponent does not contain zero on some subset of M . More precisely, let

Λ = x ∈ L : there exists 1 ≤ k(x) < s(x)

with χk(x)(x) < 0 and χk(x)+1(x) > 0. (6.4)

Note that the set Λ is f -invariant. We say that f is a dynamical systemwith nonzero Lyapunov exponents if there exists a Borel ergodic f -invariantprobability measure ν on M such that ν(Λ) = 1. The measure ν is called ahyperbolic measure for f . Note that if ν is hyperbolic then there is a number1 ≤ k = kν < s such that χν

k < 0 and χνk+1 > 0.

Consider the set Λ = Λν of those points in Λ which are Lyapunov regular

and satisfy (6.3). By the Multiplicative Ergodic Theorem we have ν(Λ) = 1.

For every x ∈ Λ, we set

Es(x) =k⊕

i=1

Ei(x) and Eu(x) =s⊕

i=k+1

Ei(x).


Theorem 6.3 (see [27]). Let ν be an ergodic hyperbolic measure ν. The

subspaces Es(x) and Eu(x), x ∈ Λ have the following properties:

L1. they depend measurably on x;L2. they form a splitting of the tangent space, i.e., TxM = Es(x) ⊕

Eu(x);L3. they are invariant:

dxfEs(x) = Es(f(x)) and dxfE

u(x) = Eu(f(x));

Furthermore, there exist ε0 > 0 and measurable functions C(x, ε) > 0 and

K(x, ε) > 0, x ∈ Λ and 0 < ε ≤ ε0 such that:

L4. the subspace Es(x) is stable: if v ∈ Es(x) and n > 0, then

‖dxfnv‖ ≤ C(x, ε)e(χk+ε)n‖v‖;

L5. the subspace Eu(x) is unstable: if v ∈ Eu(x) and n < 0, then

‖dxfnv‖ ≤ C(x, ε)e(χk+1−ε)n‖v‖;

L6.

∠(Es(x), Eu(x)) ≥ K(x, ε);

L7. for every m ∈ Z,

C(fm(x), ε) ≤ C(x, ε)eε|m| and K(fm(x), ε) ≥ K(x, ε)e−ε|m|.

Remark 6.4. Condition (L7) is crucial and is a manifestation of theregularity property (it is an analog in the discrete time case of Condition(4.14)). Roughly speaking it means that the estimates (L4), (L5), and (L6)may get worse as |m| → ∞ but only with subexponential rate. We stressthat the rate of the contraction along stable subspaces and the rate of theexpansion along unstable subspaces are exponential and hence, are substan-tially stronger.

Proof of Theorem 6.3. We begin with the following general state-ment.

Lemma 6.5. Let X ⊂M be a Borel f-invariant set and A(x, ε) a Borelpositive function on X × [0, ε0), 0 < ε0 < 1 such that for every ε0 ≥ ε > 0,x ∈ X, and m ∈ Z,

M1(x, ε)e−ε|m| ≤ A(fm(x), ε) ≤M2(x, ε)e

ε|m|,

where M1(x, ε) and M2(x, ε) are Borel functions. Then one can find Borelpositive functions B1(x, ε) and B2(x, ε) such that

B1(x, ε) ≤ A(x, ε) ≤ B2(x, ε), (6.5)

and for m ∈ Z,

B1(x, ε)e−2ε|m| ≤ B1(f

m(x), ε), B2(x, ε)e2ε|m| ≥ B2(f

m(x), ε). (6.6)


Proof of the lemma. It follows from the conditions of the lemmathat there exists m(x, ε) > 0 such that if m ∈ Z is such that |m| > m(x, ε)then

−2ε ≤ 1

|m| logA(fm(x), ε) ≤ 2ε.

Set

B1(x, ε) = min−m(x,ε)≤i≤m(x,ε)

1, A(f i(x), ε)e2ε|i|

,

B2(x, ε) = max−m(x,ε)≤i≤m(x,ε)

1, A(f i(x), ε)e−2ε|i|

.

The functions B1(x, ε) and B2(x, ε) are Borel. Moreover, if n ∈ Z then

B1(x, ε)e−2ε|n| ≤ A(fn(x), ε) ≤ B2(x, ε)e

2ε|n|. (6.7)

Furthermore, if b1 ≤ 1 ≤ b2 are such that

b1e−2ε|n| ≤ A(fn(x), ε) (6.8)

and

b2e2ε|n| ≥ A(fn(x), ε) (6.9)

for all n ∈ Z then b1 ≤ B1(x, ε) and b2 ≥ B1(x, ε). In other words,

B1(x, ε) = supb ≤ 1 : inequality (6.8) holds for all n ∈ Z,B2(x, ε) = infb ≥ 1 : inequality (6.9) holds for all n ∈ Z. (6.10)

Inequalities (6.5) follow from (6.7) (with n = 0). We also have

A(fn+m(x), ε) ≤ B2(x, ε)e2ε|n+m| ≤ B2(x, ε)e

2ε|n|+2ε|m|,

A(fn+m(x), ε) ≥ B1(x, ε)e−2ε|n+m| ≥ B1(x, ε)e

−2ε|n|−2ε|m|,

Comparing these inequalities with (6.7) written at the point fm(x) andtaking (6.10) into account we obtain (6.6). The proof of the lemma is com-plete.

We now apply Lemma 6.5 to construct the functionK(x, ε). LetK : Λ →R be a Borel function. It is called tempered at the point x if

limm→±∞

1

mlogK(fm(x)) = 0.

In other words the Lyapunov exponent of the function K(x) is exact andis 0. It follows that the function A(x, ε) = K(x) satisfies the conditions ofLemma 6.5 for all ε > 0.

Fix ε > 0 and consider the function γ(x) = ∠(Es(x), Eu(x)) for x ∈ Λ.Since this function is tempered (see Theorem 6.1) applying Lemma 6.5 weconclude that the function K(x, ε) = B1(x,

12ε) satisfies Conditions (L6) and

(L7) of the theorem.We will now show how to construct the function C(x, ε). The proof

is an elaboration for the discrete time case of arguments in the proof ofTheorem 4.3 (see (4.14)).


Lemma 6.6. There exists a Borel positive function D(x, ε) (x ∈ Λ andε > 0 is sufficiently small), such that if m ∈ Z and 1 ≤ i ≤ s, then

D(fm(x), ε) ≤ D(x, ε)2e2ε|m| (6.11)

and

‖dfnix‖ ≤ D(x, ε)e(χi+ε)n, ‖df−n

ix ‖ ≥ D(x, ε)−1e−(χi+ε)n

for any n ≥ 0, where dfnix = dxf

n|Ei(x).

Proof of the lemma. Let x ∈ Λ. By Theorem 6.1 (we use the no-tation of that theorem) there exists a number n(x, ε) ∈ N such that ifn ≥ n(x, ε), then

χi − ε ≤ 1

nlog ‖dfn

ix‖ ≤ χi + ε, −χi − ε ≤ 1

nlog ‖df−n

ix ‖ ≤ −χi + ε,

and

−χi − ε ≤ 1

nlog ‖d∗fn

ix‖ ≤ −χi + ε, χi − ε ≤ 1

nlog ‖d∗f−n

ix ‖ ≤ χi + ε,

where d∗fnix = d∗xf

n|E∗i (x) (recall that E∗

i (x) ⊂ T ∗xM is the dual space to

Ei(x) and d∗f is the co-differential). Set

D+1 (x, ε) = min

1≤i≤smin

0≤j≤n(x,ε)

1, ‖df j

ix‖e(−χi+ε)j , ‖d∗f jix‖e(χi+ε)j

,

D−1 (x, ε) = min

1≤i≤smin

−n(x,ε)≤j≤0

1, ‖df j

ix‖e(−χi−ε)j , ‖d∗f jix‖e(χi−ε)j

,

D+2 (x, ε) = max

1≤i≤smax

0≤j≤n(x,ε)

1, ‖df j

ix‖e(−χi−ε)j , ‖d∗f jix‖e(χi−ε)j

,

D−2 (x, ε) = max

1≤i≤smax

−n(x,ε)≤j≤0

1, ‖df j

ix‖e(−χi+ε)j , ‖d∗f jix‖e(χi+ε)j

,

and

D1(x, ε) = minD+1 (x, ε), D−

1 (x, ε), D2(x, ε) = maxD+2 (x, ε), D−

2 (x, ε),

D(x, ε) = maxD1(x, ε)−1, D2(x, ε).

The function D(x, ε) is measurable, and if n ≥ 0 and 1 ≤ i ≤ s then

D(x, ε)−1e(χi−ε)n ≤ ‖dfnix‖ ≤ D(x, ε)e(χi+ε)n,

D(x, ε)−1e(−χi−ε)n ≤ ‖df−nix ‖ ≤ D(x, ε)e(−χi+ε)n,

D(x, ε)−1e(−χi−ε)n ≤ ‖d∗fnix‖ ≤ D(x, ε)e(−χi+ε)n,

D(x, ε)−1e(χi−ε)n ≤ ‖d∗f−nix ‖ ≤ D(x, ε)e(χi+ε)n.

(6.12)

Moreover, if d ≥ 1 is a number for which Inequalities (6.12) hold for alln ≥ 0 and 1 ≤ i ≤ s with D(x, ε) replaced by d then d ≥ D(x, ε). Therefore,

D(x, ε) = infd ≥ 1 : the inequalities (6.12) hold for all n ≥ 0

and 1 ≤ i ≤ s with D(x, ε) replaced by d. (6.13)


We wish to compare the values of the function D(x, ε) at the points x andfmx for m ∈ Z. First, let us notice that for every x ∈ M , v ∈ TxM , andϕ ∈ T ∗

xM with ϕ(v) = 1 we have

(d∗xfϕ)(dxfv) = ϕ((d∗xf)−1dxfv) = ϕ(v) = 1. (6.14)

Second, using the Riemannian metric on the manifold M we introduce theidentification map τx : T ∗

xM → TxM such that 〈τx(ϕ), v〉 = ϕ(v) wherev ∈ TxM and ϕ ∈ T ∗

xM .Let vn

k : k = 1, . . ., p be a basis of Ei(fn(x)) and wn

k : k = 1, . . ., pbe the dual basis of E∗

i (fn(x)). We have τfn(x)(wnk ) = vn

k . Let Ain,m and

Bin,m be matrices corresponding to the linear maps dfn+1

ifm(x) and d∗fn+1ifm(x)

with respect to those bases. It follows from (6.14) that

Ai0,m(Bi

0,m)∗ = Id

and hence, for every n > 0 the matrix corresponding to the map dfnifm(x) is

Ain,m = Ai

0,m+n(Ai0,m)−1 = Ai

0,m+n(Bi0,m)∗.

It follows from here and (6.12) that:

1. if n > 0 then

‖dfnifm(x)‖ ≤ D(x, ε)2e(χi+ε)(n+m)+(−χi+ε)m = D(x, ε)2e2εme(χi+ε)n,

‖dfnifm(x)‖ ≥ D(x, ε)−2e(χi−ε)(n+m)+(−χi−ε)m = D(x, ε)−2e−2εme(χi−ε)n,

2. if n > 0 and m− n ≥ 0 then

‖df−nifm(x)‖ ≤ D(x, ε)2e(χi+ε)(m−n)+(−χi+ε)m = D(x, ε)2e2εme(−χi+ε)n,

‖df−nifm(x)‖ ≥ D(x, ε)−2e(χi−ε)(m−n)+(−χi−ε)m = D(x, ε)−2e−2εme(−χi−ε)n,

3. if n > 0 and n−m ≥ 0 then

‖df−nifm(x)‖ ≤ D(x, ε)2e(χi+ε)(n−m)+(−χi+ε)m = D(x, ε)2e2εme(−χi+ε)n,

‖df−nifm(x)‖ ≥ D(x, ε)−2e(χi−ε)(n−m)+(−χi−ε)m = D(x, ε)−2e−2εme(−χi−ε)n,

Similar inequalities hold for the maps d∗fnifm(x) for each n, m ∈ Z. Compar-

ing this with the inequalities (6.12) applied to the point fm(x) and using(6.13) we conclude that if m ≥ 0, then

D(fm(x), ε) ≤ D(x, ε)2e2εm. (6.15)

Similar arguments show that if m ≤ 0, then

D(f−m(x), ε) ≤ D(x, ε)2e−2εm. (6.16)

It follows from (6.15) and (6.16) that if m ∈ Z, then

D(fm(x), ε) ≤ D(x, ε)2e2ε|m|.



We now proceed with the proof of the theorem. Replacing in (6.11) mby −m and x by fm(x) we obtain

D(fm(x), ε) ≥√D(x, ε)e−ε|m|. (6.17)

Consider two disjoint subsets σ1, σ2 ⊂ [1, s] ∩ N and set

L1(x) =⊕

i∈σ1

Ei(x), L2(x) =⊕

i∈σ2

Ei(x)

and γσ1σ2(x) = ∠(L1(x), L2(x)). By Theorem 6.1 the function γσ1σ2 istempered and hence, in view of Lemma 6.5 one can find a function Kσ1σ2(x)satisfying Condition (L7) such that

γσ1σ2(x) ≥ Kσ1σ2(x).

SetT (x, ε) = minKσ1σ2(x),

where the minimum is taken over all pairs of disjoint subsets σ1, σ2 ⊂ [1, s]∩N. The function T (x, ε) satisfies Condition (L7).

Let v ∈ Es(x). Write v =∑k

i=1 vi where vi ∈ Ei(x). We have

‖v‖ ≤k∑

i=1

‖vi‖ ≤ LT−1(x, ε)‖v‖,

where L > 1 is a constant. Let us set C ′(x, ε) = LD(x, ε)T (x, ε)−1. It fol-lows from (6.11) and (6.17) that the function C ′(x, ε) satisfies the conditionof Lemma 6.5 with

M1(x, ε) =2

πL√D(x, ε) and M2(x, ε) = LD(x, ε)2T (x, ε)−1.

Therefore, there exists a function C1(x, ε) ≥ C ′(x, ε) for which the state-ments of Lemma 6.5 hold.

Applying the above arguments to the inverse map f−1 and the subspaceEu(x) one can construct a function C2(x, ε) for which the statements ofLemma 6.5 hold. The desired function C(x, ε) can now be defined by

C(x, ε) = maxC1(x, ε/2), C2(x, ε/2).This completes the proof of the theorem.

7. Nonuniform Hyperbolicity. Regular Sets

Let f : M → M be a diffeomorphism of a compact smooth RiemannianmanifoldM . A measurable f -invariant subset R ⊂M is called nonuniformlyhyperbolic if there exist: (a) numbers λ and µ such that 0 < λ < 1 < µ; (b)a number ε and real functions C, K : R → (0,∞); (c) subspaces Es(x) andEu(x) for each x ∈ R, which satisfy the following conditions:

H1. the subspaces Es(x) and Eu(x) depend measurably on x and forman invariant splitting of the tangent space, i.e.,

TxM = Es(x) ⊕ Eu(x), dxfEs(x) = Es(f(x)), dxfE

u(x) = Eu(f(x));


H2. the subspace Es(x) is stable: if v ∈ Es(x), m ∈ Z, and n > 0, then

‖dfm(x)fnv‖ ≤ C(fm(x))λneεn‖v‖;

H3. the subspace Eu(x) is unstable: if v ∈ Eu(x), m ∈ Z, and n < 0,then

‖dfm(x)fnv‖ ≤ C(fm(x))µneε|n|‖v‖;

H4. for n ∈ Z,

∠(Es(fn(x)), Eu(fn(x))) ≥ K(fn(x));

H5. for m, n ∈ Z,

C(fm+n(x)) ≤ C(fm(x))eε|n|, K(fm+n(x)) ≥ K(fm(x))e−ε|n|.

We can summarize the discussion in the previous section by saying that forany hyperbolic measure ν the set of Lyapunov regular points with nonzeroLyapunov exponents contains a nonuniformly hyperbolic set of full ν-mea-sure with

λ = eχνk , µ = eχ

νk+1 , C(x) = C(x, ε), K(x) = K(x, ε)

for any fixed 0 < ε ≤ ε0 (see Conditions (L1)-(L7) in Section 6). In fact,finding trajectories with nonzero Lyapunov exponents seems to be a univer-sal approach in establishing nonuniform hyperbolicity.

We emphasize that the set of points (trajectories) with nonzero Lya-punov exponents whose regularity coefficient is sufficiently small (but maynot necessarily be zero) is nonuniformly hyperbolic for some ε > 0.

We now provide a more detailed description of a nonuniformly hyperbolicset R. Fix ε > 0. Given ℓ > 0, we introduce the regular set (of level ℓ) by

Rℓ =

x ∈ R : C(x, ε) ≤ ℓ, K(x, ε) ≥ 1

ℓ

. (7.1)

Regular sets can be viewed as noninvariant uniformly hyperbolic sets for thediffeomorphism f . They have the following basic properties:

R1. Rℓ ⊂ Rℓ+1;R2. if m ∈ Z, then fm(Rℓ) ⊂ Rℓ′ , where ℓ′ = ℓ exp(|m|ε);R3. the subspaces Es(x) and Eu(x) depend continuously on x ∈ Rℓ;

moreover, they depend Holder continuously on x ∈ Rℓ (see Appen-dix A by Brin). This means that

d(Es(x), Es(y)) ≤ Cρ(x, y)α and d(Eu(x), Eu(y)) ≤ Cρ(x, y)α,

where C > 0 and α ∈ (0, 1] are constants, and d is the distancein the Grassmannian bundle of TM generated by the Riemannianmetric.

We consider the sets Qℓ = Rℓ that are the closures of the sets Rℓ.Set Q =

⋃ℓ≥1 Qℓ. It is easy to see that Qℓ ⊂ Qℓ+1 and that the set Q is

f -invariant.Given a point x ∈ Qℓ choose a sequence of points xn ∈ Rℓ which con-

verges to x. Passing to a subsequence, we may assume that the sequences


of subspaces Es(xn) and Eu(xn) converge to some subspaces at x which wedenote by Es(x) and Eu(x) respectively. It is easy to see that they satisfythe following properties:

R4. TxM = Es(x) ⊕ Eu(x);R5. if v ∈ Es(x) and n > 0, then

‖dxfnv‖ ≤ ℓλneεn‖v‖;

R6. if v ∈ Eu(x) and n < 0, then

‖dxfnv‖ ≤ ℓµneε|n|‖v‖;

R7. ∠(Es(x), Eu(x)) ≥ 1ℓ .

This implies that the subspaces Es(x) and Eu(x) are uniquely defined (inparticular, they do not depend on the choice of the sequence xn → x). Theydepend continuously on x ∈ Qℓ (and indeed, Holder continuously). Further-more, the subspaces Es(x) and Eu(x), for x ∈ Q, determine a nonuniformlyhyperbolic structure on the set Q with C(x, ε) = ℓ and K(x, ε) = 1

ℓ if

x ∈ Qℓ \ Qℓ−1.We now consider a smooth flow ϕt on a compact smooth Riemannian

manifold M which is generated by a vector field X(x). A measurable ϕt-invariant subset R ⊂ M is called nonuniformly hyperbolic if there exist:(a) numbers λ and µ such that 0 < λ < 1 < µ; (b) real functions C,K : R × (0, 1) → (0,∞); (c) subspaces Es(x) and Eu(x) for each x ∈ R,which satisfy Conditions (H2)-(H5) and the following condition:

H1’. the subspaces Es(x) and Eu(x) depend measurably on x and to-gether with the subspace E0(x) = αX(x) : α ∈ R form an invari-ant splitting of the tangent space, i.e.,

TxM = Es(x) ⊕ Eu(x) ⊕ E0(x),

with

dxfEs(x) = Es(f(x)) and dxfE

u(x) = Eu(f(x)).

One can extend the notion of regular set to flows on nonuniformly hyperbolicsubsets.

We say that a dynamical system (with discrete or continuous time) isnonuniformly hyperbolic if it possesses an invariant nonuniformly hyperbolicsubset.

Remark 7.1. One can generalize the notion of nonuniformly hyper-bolicity from complete to partial. More precisely, we say that a set R isnonuniformly partially hyperbolic if there exist: (a) numbers λ and µ suchthat 0 < λ < µ and minλ, µ−1 < 1; (b) real functions C, K : R× (0, 1) →(0,∞); (c) subspaces Es(x) and Eu(x) for each x ∈ R, which satisfy Condi-tions (H1)–(H5). Let us emphasize that in the case of partial hyperbolicitythe vectors in the unstable subspaces Eu(x) may indeed contract and havenegative Lyapunov exponents since the number µ is not necessarily greater


than one. In these lectures we consider only the case of complete hyper-bolicity although many results can be generalized (sometimes literally) tononuniformly partially hyperbolic sets.

One can further generalize nonuniform (complete or partial) hyperbolic-ity by requiring that λ and µ be f -invariant measurable functions λ : R → Rand µ : R → R such that 0 < λ(x) < 1 < µ(x) for each x ∈ R (in the case ofpartial hyperbolicity one should assume that instead 0 < λ(x) < µ(x)) andminλ(x), µ(x)−1 < 1 for each x ∈ R). In other words, the rates λ(x) andµ(x) may vary from trajectory to trajectory. In fact, one can easily reducethis more general case to the previous one by considering the (invariant)sets of points x where λ(x) ≤ λ and µ(x) ≤ µ for fixed constants λ and µ.

Remark 7.2. Let ν be a hyperbolic measure for a diffeomorphism f andR the set of Lyapunov regular points with nonzero Lyapunov exponents.The regular sets Rℓ consist of Lyapunov regular points which satisfy (7.1)while the sets Qℓ may contain some nonregular points with sufficiently smallregularity coefficient (compare to (4.13) and (4.14)).

8. Examples of Nonuniformly Hyperbolic Systems

We present some examples of dynamical systems with continuous timewhich are nonuniformly hyperbolic. The first such example was constructedin [26] by a “surgery” of an Anosov flow.

8.1. Let ϕt be an Anosov flow on a compact 3-dimensional manifold Mwhich is defined by a vector field X and preserves a smooth ergodic measureµ. Fix a point p0 ∈ M . One can introduce a coordinate system x, y, z ina ball Bd(p0) (for some d > 0) such that p0 is the origin (i.e., p0 = 0) andX = ∂/∂z.

For each ε > 0, let Tε = S1 ×Dε ⊂ Bd(0) be the solid torus obtained byrotating the disk

Dε = (x, y, z) ∈ Bd(0) : x = 0 and (y − d/2)2 + z2 ≤ (εd)2around the z-axis. Every point on the solid torus can be represented as apair (θ, y, z) with θ ∈ S1 and (y, z) ∈ Dε.

For every 0 ≤ α ≤ 2π, we consider the cross-section of the solid torusΠα = (θ, y, z) : θ = α.

Let X be a smooth vector field on M \ Tε and ϕt the flow generated by

X. One can construct a vector field X such that:

1. X|(M \ T2ε) = X|(M \ T2ε);

2. for any 0 ≤ α, β ≤ 2π, the vector field X|Πβ is the image of the

vector field X|Πα under the rotation around the z-axis that movesΠα onto Πβ ;

3. for every 0 ≤ α ≤ 2π, the unique two fixed points of the flow ϕt|Πα

are those in the intersection of Πα with the hyperplanes z = ±εd;


Πα

Figure 1. A cross-section Πα of the solid torus and the flow ϕt

4. for every 0 ≤ α ≤ 2π and (y, z) ∈ D2ε \ intDε, the trajectory ofthe flow ϕ|Πα passing through the point (y, z) is invariant under thesymmetry (α, y, z) 7→ (α, y,−z);

5. the flow ϕt|Πα preserves the measure µα that is the conditional mea-sure generated by the measure µ on the set Πα (see [5] for details).

See Figure 1. One can see that the orbits of the flows ϕt and ϕt coincideon M \ T2ε, that the flow ϕt preserves the measure µ, and that the onlyfixed points of this flow are those on the circles (θ, y, z) : z = −εd and(θ, y, z) : z = εd.

On the set T2ε \ intTε we introduce natural coordinates θ1, θ2, r with0 ≤ θ1, θ2 < 2π and εd ≤ r ≤ 2εd such that the set of fixed points of ϕt iscomposed of those for which r = εd, and θ1 = 0 or θ1 = π.

Consider the flow on T2ε \ intTε defined by

(θ1, θ2, r, t) 7→ (θ1, θ2 + [2 − r/(εd)]4t cos θ1, r),

and let X be the corresponding vector field. Consider now the flow ψt onM \ intTε generated by the vector field Y on M \ intTε defined by

Y (x) =

X(x), x ∈M \ intT2ε

X(x) + X(x), x ∈ intT2ε \ intTε.

Proposition 8.1. The following properties hold:

1. The flow ψt preserves the measure µ and is ergodic.2. The flow ψt has no fixed points.


3. The flow ψt is nonuniformly hyperbolic (but not uniformly hyper-bolic). Moreover, for µ-almost every x ∈M \ T2ε,

χ(x, v) < 0 if v ∈ Es(x), and χ(x, v) > 0 if v ∈ Eu(x),

where Eu(x) and Es(x) are respectively stable and unstable subspacesof the flow ϕt at the point x.

Proof. By the construction of ψt, it preserves µ. Since the vector fields

X and X commute the flow is ergodic. Therefore, the first statement holds.The second statement follows from the construction of the flow ψt. In orderto prove the third statement consider the function

T (x, t) =

∫ t

0IT2ε

(ϕτx) dτ,

where IT2εdenotes the indicator (the characteristic function) of the set T2ε.

By the Birkhoff Ergodic Theorem,

limt→+∞

T (x, t)

t= µ(T2ε)

for µ-almost every x ∈M .Fix a point x ∈ M \ T2ε. Consider a moment of time t1 at which the

trajectory ψtx enters the set T2ε and the next moment of time t2 at whichthis trajectory exits the set T2ε. Given a vector v ∈ Eu(x) denote by vi theorthogonal projection of the vector dxψtiv onto the (x, y) plane for i = 1, 2.It follows from the construction of the flows ϕt and ψt (see Property 4) that‖v1‖ ≥ ‖v2‖. Since the unstable subspaces Eu(x) depend continuously on xthere exists K ≥ 1 (independent of x, t1, and t2) such that

‖dxψtv‖ ≥ K‖dxϕt−T (x,t)v‖.It follows that for µ-almost every x ∈M \ T2ε and v ∈ Eu(x),

χ(x, v) = lim supt→+∞

1

tlog ‖dxψtv‖ ≥ (1 − µ(T2ε)) lim sup

t→+∞

1

tlog ‖dxϕtv‖ > 0.

provided ε is sufficiently small. Repeating the above argument with respectto the inverse flow ψ−t one can show that χ(x, v) < 0 for µ-almost everyx ∈M \ T2ε and v ∈ Es(x).

Set M1 = M \Tε and consider a copy (M1, ψt) of the flow (M1, ψt). One

can glue the manifolds M1 and M1 along their boundaries ∂Tε and obtain athree-dimensional smooth Riemannian manifold D without boundary. Wedefine a flow Ft on D by

Ftx =

ψtx, x ∈M1

ψtx, x ∈ M1.

It is clear that the flow Ft is nonuniformly hyperbolic and preserves themeasure µ.


8.2. Our next example is geodesic flows on compact smooth Riemann-ian manifolds of nonpositive curvature. Let M be a compact smooth p-dimensional Riemannian manifold. We assume that for any x ∈M and anytwo vectors v1, v2 ∈ TxM the sectional curvature Kx(v1, v2) satisfies

Kx(v1, v2) ≤ 0. (8.1)

We then say that M has nonpositive curvature.The geodesic flow gt acts on the tangent bundle TM by the formula

gtv = γv(t),

where γv(t) is the unit tangent vector along the geodesic γv(t) defined bythe vector v (i.e., such that γv(0) = v; this geodesic is uniquely defined).The geodesic flow generates a vector field V on TM given by

V (v) =d(gtv)

dt

∣∣∣t=0

.

Since M is compact the flow gt is well-defined for all t ∈ R.We endow the second tangent space T (TM) with a special Riemannian

metric. Let π : TM →M be the natural projection (i.e., π(x, v) = x for eachx ∈ M and each v ∈ TxM) and K : T (TM) → TM the linear (connection)operator defined by Kξ = (∇Z)(t)|t=0, where Z(t) is any curve in TM suchthat Z(0) = dπξ, d

dtZ(t)|t=0 = ξ and ∇ is the covariant derivative. Thecanonical metric on T (TM) is given by

〈ξ, η〉v = 〈dvπξ, dvπη〉πv + 〈Kξ,Kη〉πv.

The set SM ⊂ TM of the unit vectors is invariant with respect to thegeodesic flow, and is a compact manifold of dimension 2p−1. In what followswe consider the geodesic flow restricted to the subset SM .

The study of hyperbolic properties of the geodesic flow is based uponthe description of solutions of the variational equation (6.2) for the flow.One can show that this equation along a given trajectory gtv of the flow isthe Jacobi equation along the geodesic γv(t):

Y ′′(t) +RXYX(t) = 0. (8.2)

Here Y (t) is a vector field along γv(t), X(t) = γ(t), and RXY is the curvaturetensor. More precisely, the relation between the variational equations (6.2)and the Jacobi equation (8.2) can be described as follows. Fix a vectorv ∈ SM and an element ξ ∈ TvSM . Let Yξ(t) be the unique solution ofEquation (8.2) satisfying the initial conditions Yξ(0) = dvπξ and Y ′

ξ (0) =

Kξ. One can show that the map ξ 7→ Yξ(t) is an isomorphism for whichdgtvπdvgtξ = Yξ(t) and Kdvgtξ = Y ′

ξ (t) (see [9]). This map establishes

the identification between solutions of the variational equation (6.2) andsolutions of the Jacobi equation (8.2).

Recall that the Fermi coordinates ei(t), for i = 1, . . ., p along thegeodesic γv(t) are obtained by the time t parallel translation along γv(t) of


an orthonormal basis ei(0) in Tγv(0)M where e1(t) = γ(t). Using thesecoordinates we can rewrite Equation (8.2) in the matrix form

d2

dt2A(t) +K(t)A(t) = 0, (8.3)

where A(t) = (aij(t)) and K(t) = (kij(t)) are matrix functions, with entrieskij(t) = Kγv(t)(ei(t), ej(t)). The boundary value problem for Equation (8.3)has a unique solution, i.e., for any numbers s1, s2 and any matrices A1, A2

there exists a unique solution A(t) of Equation (8.3) satisfying A(s1) = A1

and A(s2) = A2.

Proposition 8.2 (see [9, 29]). Given s ∈ R, let As(t) be the uniquesolution of Equation (8.3) satisfying the boundary conditions: As(0) = Id(where Id is the identity matrix) and As(s) = 0. Then there exists a limit

limt→∞

d

dtAs(t)

∣∣∣t=0

= A+.

We define the positive limit solution A+(t) of Equation (8.3) as thesolution that satisfies the initial conditions:

A+(0) = Id andd

dtA+(t)

∣∣∣t=0

= A+.

One can show (see [9, 29]) that this solution is nondegenerate (i.e., thatdetA+(t) 6= 0 for every t ∈ R) and that A+(t) = lims→+∞As(t). Moreover,it can be written in the form

A+(t) = C(t)

∫ ∞

tC(s)−1(C(s)−1)∗ds,

where C(t) is the solution of Equation (8.3) satisfying the initial conditions

C(0) = 0 andd

dtC(t)

∣∣∣t=0

= Id .

Similarly, letting s → −∞ we can define the negative limit solution A−(t)of Equation (8.3).

For every v ∈ SM let us set

E+(v) = ξ ∈ TvSM : 〈ξ, V (v)〉 = 0 and Yξ(t) = A+(t)dvπξ,E−(v) = ξ ∈ TvSM : 〈ξ, V (v)〉 = 0 and Yξ(t) = A−(t)dvπξ,

where V is the vector field generated by the geodesic flow.

Proposition 8.3 (see [9]). The sets E−(v) and E+(v) are subspaces ofTvSM and satisfy the following properties:

1. dimE−(v) = dimE+(v) = p− 1;2. dvπE

−(v) = dvπE+(v) = w ∈ TπvM : w is orthogonal to v;

3. the subspaces E−(v) and E+(v) are invariant under the differentialdvgt, i.e., dvgtE

−(v) = E−(gtv) and dvgtE+(v) = E+(gtv);

4. if τ : SM → SM is the involution defined by τv = −v, then

E+(−v) = dvτE−(v) and E−(−v) = dvτE

+(v);


5. if Kx(v1, v2) ≥ −a2 for some a > 0 and all x ∈ M , then ‖Kξ‖ ≤a‖dvπξ‖ for every ξ ∈ E+(v) and every ξ ∈ E−(v);

6. if ξ ∈ E+(v) or ξ ∈ E−(v), then Yξ(t) 6= 0 for every t ∈ R;7. ξ ∈ E+(v) (respectively, ξ ∈ E−(v)) if and only if

〈ξ, V (v)〉 = 0 and ‖dgtvπdvgtξ‖ ≤ c

for every t > 0 (respectively, t < 0), for some constant c > 0;8. if ξ ∈ E+(v) (respectively, ξ ∈ E−(v)) then the function t 7→ ‖Yξ(t)‖

is nonincreasing (respectively, nondecreasing).

In view of Properties 5 and 7 we have ξ ∈ E+(v) (respectively, ξ ∈E−(v)) if and only if 〈ξ, V (v)〉 = 0 and ‖dvgtξ‖ ≤ c for t > 0 (respectively,t < 0), for some constant c > 0. This observation and Property 3 justify tocall E+(v) and E−(v) the stable and unstable subspaces.

In general, the subspaces E−(v) and E+(v) do not span the whole secondtangent space TvSM . If they do span TvSM for every v ∈ SM , then thegeodesic flow is Anosov (see [9]). This is the case when the curvature isstrictly negative. However, for a general manifold of nonpositive curvatureone can only expect that the geodesic flow is nonuniformly hyperbolic. Tosee this consider the set

∆ =

v ∈ SM : lim sup

t→∞

1

t

∫ t

0Kπgsv(v, w) ds < 0

for every w ∈ SM orthogonal to v

. (8.4)

It is easy to see that the set ∆ is measurable and invariant under the flow gt.

Theorem 8.4 (see [28, 29]). If the Riemannian manifold M has non-positive curvature then for every v ∈ ∆, we have χ(v, ξ) < 0 if ξ ∈ E+(v)and χ(v, ξ) > 0 if ξ ∈ E−(v).

Proof. Let ψ : R+ → R be a continuous function. Set

ψ = lim supt→∞

1

t

∫ t

0ψ(s) ds, ψ = lim inf

t→∞

1

t

∫ t

0ψ(s) ds,

ψ = lim inft→∞

1

t

∫ t

0ψ(s)2 ds.

We need the following lemma.

Lemma 8.5. Assume that c = supt≥0 |ψ(t)| <∞. Then:

1. if ψ(t) ≤ 0 for all t ≥ 0 and ψ > 0 then ψ < 0;

2. if ψ(t) ≥ 0 for all t ≥ 0 and ψ > 0 then ψ > 0.

Proof of the lemma. Assume that ψ(t) ≤ 0. Then ψ ≤ 0. On theother hand, if c > 0 then

−ψc

=

∣∣∣∣ψ

c

∣∣∣∣ ≥(ψ

c

)=ψ

c2> 0.


This implies that ψ < 0 and completes the proof of the first statement. Theproof of the second statement is similar.

We proceed with the proof of the theorem. Fix v ∈ ∆, ξ ∈ E+(v), andconsider the function ϕ(t) = 1

2‖Yξ(t)‖2. Using (8.3) we obtain

d2

dt2ϕ(t) =

1

2

d2

dt2〈Yξ, Yξ〉 = −K(t)ϕ(t) + ‖Y ′

ξ (t)‖2,

where K(t) = Kγv(t)(Yξ(t), w(t)) and w(t) = γv(t). It follows from Proposi-

tion 8.3 that ϕ(t) 6= 0, and ddtϕ(t) ≤ 0 for all t ≥ 0. Set

z(t) = (ϕ(t))−1 d

dtϕ(t).

It is easy to check that the function z(t) satisfies the Ricatti equation

d

dtz(t) + z(t)2 − (ϕ(t))−1‖Y ′

ξ (t)‖2 +K(t) = 0. (8.5)

By Proposition 8.3,∣∣∣∣d

dtϕ(t)

∣∣∣∣ =1

2

∣∣∣∣d

dt〈Yξ, Yξ〉

∣∣∣∣ = |〈Yξ, Y′ξ 〉|

= |〈dgtvπdvgtξ,Kdvgtξ〉| ≤ a‖dgtvπdvgtξ‖2 = 2aϕ(t).

It follows that supt≥0 |z(t)| ≤ 2a. Integrating the Ricatti equation (8.5) onthe interval [0, t] we obtain that

z(t) − z(0) +

∫ t

0z(s)2 ds =

∫ t

0(ϕ(s))−1‖Y ′

ξ (s)‖2 ds−∫ t

0K(s) ds.

It follows that for v ∈ ∆ (see (8.4)) we have

lim inft→∞

1

t

∫ t

0z(s)2 ds ≥ lim inf

t→∞

1

t

∫ t

0(ϕ(s))−1‖Y ′

ξ (s)‖2 ds

− lim supt→∞

1

t

∫ t

0K(s) ds > 0.

Therefore, in view of Lemma 8.5 we conclude that

lim supt→∞

1

t

∫ t

0z(s) ds < 0.

On the other hand, using Proposition 8.3 we find that

χ(v, ξ) = lim supt→∞

1

tlog ‖dvgtξ‖ = lim sup

t→∞

1

tlog ‖dgtvπdvgtξ‖

= lim supt→∞

1

tlog ‖Yξ(t)‖ =

1

2lim sup

t→∞

1

t

∫ t

0z(s) ds.

This completes the proof of the first statement of the theorem. The secondstatement can be proved in a similar way.


The geodesic flow preserves the Liouville measure µ on the tangentbundle which is induced by the Riemannian metric. We denote by ν theLebesgue measure on M . It follows from Theorem 8.4 that if the set ∆has positive Liouville measure then the geodesic flow gt|∆ is nonuniformlyhyperbolic. It is, therefore, crucial to find conditions which would guaranteethat ∆ has positive Liouville measure.

We first consider the two-dimensional case.

Theorem 8.6. Let M be a smooth compact surface of nonpositive cur-vature K(x) and genus greater than 1. Then µ(∆) > 0.

Proof. By the Gauss–Bonnet formula the Euler characteristic of M is12π

∫M K(x) dν(x). It follows from the condition of the theorem that

∫

MK(x) dν(x) < 0. (8.6)

Choose two orthogonal vectors v, w ∈ SM . It is easy to see that

lim supt→∞

1

t

∫ t

0Kπ(gsv)(v, w) ds = lim sup

t→∞

1

t

∫ t

0K(π(gsv)) ds.

By the Birkhoff ergodic theorem we obtain that for µ-almost every v ∈ SMthere exists the limit

limt→∞

1

t

∫ t

0K(π(gsv)) ds = Φ(v)

and that ∫

SMΦ(v) dµ(v) =

∫

MK(x) dν(x).

The desired result follows from (8.6).

In the multidimensional case one can establish the following criterion forpositivity of the Liouville measure of the set ∆.

Theorem 8.7. Let M be a smooth compact Riemannian manifold ofnonpositive curvature. Assume that there exist x ∈M and a vector v ∈ SxMsuch that

Kx(v, w) < 0 (8.7)

for any vector w ∈ SxM which is orthogonal to v. Then µ(∆) > 0.

Proof. Condition (8.7) holds in a small open neighborhood of (x, v).The desired result now follows from the Birkhoff ergodic theorem.

We remark that Condition (8.7) is a multidimensional version of (8.6).It is easy to see that the set ∆ is everywhere dense. One can also show thatit is open (see Theorem 16.4 below).


9. Existence of Local Stable Manifolds

Consider a nonuniformly (partially) hyperbolic f -invariant set R, andthe associated sets Rℓ defined by (7.1). According to the results in Section 7we may replace everywhere in the sequel the sets Rℓ by the compact sets

Rℓ.The hyperbolicity conditions allow one to describe the asymptotic be-

havior of trajectories which start in a small neighborhood of a hyperbolictrajectory. More precisely, applying Theorem 4.4 one can construct at everypoint x ∈ R local stable and unstable manifolds V s(x) and V u(x) such thatevery trajectory fn(y) with y ∈ V s(x) approaches fn(x) with an exponen-tial rate, and every trajectory f−n(y) with y ∈ V u(x) approaches f−n(x)with an exponential rate. Roughly speaking, the behavior of trajectoriesthat start in a neighborhood of x resembles that of the trajectories in aneighborhood of a fixed (or periodic) hyperbolic point. However, the rateof exponential convergence may vary from orbit to orbit and may also getworse along the orbit but with subexponential rate (due to Condition (H5)in Section 7).

A more precise description is given by the following theorem which isone of the key results in the hyperbolic theory.

Theorem 9.1 (Stable Manifold Theorem). Let R be a nonuniformlyhyperbolic set for a C1+α diffeomorphism f . Then for every x ∈ R thereexists a local stable manifold V s(x) such that x ∈ V s(x), TxV

s(x) = Es(x),and if y ∈ V s(x) and n ≥ 0 then

ρ(fn(x), fn(y)) ≤ T (x)λneεnρ(x, y), (9.1)

where ρ is the distance induced in M by the Riemannian metric and T : R →(0,∞) is a Borel function such that if m ∈ Z then

T (fm(x)) ≤ T (x)e10ε|m|. (9.2)

Inequality (9.2) should be compared to Condition (H5) in Section 7.The Stable Manifold Theorem was first established by Pesin in [27]. His

approach was an elaboration of the classical work of Perron. This approachwas extended by Katok and Strelcyn in [15] to smooth maps with singular-ities (they include billiard systems and other physical models). Ruelle [34]provided another proof of Theorem 9.1, based on studying the perturbationsof the matrix products in the Multiplicative Ergodic Theorem 6.2. Fathi,Herman, and Yoccoz [10] provided a detailed exposition of Theorem 9.1which essentially follows the approaches of Pesin and Ruelle. Another proofof the Stable Manifold Theorem was provided by Pugh and Shub in [32]using graph transform techniques.

In [30], Pugh constructed an explicit example of a C1 diffeomorphism(which is not C1+α for any α > 0) on a 4-dimensional manifold for whichthe statement of Theorem 9.1 does not hold. More precisely, there exists


no manifold tangent to Es(x) such that (9.1) holds on some open neighbor-hood of x. This example illustrates that the hypothesis α > 0 is crucial inTheorem 9.1.

On another direction, Liu and Qian [18] established a version of Theo-rem 9.1 for random maps. One can extend the Stable Manifold Theorem 9.1to infinite-dimensional spaces. Ruelle [35] proved this theorem for Hilbertspaces, closely following his approach in [34], and Mane [22] consideredBanach spaces (under certain compactness assumptions on the dynamics).

The local stable manifold V s(x) in Theorem 9.1 is constructed via asmooth map

ψs : Bs(r) → Eu(x)

which satisfiesψs(0) = 0 and dψs(0) = 0. (9.3)

Here Bs(r) is the ball of radius r in Es(x) centered at the origin; r = r(x) iscalled the size of the local stable manifold. One now obtains the local stablemanifold by projecting the graph of ψs into M by the exponential map

V s(x) = expx(x, ψs(x)) : x ∈ Bs(r).It follows from (9.3) that

x ∈ V s(x) and TxVs(x) = Es(x).

We now describe a construction of the function ψs. Fix x ∈M and considerthe map

fx = exp−1f(x) f expx : Bs(r) ×Bu(r) → Tf(x)M, (9.4)

which is well-defined if r is sufficiently small (here Bu(r) is the ball of radiusr in Eu(x) centered at the origin). By Condition (H1) in Section 7, the map

f can be written in the following form:

fx(v, w) = (Axv + gx(v, w), Bxw + hx(v, w)), (9.5)

where v ∈ Es(x) and w ∈ Eu(x). Furthermore,

Ax : Es(x) → Es(f(x)) and Bx : Eu(x) → Eu(f(x))

are linear maps. In view of Conditions (H2) and (H3) in Section 7 the mapAx is a contraction and the map Bx is an expansion. Since f is of classC1+α we also have

‖gx(v, w)‖ ≤ C1(‖v‖ + ‖w‖)1+α, ‖hx(v, w)‖ ≤ C2(‖v‖ + ‖w‖)1+α, (9.6)

and

‖dgx(v1, w1) − dgx(v2, w2)‖ ≤ C1(‖v1 − v2‖ + ‖w1 − w2‖)α,

‖dhx(v1, w1) − dhx(v2, w2)‖ ≤ C2(‖v1 − v2‖ + ‖w1 − w2‖)α (9.7)

where C1 > 0 and C2 > 0 are constants (which may depend on x).

In other words the map fx can be viewed as a small perturbation ofthe linear map (v, w) 7→ (Axv,Bxw) by the perturbation (gx(v, w), hx(v, w))satisfying Conditions (9.6) and (9.7), which are analogous to Condition (1.7).


Therefore, if the point x is Lyapunov regular then an appropriate discretetime version of Theorem 4.4 applies and gives the existence of a local stablemanifold at x.

Note that if the point x is nonuniformly hyperbolic (i.e., x ∈ R) but isnot Lyapunov regular then Theorem 4.4 cannot be used. Furthermore, The-orem 4.4 does not provide any information on the size of the local manifoldand how it varies with x.

Therefore, we describe another approach for constructing local stablemanifolds. First we construct a special inner product in the tangent bundleTR which is known as the Lyapunov inner product. It provides an importanttechnical tool in studying nonuniform hyperbolicity.

Choose numbers 0 < λ′ < µ′ <∞ such that

λeε < λ′, µ′ < µe−ε. (9.8)

We define a new inner product 〈·, ·〉′x, called Lyapunov inner product, asfollows. Set

〈v, w〉′x =

∞∑

k=0

〈dfkv, dfkw〉fk(x)λ′−2k

(9.9)

if v, w ∈ Es(x), and

〈v, w〉′x =∞∑

k=0

〈df−kv, df−kw〉f−k(x)µ′2k

(9.10)

if v, w ∈ Eu(x).Notice that each series converges. Indeed, by the Cauchy–Schwarz in-

equality, Condition (H5) in Section 7, and (9.8), if v, w ∈ Es(x) then

〈v, w〉′x ≤∞∑

k=0

C(fk(x))2λ2kλ′

−2k‖v‖x‖w‖x

≤ C(x)2(1 − λeε/λ′)−1‖v‖x‖w‖x <∞,

(9.11)

and if v, w ∈ Eu(x) then

〈v, w〉′x ≤∞∑

k=0

C(f−k(x))2µ−2kµ′

2k‖v‖x‖w‖x

≤ C(x)2(1 − µ′/(µe−ε))−1‖v‖x‖w‖x <∞.

(9.12)

We extend 〈·, ·〉′x to all vectors in TxM by declaring the subspaces Es(x) andEu(x) to be mutually orthogonal with respect to 〈·, ·〉′x, i.e., we set

〈v, w〉′x = 〈vs, ws〉′x + 〈vu, wu〉′x,where v = vs+vu and w = ws+wu with vs, ws ∈ Es(x) and vu, wu ∈ Eu(x).

We emphasize that the Lyapunov inner product, and hence, the associ-ated norm ‖ · ‖′, called Lyapunov norm, depend on the choice of numbersλ′ and µ′. The Lyapunov inner product has several important propertieswhich determine its use in the study of nonuniform hyperbolicity.


N1. the angle between the subspaces Es(x) and Eu(x) in the inner prod-uct 〈·, ·〉′x is π/2 for each x ∈ R;

N2. ‖Ax‖′ ≤ λ′ and ‖Bx−1‖′ ≤ (µ′)−1;

N3. the relation between the Lyapunov inner product and the Riemann-ian inner product is given by

1√2‖w‖x ≤ ‖w‖′x ≤ D(x)‖w‖x,

where w ∈ TxM , and

D(x) = C(x)K(x)−1[(1 − λeε/λ′)−1 + (1 − µ′/(µe−ε))−1]1/2

is a measurable function satisfying for m ∈ Z,

D(fm(x)) ≤ D(x)e2ε|m|. (9.13)

Property (N1) holds true due to the construction of the Lyapunov innerproduct. To show Property (N2) we use (9.9) and write for v ∈ Es(x),

‖Axv‖′2 = 〈dfv, dfv〉′ =∞∑

k=0

〈dfkdfv, dfkdfv〉λ′−2k

=∞∑

k=1

〈dfkv, dfkv〉λ′−2(k−1) = λ′2∞∑

k=1

〈dfkv, dfkv〉λ′−2k

= λ′2(‖v‖′2 − ‖v‖2) ≤ λ′2‖v‖′2.

The proof of the second inequality is similar and uses (9.10). Property (N3)follows from Condition (H4) in Section 7, (9.11), and (9.12).

Properties (N1) and (N2) show that the action of the differential df onTR is uniformly hyperbolic with respect to the Lyapunov inner product.One can now use either the Hadamard method or the Perron method to-gether with the Lyapunov inner product to construct local stable manifolds,well-known in the uniform hyperbolic theory. Note that the “perturbation

map” fx (see (9.4) and (9.5)) satisfies Condition (9.6) in a neighborhood Ux

of the point x whose size depends on x. Moreover, the size of Ux decaysalong the trajectory of x with subexponential rate (see (9.13)). This requiresa substantial modification of the classical Hadamard–Perron approach andforces the size of the local stable manifolds to decay along the trajectory ofx with subexponential rate (see (9.2)).

We now briefly describe a modification of the Perron method.Fix x ∈ R. Consider the trajectory fm(x) and the family of maps

Fm = ffm(x). We identify the tangent spaces Tfm(x)M with Rn = Rk×Rp−k

(recall that p = dimM and 1 ≤ k < p) via an isomorphism τm such that

τm(Es(x)) = Rk and τm(Eu(x)) = Rp−k. We can write Fm = τm+1 Fm τm

−1 in the form

Fm(v, w) = (Amv + gm(v, w), Bmw + hm(v, w)), (9.14)


where Am : Rk → Rk and Bm : Rp−k → Rp−k are linear maps, and g : Rn →Rk and h : Rn → Rn−k are nonlinear maps defined for each v ∈ Bs(r0) ⊂ Rk

and w ∈ Bu(r0) ⊂ Rp−k (recall that Bs(r0) and Bu(r0) are balls centeredat 0 of radius r0). With respect to the Lyapunov metric these maps satisfy:

‖Am‖′ ≤ λ′ and (‖Bm−1‖′)−1 ≥ µ′, where 0 < λ′ < min1, µ′

(9.15)and

gm(0, 0) = 0, dgm(0, 0) = 0, hm(0, 0) = 0, dhm(0, 0) = 0, (9.16)

‖dgm(v1, w1) − dgm(v2, w2)‖′ ≤ Cγ−m(‖v1 − v2‖′ + ‖w1 − w2‖′)α,

‖dhm(v1, w1) − dhm(v2, w2)‖′ ≤ Cγ−m(‖v1 − v2‖′ + ‖w1 − w2‖′)α,(9.17)

whereλ′

α< γ < 1, 0 < α ≤ 1, C > 0 (9.18)

(see (9.7) and Conditions (N2) and (N3)).

Theorem 9.2 (see [27]). Let κ be any number satisfying

λ′ < κ < minµ′, γ 1α . (9.19)

Then there exist constants D > 0 and r0 > r > 0, and a map ψs : Bs(r) →Rp−k such that:

1. ψs is of class C1+α, and ψs(0) = 0 and dψs(0) = 0;2. ‖dψs(v1) − dψs(v2)‖′ ≤ D(‖v1 − v2‖′)α for any v1, v2 ∈ Bs(r);3. if m ≥ 0 and v ∈ Bs(r) then

(m−1∏

i=0

Fi

)(v, ψs(v)) ∈ Bs(r) ×Bu(r),

∥∥∥∥∥

(m−1∏

i=0

Fi

)(v, ψs(v))

∥∥∥∥∥

′

≤ Dκm‖(v, ψs(v))‖′,

where∏m−1

i=0 Fi denotes the composition Fm−1 · · · F0;4. given v ∈ Bs(r) and w ∈ Bu(r), if there is a number K > 0 such

that(m−1∏

i=0

Fi

)(v, w) ∈ Bs(r) ×Bu(r) and

∥∥∥∥∥

(m−1∏

i=0

Fi

)(v, w)

∥∥∥∥∥

′

≤ Kκm

for every m ≥ 0, then w = ψs(v);5. the numbers D and r depend only on the numbers λ′, µ′, γ, α, κ,

and C.

Proof. Consider the linear space Γκ of sequences of vectors

z = z(m) ∈ Rpm∈N,

satisfying the following condition:

‖z‖κ = supm≥0

(κ−m‖z(m)‖′) <∞.


It is easy to verify that Γκ is a Banach space with the norm ‖z‖κ. Considerthe open set

W = z ∈ Γκ : z(m) ∈ Bs(r) ×Bu(r) for every m ∈ N

and the map Φκ : Bs(r0) ×W → Γκ defined by

Φκ(y, z)(0) =

y,−

∞∑

k=0

(k∏

i=0

Bi

)−1

hk(z(k))

,

and if m > 0 then

Φκ(y, z)(m) = −z(m) +

((m−1∏

i=0

Ai

)y, 0

)

+

m−1∑

n=0

(m−1∏

i=n+1

Ai

)gn(z(n)),−

∞∑

n=0

(n∏

i=0

Bi+m

)−1

hn+m(z(n+m))

.

Using Conditions (9.15)–(9.19) we will show that the map Φκ is well-definedand is continuously differentiable over y and z. Indeed, by (9.16) and (9.17),if z ∈ Bs(r0) ×Bu(r0) and n ≥ 0, then

‖gn(z)‖′ = ‖gn(z) − gn(0)‖′ ≤ ‖dgn(ξ)‖′ · ‖z‖′

= ‖dgn(ξ) − dgn(0)‖′ · ‖z‖′

≤ Cγ−n(‖ξ‖′)α‖z‖′ ≤ Cγ−n(‖z‖′)1+α,

(9.20)

where ξ lies on the interval that joins the point 0 and z. Similarly, we havethat

‖hn(z)‖′ ≤ Cγ−n(‖z‖′)1+α. (9.21)


Using (9.15), (9.20), and (9.21) we obtain

‖Φκ(y, z)‖κ = supm≥0

(κ−m‖Φκ(y, z)(m)‖′)

≤ supm≥0

(κ−m‖z(m)‖′)

+ supm≥0

κ−m

[m−1∏

i=0

‖Ai‖′ · ‖y‖′ +m−1∑

n=0

(m−1∏

i=n+1

‖Ai‖′)

× Cγ−n(‖z(n)‖′)1+α

+∞∑

n=0

(n∏

i=0

‖B−1i+m‖′

)Cγ−(m+n)(‖z(n+m)‖′)1+α)

]

≤‖z‖κ + supm≥0

(κ−mλ′m

)‖y‖′

+ supm≥0

(κ−mC‖z‖κ

1+α

[m−1∑

n=0

λ′m−n−1

γ−nκ(1+α)n

+∞∑

n=0

µ′−(n+1)

γ−(m+n)κ(1+α)(m+n)

]).

(9.22)

In view of (9.19) we have

supm≥0

(κ−mλ′m

) = 1.

Since the function x 7→ xax reaches its maximum −1/e log a at x = −1/ log awe obtain

supm≥0

[κ−mλ′

m−1m−1∑

n=0

(λ′

−1γ−1κ(1+α)

)n]

≤λ′−1(κ−1λ′)mm if λ′−1γ−1κ(1+α) ≤ 1

λ′−1(γ−1κα)mm if λ′−1γ−1κ(1+α) ≥ 1

≤ λ′−1e−1

(log max

κα

γ,λ′

κ

)−1def= M1.

Furthermore, since µ′−1γ−1κ1+α = (µ′−1κ)(καγ−1) < 1, we have

supm≥0

(κ−mγ−mκ(1+α)mµ′

−1m−1∑

n=0

(µ′

−1γ−1κ(1+α)

)n)

=1

µ′ − γ−1κ1+α

def= M2.

(9.23)

Setting M = M1 +M2 we conclude that

‖Φκ(y, z)‖κ ≤ ‖z‖κ + ‖y‖′ + CM‖z‖κ1+α.


This implies that the map Φκ is well-defined. Moreover,

Φκ(0, 0) = (0, 0).

We now show that the map Φκ is of class C1. Indeed, for any y ∈ Bs(r0)and t ∈ Es such that y + t ∈ Bs(r0), given z ∈W and m ≥ 0 we have

Φκ(y + t, z)(m) − Φκ(y, z)(m) =

((m−1∏

i=0

Ai

)t, 0

).

It follows that

∂yΦκ(y, z)(m) =

(m−1∏

i=0

Ai, 0

).

Now for any y ∈ Bs(r0), z ∈W , t ∈ Γκ such that z + t ∈W we can write

Φκ(y, z + t) − Φκ(y, z) = (Aκ(z) − Id)t+ o(z, t),

where Id is the identity map,

(Aκ(z))t(m) =

m−1∑

n=0

(m−1∏

i=n+1

Ai

)dgn(z(n))t(n),

−∞∑

n=0

(n∏

i=0

Bi+m

)−1

dhn+m(z(n+m))t(m+ n)

,

and

o(z, t)(m) =

m−1∑

n=0

(m−1∏

i=n+1

Ai

)o1(z, t)(n),−

∞∑

n=0

(n∏

i=0

Bi+m

)−1

o2(z, t)(m+ n)

.

Here oi(z, t)(m) for i = 1, 2 are defined by

o1(z, t)(m) = gm((z + t)(m)) − gm(z(m)) − dgm(z(m))t(m),

o2(z, t)(m) = hm((z + t)(m)) − hm(z(m)) − dhm(z(m))t(m).


If z1, z2 ∈W and t ∈ Γκ then

‖(Aκ(z1) − Aκ(z2))t‖κ =

≤ supm≥0

κ−m

[m−1∑

n=0

(m−1∏

i=n+1

‖Ai‖′)Cγ−n(‖z1(n) − z2(n)‖′)α‖t(n)‖′

+∞∑

n=0

(n∏

i=0

‖B−1i+m‖′

)Cγ−(n+m)(‖z1(n+m) − z2(n+m)‖′)α

× ‖t(n+m)‖′]

≤ supm≥0

κ−mC

[m−1∑

n=0

λ′m−n−1

γ−nκ(1+α)n

+∞∑

n=0

µ′−(n+1)

γ−(n+m)κ(1+α)(n+m)

]× ‖z1 − z2‖κ

α‖t‖κ.

It follows that (see (9.22)–(9.23))

‖(Aκ(z1) − Aκ(z2))t‖κ ≤ CM‖z1 − z2‖κα‖t‖κ. (9.24)

It follows from (9.17) and the mean value theorem that

‖oi(z, t)(m)‖′ ≤ Cγ−m(‖t(m)‖′)1+α.

This implies that

‖o(z, t)‖κ ≤ CM‖t‖κ1+α.

We conclude that ∂zΦκ(y, z) = Aκ(z)− Id. In particular, ∂zΦκ(y, 0) = − Id.By (9.24), the map ∂zΦκ(y, z) is continuous. Therefore, the map Φκ satisfiesthe conditions of the Implicit Function Theorem, and hence, there exist anumber r ≤ r0 and a map ϕ : Bs(r) →W of class C1 with

ϕ(0) = 0 and Φκ(y, ϕ(y)) = 0. (9.25)

Let us notice that by applying an appropriate version of the Implicit Func-tion Theorem one can obtain an explicit estimate of the number r and thusshow that it depends only on the numbers λ′, µ′, γ, α, κ, and C. Moreprecisely, the following statement holds.

Lemma 9.3 (Implicit Function Theorem). Let E1, E2, and G be Banachspaces and g : A1 × A2 → G a C1 map, where Ai ⊂ Ei is a ball centeredat 0 of radius Ri for i = 1, 2. Assume that g(0, 0) = 0, and that thepartial derivative (over the second coordinate) D2g(0, 0) : E2 → G is a linearhomeomorphism. Assume also that D2g is Holder continuous in A1 × A2

with Holder constant a and Holder exponent α. Let B be a ball in E1 centeredat 0 of radius R where

r0 = min

R1, R2,

R2

2bc,

1

(1 + 2bc)(2ac)1/α

.


Set

b = maxx∈A1

‖D1g(x, 0)‖, c = ‖(D2g(0, 0))−1‖.

Then there exists a unique map u : B → A2 satisfying the following proper-ties:

1. u is of class C1+α, g(x, u(x)) = 0 for every x ∈ B, and u(0) = 0;2. if x1, x2 ∈ B then

∥∥∥∥du

dx(x1) −

du

dx(x2)

∥∥∥∥ ≤ 8ac(1 + 2bc)2‖x1 − x2‖α;

3. if x ∈ B then ∥∥∥∥du

dx(x)

∥∥∥∥ ≤ 1 + 2bc.

Proof of the lemma. See [27].

Let us notice that the map Φκ satisfies the conditions of Lemma 9.3 with

c = 1, b = 1, a = CM. (9.26)

To show this we observe that the map ∂zΦκ is Holder continuous. Indeed,by (9.24),

‖∂zΦκ(y1, z1) − ∂zΦκ(y2, z2)‖ ≤ ‖∂zΦκ(y1, z1) − ∂zΦκ(y1, z2)‖+ ‖∂zΦκ(y1, z2) − ∂zΦκ(y2, z2)‖

= 2‖Aκ(z1) − Aκ(z2)‖ ≤ CM‖z1 − z2‖κα.

We now describe some properties of the map ϕ. Differentiating the secondequality in (9.25) with respect to y we obtain

dϕ(y) = −[∂zΦκ(y, ϕ(y))]−1∂yΦκ(y, ϕ(y)).

Setting y = 0 in this equality yields

dϕ(0)(m) =

(m−1∏

i=0

Ai, 0

). (9.27)

One can write the vector ϕ(y)(m) in the form

ϕ(y)(m) = (ϕ1(y)(m), ϕ2(y)(m)),

where ϕ1(y)(m) ∈ Rk and ϕ2(y)(m) ∈ Rp−k. It follows from (9.25) that ifm ≥ 0 then

ϕ1(y)(m) =

(m−1∏

i=0

Ai

)y +

m−1∑

n=0

(m−1∏

i=n+1

Ai

)gn(ϕ(y)(n)), (9.28)

and

ϕ2(y)(m) = −∞∑

n=0

(n∏

i=0

Bi+m

)−1

hn+m(ϕ(y)(n+m)). (9.29)


These equalities imply that

ϕ1(y)(m+ 1) = Amϕ1(y)(m) + gm(ϕ1(y)(m), ϕ2(y)(m)),

ϕ2(y)(m+ 1) = Bmϕ2(y)(m) + hm(ϕ1(y)(m), ϕ2(y)(m)).

Indeed, iterating the first equality “forward” one easily obtains (9.28). Re-writing the second equality in the form

ϕ2(y)(m) = B−1m ϕ2(y)(m+ 1) −B−1

m hm(ϕ1(y)(m), ϕ2(y)(m)

and iterating it “backward” yields (9.29).This, we obtain that the function ϕ(y) is invariant under the family of

maps Fm, i.e.,

Fm(ϕ(y)(m)) = ϕ(y)(m+ 1). (9.30)

The desired map ψs is now defined by

ψs(v) = ϕ2(v)(0)

for each v ∈ Bs(r). Note that ϕ1(v)(0) = v. It follows from (9.25), (9.27),and Lemma 9.3 that the map ψs satisfies Statement 1 of the theorem. Itfollows from (9.30) that

m−1∏

i=0

Fi(v, ψs(v)) =

m−1∏

i=0

Fi(ϕ1(v)(0), ϕ2(v)(0))

=m−1∏

i=0

Fi(ϕ(v)(0) = ϕ(v)(m).

Applying Lemma 9.3 and (9.26) we find that∥∥∥∥∥

m−1∏

i=0

Fi(v, ψs(v))

∥∥∥∥∥

′

≤ κm‖ϕ(v)‖κ = κm‖ϕ(v) − ϕ(0)‖κ

≤ κm supξ∈Bs(r)

‖dϕ(ξ)‖κ‖v‖′ ≤ 3κm‖v‖′ ≤ 3κm‖(v, ψs(v))‖′

(we use here the fact that for every v = (v1, v2) ∈ R we have ‖v1‖, ‖v2‖ ≤‖v‖). This proves Statement 3. Furthermore, for any u1, u2 ∈ Bs(r) we havein view of Lemma 9.3 and (9.26) that D depends only on the numbers λ′,µ′, γ, α, κ, and C, and that

‖dψs(v1) − dψs(v2)‖′ = ‖dϕ2(v1)(0) − dϕ2(v2)(0)‖′

≤ ‖dϕ(v1)(0) − dϕ(v2)(0)‖′ ≤ 72CM(‖v1 − v2‖′)α.

This establishes Statements 2 and 5. Let (v, w) ∈ Bs(r) × Bu(r0) satisfiesthe assumptions of Statement 4 in the theorem. Set

z(m) =

(m−1∏

i=0

Fi(v, w)

).


It follows that z ∈ Γκ (with ‖z‖κ ≤ K) and that Φk(v, z) = 0. The unique-ness of the the map ϕ implies that z = ϕ(v) and hence,

w = ϕ2(v)(0) = ψs(v).

This establishes Statement 4. The desired result now follows.

Remark 9.4. Theorem 9.1 holds for nonuniformly partially hyperbolicsets. The proof does not require any changes.

10. Basic Properties of Local Stable and Unstable Manifolds

In the following series of remarks we describe some basic properties oflocal manifolds.

Remark 10.1. In the hyperbolic theory there is a symmetry betweenthe objects marked by the index “s” and those marked by the index “u”.Namely, when the time direction is reversed the statements concerning ob-jects with index “s” become the statements about the corresponding objectswith index “u”. In particular, this allows one to define a local unstable man-ifold V u(x) at a point x as a local stable manifold for f−1. Its propertiesare similar to those of V s(x).

Remark 10.2. It follows from Statement 5 of Theorem 9.2 and Condi-tion (N3) that the sizes of local stable and unstable manifolds are boundedfrom below on any regular set Rℓ, i.e.,

r(x) ≥ rℓ > 0 for x ∈ Rℓ,

where rℓ depends only on ℓ. Moreover, local stable and unstable manifolds

depend continuously on x ∈ Rℓ in the C1 topology, i.e., if xn ∈ Rℓ is a se-quence of points converging to x then the sequence of local stable manifoldsV s(xn) converges to V s(x) and the sequence of local unstable manifoldsV u(xn) converges to V u(x) in the C1 topology. The existence of the man-

ifolds V s(x) and V u(x) for every point x ∈ Rℓ follows from Theorem 9.1

and the existence of a nonuniformly hyperbolic structure on the set Rℓ (seeSection 7).

In addition, there exists a number δℓ > 0 such that for every x ∈ Rℓ andy ∈ Rℓ ∩ B(x, δℓ) the intersection V s(x) ∩ V u(y) is not empty and consistsof a single point which depends continuously on x and y.

Remark 10.3. It also follows from Statement 5 of Theorem 9.2 andCondition (N3) in Section 9 that the sizes of the local stable and unstablemanifolds at a point x ∈ R and any point y = fm(x) for m ∈ Z along thetrajectory of x are related by

r(fm(x)) ≥ Ke−ε|m|r(x), (10.1)

where K > 0 is a constant.If µ is an invariant Borel ergodic measure for f , then for all sufficiently

large ℓ the regular set Rℓ has positive measure. Therefore, the trajectory of


almost every point visits the set Rℓ infinitely many times. It follows thatfor typical points x the function r(fm(x)) is an oscillating function of mwhich is of the same order as r(x) for many values of m. Nevertheless, forsome integers m the value r(fm(x)) may become as small as it is allowed by(10.1). Let us emphasize that the rate of the decrease of the size of the localstable manifolds V s(fm(x)) as m→ ∞ is smaller than the rate of approachof the trajectories fm(x) and fm(y) for y ∈ V s(x).

Remark 10.4. Local stable and unstable manifolds depend Holder con-tinuously on x ∈ Rℓ; see Property (R3) in Section 7. For every ℓ ≥ 1, x ∈ Rℓ,and points z1, z2 ∈ V s(x) or z1, z2 ∈ V u(x) we have

d(Tz1Vs(x), Tz2V

s(x)) ≤ Cρ(z1, z2)α,

d(Tz1Vu(x), Tz2V

u(x)) ≤ Cρ(z1, z2)α,

where C > 0 is a constant and d is the distance in the Grassmannian bundleof TM generated by the Riemannian metric.

Remark 10.5. One can obtain a more refined information about thesmoothness of the local stable manifold. More precisely, if f is of classCp+α, with p ≥ 1 and 0 < α ≤ 1 (i.e., dpf is Holder continuous with Holderexponent α), then V s(x) is of class Cp; in particular, if f is of class Cp

for some p ≥ 2, then V s(x) is of class Cp−1 (and even of class Cp−1+α forany 0 < α < 1). These results are immediate consequences of the followingversion of Theorem 9.2.

Theorem 10.6. Assume that the conditions of Theorem 9.2 hold. Inaddition, assume that:

1. gm and hm are of class Cp for some p ≥ 2;2. there exists a constant K such that for ℓ = 1, . . ., p,

supz∈B

‖dℓgm(z)‖′ ≤ Kγ−m, supz∈B

‖dℓhm(z)‖′ ≤ Kγ−m,

where B = Bs(r0) ×Bu(r0) (see (9.14));3. if z1, z2 ∈ B, then

‖dpgm(z1) − dpgm(z2)‖′ ≤ Kγ−m(‖z1 − z2‖′)α,

‖dphm(z1) − dphm(z2)‖′ ≤ Kγ−m(‖z1 − z2‖′)α

for some α ∈ (0, 1).

If ψs(u) is the map constructed in Theorem 9.2, then there exists a numberN > 0, depending only on the numbers λ′, µ′, γ, α, κ, and K, such that:

1. ψs is of class Cp+α;2. supu∈Bs(r) ‖dℓψs(u)‖′ ≤ Nℓ for ℓ = 1, . . ., p.

Proof. It is sufficient to show that Φκ is of class Cp. Indeed, a simplemodification of arguments in the proof of Theorem 9.2 allows one to showthat

dℓyΦκ(y, z) = (0, 0), 2 ≤ ℓ ≤ p


and

dℓzΦκ(y, z)(m) =(

m−1∑

n=0

(m−1∏

i=n+1

Ai

)dℓgn(z(n)),−

∞∑

n=0

(n∏

i=0

B−1i+m

)dℓhm+n(z(m+ n))

)

(see [27] for more details).

In [32], Pugh and Shub strengthened the above result and showed thatin fact, if f is of class Cp for some p ≥ 2, then V s(x) is also of class Cp.

Remark 10.7. Consider a nonuniformly hyperbolic set R which con-sists of a single nonuniformly hyperbolic trajectory fnxn∈Z. The StableManifold Theorem 9.1 applies. It characterizes the behavior of trajectoriesnearby an individual nonuniformly hyperbolic trajectory, and does not needthe presence of any other nonuniformly hyperbolic trajectories. Note alsothat one can replace the assumption that the manifold M is compact bythe assumption that the diffeomorphism f satisfies (9.4)–(9.7) along a givennonuniformly hyperbolic trajectory.

Remark 10.8. There is another proof of the Stable Manifold Theoremwhich is based on a version of the Graph Transform Property — a state-ment that is well-known in the uniform hyperbolic theory. This approach isessentially an elaboration of the Hadamard method.

Let x ∈ R. Choose numbers r0, b0, c0, and d0 and for every m ≥ 0, set

rm = r0e−εm, bm = b0e

−εm, cm = c0e−εm, dm = d0λ

′meεm.

Consider the set Ψ of C1+α functions on (m, v) : m ∈ N, v ∈ Bs(rm) suchthat

ψ(m, v) ∈ Eu(f−m(x)) for every m ≥ 0 and v ∈ Bs(rm)

(where Bs(rm) is the ball in Es(f−m(x)) centered at 0 of radius rm), andsatisfying the following conditions:

‖ψ(m, 0)‖ ≤ bm, maxv∈Bs(rm)

‖dψ(m, v)‖ ≤ cm,

and if v1, v2 ∈ Bs(rm), then

‖dψ(m, v1) − dψ(m, v2)‖ ≤ dm‖v1 − v2‖α. (10.2)

Theorem 10.9 (Graph Transform Property). For every ℓ ≥ 1 there arepositive constants r0, b0, c0, and d0, which depend only on ℓ, such that for

every x ∈ Rℓ and every function ψ ∈ Ψ there exists a function ψ ∈ Ψ forwhich

F−1m ((v, ψ(m, v)) : v ∈ Bs(rm)) ⊃ (v, ψ(m+ 1, v)) : v ∈ Bs(rm+1)

(10.3)for all m ∈ N.


Sketch of the proof. Let (v(m+1), w(m+1)) = F−1m (v, ψ(m, v)). One

can write

(v(m+1), w(m+1)) = (A−1m v + gm(v, ψ(m, v)), B−1

m ψ(m, v) + hm(v, ψ(m, v))),

where (compare with Section 9)

gm(0, 0) = 0, dgm(0, 0) = 0, hm(0, 0) = 0, dhm(0, 0) = 0,

‖dgm(v1, w1) − dgm(v2, w2)‖′ ≤ Cγ−m(‖v1 − v2‖′ + ‖w1 − w2‖′)α,

‖dhm(v1, w1) − dhm(v2, w2)‖′ ≤ Cγ−m(‖v1 − v2‖′ + ‖w1 − w2‖′)α

for some constants satisfying (9.18). We have

‖v(m+1)1 − v

(m+1)2 ‖ ≥ (λ′)−1‖v1 − v2‖

− ‖gm(v1, ψ(m, v1)) − gm(v2, ψ(m, v2))‖= [(λ′)−1 − c(1 + cm)]‖v1 − v2‖

for some constant c > 0. Therefore, by choosing r0, b0, c0 > 0 sufficiently

small, one can define a function v 7→ ψ(m+ 1, v) on Bs(rm+1) by

ψ(m+ 1, v(m+1)) = w(m+1), (10.4)

which satisfies (10.3). Furthermore, we have

‖w(m+1)1 − w

(m+1)2 ‖ ≤ (µ′)−1cm‖v1 − v2‖ + c(1 + cm)‖v1 − v2‖

≤ (µ′)−1cm + c(1 + cm)

(λ′)−1 − c(1 + cm)‖v(m+1)

1 − v(m+1)2 ‖.

By eventually choosing a smaller c0 > 0, this implies that

‖ψ(m+ 1, 0)‖ ≤ bm+1, maxv∈Bs(rm+1)

‖dψ(m+ 1, v)‖ ≤ cm+1,

Taking derivatives in (10.4) and using (10.2) one can show that if v1, v2 ∈Bs(rm+1), then

‖dψ(m+ 1, v1) − dψ(m+ 1, v2)‖ ≤ dm+1‖v1 − v2‖α.

This completes the proof of the theorem.

Remark 10.10. Let E = E1 × E2 be a Banach space which is theproduct of two Banach spaces E1 and E2. Let also Fm : E → E be a familyof C1+α maps of the form (9.14) satisfying Conditions (9.15)–(9.17). Thenthe statement of Theorem 9.2 holds (with the obvious modifications).

Remark 10.11. For every x ∈ R we define the global stable and unstablemanifolds by

W s(x) =∞⋃

n=0

f−n(V s(fn(x))), W u(x) =∞⋃

n=0

fn(V u(f−n(x))). (10.5)

They are finite-dimensional immersed smooth submanifolds (of class Cr+α

if f is of class Cr+α) invariant under f . They have the following propertieswhich are immediate consequences of the Stable Manifold Theorem 9.2.


Theorem 10.12. If x, y ∈ R, then:

1. W s(x)∩W s(y) = ∅ if y /∈W s(x); W u(x)∩W u(y) = ∅ if y /∈W u(x);2. W s(x) = W s(y) if y ∈W s(x); W u(x) = W u(y) if y ∈W u(x);3. for every y ∈W s(x) (or y ∈W u(x)) we have that ρ(fn(x), fn(y)) →

0 as n→ +∞ (respectively, n→ −∞) with an exponential rate.

We emphasize that for dynamical systems satisfying uniform hyperbol-icity conditions (Anosov diffeomorphisms or Axiom A diffeomorphisms) theglobal stable and unstable manifolds have the following characterization:

W s(x) = y ∈M : ρ(fn(x), fn(y)) → 0 as n→ +∞ ,W u(x) = y ∈M : ρ(fn(x), fn(y)) → 0 as n→ −∞ .

This characterization may not hold for dynamical systems satisfying nonuni-form hyperbolicity conditions

Remark 10.13. Let ϕt be a smooth flow on a compact smooth Rie-mannian manifold M . The following is an analog of Theorem 9.1 for flowson nonuniformly hyperbolic sets (see Section 7).

Theorem 10.14 (Stable Manifold Theorem for Flows). Let R be a non-uniformly hyperbolic set for a C1+α flow ϕt. Then for every x ∈ R thereexists a local stable manifold V s(x) such that x ∈ V s(x), TxV

s(x) = Es(x),and if y ∈ V s(x) and t > 0 then

ρ(ϕt(x), ϕt(y)) ≤ T (x)λteεtρ(x, y), (10.6)

where T : R → (0,∞) is a Borel function such that if s ∈ R then

T (ϕs(x)) ≤ T (x)e10ε|s|.

The proof of Theorem 10.14 can be obtained by applying Theorem 9.1(see also Remark 9.4) to the diffeomorphism f = ϕ1 and verifying that thelocal stable manifold obtained in this way satisfies (10.6). We call V s(x) alocal stable manifold at x. In a similar fashion, by reversing the time onecan show that there exists a local unstable manifold V u(x) at x such thatTxV

u(x) = Eu(x).For every x ∈ R we define the global stable and unstable manifolds at x

by

W s(x) =⋃

t>0

ϕ−t(Vs(ϕt(x))), W u(x) =

⋃

t>0

ϕt(Vu(ϕ−t(x))). (10.7)

These are finite-dimensional immersed smooth submanifolds (of class Cr+α

if ϕt is of class Cr+α). They satisfy Properties 1 and 2 in Theorem 10.12.Furthermore, for every y ∈W s(x) (or y ∈W u(x)) we have ρ(ϕt(x), ϕt(y)) →0 as t→ +∞ (respectively, t→ −∞) with an exponential rate.

We also define the global weakly stable and unstable manifolds at x by

W s0(x) =⋃

t∈R

W s(ϕt(x)), W u0(x) =⋃

t∈R

W u(ϕt(x)).


It follows from (10.7) that

W s0(x) =⋃

t∈R

ϕt(Ws(x)), W u0(x) =

⋃

t∈R

ϕt(Wu(x)).

We remark that each of these two families of invariant immersed manifoldsforms a partition of the set R.

11. Absolute Continuity. Holonomy Map

Let f be a C1+α diffeomorphism of a compact smooth Riemannian man-ifold M . We describe one of the most crucial properties of local stable andunstable manifolds which is known as absolute continuity.

Let R be the set of nonuniformly hyperbolic points for f and Rℓ : ℓ ≥ 1the associated collection of regular sets (see Section 7). We assume that R

is not empty. Without loss of generality we may assume that each set Rℓ is

compact (otherwise we can replace them with sets Qℓ = Rℓ). We have Rℓ ⊂Rℓ+1 for every ℓ. Furthermore, the stable and unstable subspaces Es(x)and Eu(x) as well as the local stable manifolds V s(x) and local unstablemanifolds V u(x) depend continuously on x ∈ Rℓ and their sizes are boundedaway from zero by a number rℓ (see Remark 10.2).

Fix x ∈ Rℓ, a number r, 0 < r ≤ rℓ and set

Qℓ(x) =⋃

w∈Rℓ∩B(x,r)

V s(w), (11.1)

where B(x, r) is the ball at x of radius r.Consider a local open submanifold W which is uniformly transverse to

the family of local stable manifolds L(x) = V s(w) : w ∈ Rℓ ∩ B(x, r).If r is sufficiently small (in accordance with Remarks 10.3 and 10.2) thelatter can be assured provided that W is chosen so that the set exp−1

x Wis the graph of a smooth map ψ : Bu(q) ⊂ Eu(x) → Es(x), for some q,with a sufficiently small C1 norm. In this case W intersects each localstable manifold V s(w) ∈ L(x) and this intersection is transverse. We willconsider only local open submanifolds constructed in this way and call themtransversals to the family L(x). We also say that the map ψ represents W .

Let W 1 and W 2 be two transversals to the family L(x). We define theholonomy map

π : Qℓ(x) ∩W 1 → Qℓ(x) ∩W 2

using the relation

π(y) = W 2 ∩ V s(w), where y = W 1 ∩ V s(w) and w ∈ Qℓ(x) ∩B(x, r).

See Figure 2. Note that the holonomy map π is a homeomorphism onto itsimage.

We set

∆ = ∆(W 1,W 2) = ‖ψ1‖C1 + ‖ψ2‖C1 ,

where the maps ψ1 and ψ2 represent W 1 and W 2 respectively.


y

π(y)

W 1

W 2

V s(w)

Figure 2. Family L(x) of local stable leaves V s(w) for eachw ∈ Rℓ ∩B(x, r) and transversals W 1 and W 2

We recall that if ν and µ are two measures on a measurable space Xthan ν is said to be absolutely continuous with respect to µ if for every ε > 0there exists δ > 0 such that ν(E) < ε for every measurable set E for whichµ(E) < δ. A measurable invertible transformation T : X → Y of measurablespaces (X, ν) and (Y, µ) is said to be absolutely continuous if the measureµ is absolutely continuous with respect to the measure T∗ν. In this caseone defines the Jacobian J(T )(x) of T at a point x ∈ X (specified by the

measures ν and µ) to be the Radon–Nikodym derivative dµd(T∗ν) . If X is a

metric space with metric ρ then for µ-almost every x ∈ X one has

J(T )(x) = limr→0

µ(T (B(x, r)))

ν(B(x, r)). (11.2)

Given a submanifold W in M we denote by νW the Riemannian volume onW induced by the restriction of the Riemannian metric to W . We denote byJs(π)(y) the Jacobian of the holonomy map π at the point y ∈ Qℓ(x) ∩W 1

specified by the measures νW 1 and νW 2 .

Theorem 11.1 (Absolute Continuity; see [27]). Given ℓ ≥ 1, x ∈ Rℓ,and transversals W 1 and W 2 to the family L(x), the holonomy map π isabsolutely continuous (with respect to the measures νW 1 and νW 2) and theJacobian Js(π)(y) is bounded from above and bounded away from zero.

The first proof of the absolute continuity for Anosov diffeomorphism wasobtained by Anosov in [1] (see also [2]). For nonuniformly hyperbolic sys-tems the absolute continuity property was established by Pesin in [27]. Weshall present a proof of Theorem 11.1 which is an elaboration of the proof ofPesin. In [15], Katok and Strelcyn provided an extension of Pesin’s approachto smooth maps with singularities. Pugh and Shub [32] presented a proofof the absolute continuity of the holonomy map, modifying their approach


for the horocycle foliations of Anosov actions [31]. Liu and Qian, [18] es-tablished a version of Theorem 11.1 for random diffeomorphisms followingessentially the approach of Pesin.

We remark that the absolute continuity of the holonomy map does notimply directly that the Jacobian of the holonomy map is bounded (to provethis one needs the Holder property of the family of local stable manifolds oran equivalent statement).

In the appendix to [4], Brin provided a detailed presentation of the proofof absolute continuity for geodesic flows on manifolds of strictly negativesectional curvature. The presentation can be considered as a distillation ofthe original approach by Anosov and Sinai in [1, 2]. In [21], Mane presenteda proof of the absolute continuity for Anosov diffeomorphisms, although hisarguments contain a serious gap.

Remark 11.2. It is easy to see that the holonomy map π transfers themeasure νW 1 |Qℓ(x) ∩W 1 into a measure which is indeed, equivalent to themeasure νW 2 |Qℓ(x) ∩W 2.

Remark 11.3. The family of local unstable manifolds also satisfies theabsolute continuity property.

Remark 11.4. It follows from Theorem 11.1 that the stable and unstablefoliations of Anosov diffeomorphisms have the absolute continuity property.The same holds true for the stable and unstable foliations of partially hy-perbolic diffeomorphisms. Note that these foliations are Cr-continuous butin general, not smooth (in the latter case Theorem 11.1 will be a trivialcorollary of the Fubini theorem).

Remark 11.5. A. Katok constructed an example of a partially hyper-bolic volume-preserving diffeomorphism f of a compact manifold whose cen-tral distribution is integrable. The corresponding central foliation is invari-ant and continuous but not absolutely continuous. The leaves of the foliationare smooth compact submanifolds (see the description of this example in[23]). The map f is not ergodic and the Lyapunov exponent in the centraldirection is zero. Shub and Wilkinson (see [36]) constructed an open setof partially hyperbolic (but nonuniformly hyperbolic) volume-preserving er-godic diffeomorphisms of the three-dimensional torus whose central foliationis continuous but not absolutely continuous. The leaves of this foliation arediffeomorphic to the unit circle and the Lyapunov exponent in the centraldirection is positive almost everywhere.

Proof of Theorem 11.1. We split the proof into several steps.Step 1. Fix a point w ∈ Rℓ ∩ B(x, r) and let yi = V s(w) ∩ W i for

i = 1, 2. Choose m ≥ 0, q > 0 and set for i = 1, 2

W im = fm(W i), wm = fm(w), yi

m = fm(yi), qm = qeεm. (11.3)

Note thatW i0 = W i, w0 = w, yi

0 = yi, and q0 = q. Consider the open smoothsubmanifolds W 1

m and W 2m and the point wm. In view of Remark 10.5 for


i = 1, 2, there exists an open neighborhood W im(w, q) ⊂W i

m of the point yim

such thatW i

m(w, q) = expwm(ψi

m(v), v) : v ∈ Bu(qm). (11.4)

If q = q(m) is sufficiently small than for any w ∈ Rℓ ∩ B(x, r) and k =0, . . . ,m, we have that

f−1(W ik(w, q)) ⊂W i

k−1(w, q), i = 1, 2. (11.5)

We wish to compare the measures νW 1m|W 1

m(w, q) and νW 2m|W 2

m(w, q) forsufficiently large m.

Lemma 11.6. There exists a constant K1 > 0 such that the followingholds: for any m > 0 there exists q0 = q0(m) > 0 such that for any 0 < q ≤q0, we have ∣∣∣∣∣

νW 1m

(W 1m(w, q))

νW 2m

(W 2m(w, q))

− 1

∣∣∣∣∣ ≤ K1∆.

Proof of the lemma. The result follows from Theorem 10.9.

Step 2. We continue with the following Covering Lemma.

Lemma 11.7. For any m > 0 there are points wj ∈ Rℓ ∩B(x, r), j = 1,. . ., p = p(m) and q = q(m) > 0 such that the sets W i

m(wj , q) form an open

cover of the set fm(Qℓ(x) ∩ W i) (see (11.1)) of finite multiplicity whichdepends only on the dimension of W i for i = 1, 2.

Proof of the lemma. We recall the Besicovich Covering Lemma. Itstates that for each Z ⊂ Rk, if r : Z → R+ is a bounded function, then thecover B(x, r(x)) : x ∈ Z of Z contains a countable (Besicovich) subcoverof finite multiplicity depending only on k. This statements readily extendsto smooth manifolds via Whitneys’s Embedding Theorem.

It follows from the definition of sets W im(w, q) (see (11.4)) that for each

sufficiently small q = q(m) > 0 there is a number R = R(ℓ,m) > 0 suchthat for every w ∈ Rℓ ∩B(x, r),

Bim(w,

1

2R) ⊂W i

m(w, q) ⊂ Bim(w,R),

where Bim(w,R) is the ball in W i

m centered at w of radius R. Consider thecover of the set Zi = fm(Qℓ(x)∩W i) by balls Bi

m(w, 12R), w ∈ Rℓ∩B(x, r).

Applying the Besicovich Covering Lemma we obtain a Besicovich subcoverBi

m(wj ,12R), j = 1, . . . , p, p = p(m) of the set Zi of finite multiplicity M

(which does not depend on m). The sets W im(wj , q) also cover Zi and the

multiplicity M1 of this cover does not depend on m. Indeed, M1 does notexceed the multiplicity M2 of the cover of Zi by balls Bi

m(wj , R). Note thatevery ball Bi

m(wj , R) can be covered by not more than CM balls Bim(w, 1

2R)

where C depends only on the dimension of W i. Furthermore, given w wehave wj ∈ Bi

m(w, 12R) for at most a number M of points wj (otherwise at

least M + 1 balls Bim(wj ,

12R) would contain w). Therefore wj ∈ Bi

m(w,R)for at most CM points wj . This implies that M2 ≤ CM .


Step 3. We now compute the measures that are the pullbacks underf−m of the measures νW i

m|W i

m(w, q) for i = 1, 2. Note that for every w ∈Rℓ ∩ B(x, r) the points y1

m and y2m (see (11.3)) lie on the stable manifold

V s(wm). Choose z ∈ f−m(W im(w, q)) (with i = 1 or i = 2). Set zm = fm(z)

and

T i(z,m) = Jac(dzmf−m|TzmW

im(w, q)).

We need the following lemma.

Lemma 11.8. There exist K2 > 0 and m1(ℓ) > 0 such that for everyw ∈ Rℓ ∩B(x, r) and m ≥ m1(ℓ) one can find q = q(m) such that

∣∣∣∣T 2(y2

m,m)

T 1(y1m,m)

− 1

∣∣∣∣ ≤ K2∆,

and if z ∈W 10 (w, q) then

∣∣∣∣T 1(zm,m)

T 1(y1m,m)

− 1

∣∣∣∣ ≤ K2∆.

Proof of the lemma. For any 0 < k ≤ m and z ∈W 10 (w, q) we set

V (k, z) = TzkW 1

k (w, q).

We transport parallelly the space V (k, z) ⊂ TzkM along the geodesic that

connects the points zk and y1k (this geodesic is uniquely defined since these

points are sufficiently close to each other). We obtain a new subspace

V (k, z) ⊂ Ty1kM and we have∣∣∣Jac(dzk

f−1|V (k, z)) − Jac(dy1kf−1|V (k, y1))

∣∣∣

≤∣∣∣Jac(dzk

f−1|V (k, z)) − Jac(dy1kf−1|V (k, z))

∣∣∣

+∣∣∣Jac(dy1

kf−1|V (k, z)) − Jac(dy1

kf−1|V (k, y1))

∣∣∣ .

Since f ∈ C1+α in view of (11.5), we obtain∣∣∣Jac(dzk

f−1|V (k, z)) − Jac(dy1kf−1|V (k, y1))

∣∣∣≤ C1ρ(zk, y

1k)

α + C2d(V (k, z), V (k, y1)),

where C1 > 0 and C2 > 0 are constants (recall that ρ is the distance in Mand d is the distance in the Grassmannian bundle of TM generated by theRiemannian metric). It follows from Remark 10.4 (see also Property (R3)in Section 7) that

d(V (k, z), V (k, y1)) ≤ C3ρ(zk, y1k)

α,

where C3 > 0 is a constant. This and (9.1) imply that∣∣∣Jac(dzk

f−1|V (k, z)) − Jac(dy1kf−1|V (k, y1))

∣∣∣ ≤ C4(λkeεkρ(z, y1))

α,


where C4 > 0 is a constant. Note that for any 0 < k ≤ m and z ∈W 10 (w, q),

we have

C5−1 ≤ |Jac(dzk

f−1|V (k, z))| ≤ C5,

where C5 > 0 is a constant. We have

T 1(zm,m)

T 1(y1m,m)

=m∏

k=1

Jac(dzkf−1|V (k, z))

Jac(dy1kf−1|V (k, y1))

= expm∑

k=1

logJac(dzk

f−1|V (k, z))


≤ expm∑

k=1

(Jac(dzk

f−1|V (k, z))


− 1

)

≤ exp

(m∑

k=1

C4C5(λkeεkρ(z, y1))

α

)

≤ exp

(C4C5(λe

ερ(z, y1))α

1 − (λeε)α

).

The last expression can be made arbitrarily close to 1 by choosing q suffi-ciently small. This proves the second inequality. The first inequality can beproven in a similar fashion.

Lemma 11.8 allows one to compare the measures of the preimages underf−m of W 1

m(w, q) and W 2m(w, q). More precisely, the following statement

holds.

Lemma 11.9. There exist K3 > 0 and m2(ℓ) > 0 such that if w ∈Rℓ ∩B(x, r) and m ≥ m2(ℓ), then one can find q = q(m) such that

∣∣∣∣νW 1(f−m(W 1

m(w, q)))

νW 2(f−m(W 2m(w, q)))

− 1

∣∣∣∣ ≤ K3∆.

Proof of the lemma. For i = 1, 2 we have

νW i(f−m(W im(w, q))) =

∫

W im(w,q)

T i(z,m) dνW im

(z)

= T i(zim,m)νW i

m(W i

m(w, q)),

where zim ∈ W i

m(w, q) are some points. It follows from the assumptions ofthe lemma, (11.5), and Lemmas 11.6 and 11.8 that for sufficiently large m


fm(Qℓ(x) ∩W 1) fm(Qℓ(x) ∩W 2)

W 1m(w1, q)

W 2m(w1, q)

W 1m(w2, q)

W 2m(w2, q)

Figure 3. Sets fm(Qℓ(x) ∩ W i) and their covers by setsW i

m(wj , q) for i = 1, 2

and small q (assuming without loss of generality that ∆ ≤ 1),

∣∣∣∣νW 1(f−m(W 1

m(w, q)))

νW 2(f−m(W 2m(w, q)))

− 1

∣∣∣∣

≤∣∣∣∣T 1(z1

m,m)

T 2(z2m,m)

− 1

∣∣∣∣×νW 1

m(W 1

m(w, q))

νW 2m

(W 1m(w, q))

+

∣∣∣∣∣νW 1

m(W 1

m(w, q))

νW 2m

(W 2m(w, q))

− 1

∣∣∣∣∣

≤∣∣∣∣T 1(z1

m,m)

T 1(y1m,m)

T 2(y2m,m)

T 2(z2m,m)

T 1(y1m,m)

T 2(y2m,m)

− 1

∣∣∣∣× (1 +K1∆) +K1∆

≤ ((1 +K2∆)3 − 1)(1 +K1∆) +K1∆ ≤ K3∆.

The lemma follows.

Step 4. Given m > 0, choose points wj ∈ Rℓ ∩ B(x, r) and numberq = q(m) as in Lemma 11.7. Consider two sets

W 1m =

p⋃

j=1

W 1m(wj , q), W 2

m =

p⋃

j=1

W 2m(wj , q).

Note that W im ⊃ fm(Qℓ(x) ∩W i). See Figure 3.

We wish to compare the measures νW 1 |f−m(W 1m) and νW 2 |f−m(W 2

m).Let Li be the multiplicities of the covers W i

m(wj , q), i = 1, 2 constructed

in Lemma 11.7. Observe that the cover of the set Qℓ(x) ∩W i by the sets


f−m(W im(wj , q)) is also of multiplicity Li. Set L = maxL1, L2. We have

1

L

p∑

j=1

νW i(f−m(W im(wj , q))) ≤ νW i(f−m(W i

m))

≤p∑

j=1

νW i(f−m(W im(wj , q))).

It follows from Lemma 11.9 that

νW 1(f−m(W 1m))

νW 2(f−m(W 2m))

≤ L

∑pj=1 νW 1(f−m(W 1

m(wj , q)))∑pj=1 νW 2(f−m(W 2

m(wj , q)))

≤ Lmax

νW 1(f−m(W 1

m(wj , q)))

νW 2(f−m(W 2m(wj , q)))

: j = 1, . . . , p

≤ L (1 +K3∆) ,

(11.6)

with a similar bound for the inverse ratio. We conclude that

K−14 ≤ νW 1(f−m(W 1

m))

νW 2(f−m(W 2m))

≤ K4, (11.7)

where K4 > 0 is a constant independent of m.Step 5. Without loss of generality we may assume that νW i(Qℓ(x) ∩

W i) > 0. Given β > 0, denote by U iβ the β-neighborhood of the set Qℓ(x)∩

W i for i = 1, 2. We need the following lemma from measure theory.

Lemma 11.10. There exists β0 > 0 such that for every 0 < β ≤ β0 andi = 1, 2, we have

1 − ∆ ≤ νW i(Qℓ(x) ∩W i)

νW i(U iβ)

.

For any β > 0 and any sufficiently large m > 0 we have Qℓ(x) ∩W i ⊂f−m(W i

m) ⊂ U iβ for i = 1, 2. It follows from Lemma 11.10 and (11.7) that

1 − ∆

K4≤ νW 1(Qℓ(x) ∩W 1)

νW 2(Qℓ(x) ∩W 2)≤ K4

1 − ∆.

We emphasize that the constant K4 does not depend on the size of thetransversals W 1 and W 2 but only on their dimension. In particular, foreach y ∈ Qℓ(x) ∩W 1 we have

1 − ∆

K4≤ νW 2(π(Qℓ(x) ∩B1(y, r)))

νW 1(Qℓ(x) ∩B1(y, r))≤ K4

1 − ∆,

where B1(y, r) ⊂W 1 is the ball of radius r centered at y. Letting r → 0 weconclude from (11.2) that

1 − ∆

K4≤ Js(π)(y) ≤ K4

1 − ∆


for νW 1-almost every y ∈ W 1. Therefore the Jacobian is bounded and isbounded away from zero. This completes the proof of the theorem.

Remark 11.11. One can strengthen the statement of Lemma 11.7 andfor each i = 1, 2, construct a cover of the set fm(Qℓ(x) ∩W i) by the setsW i

m(wj , q) for j = 1, . . ., p such that this cover is of multiplicity 1 ona subset Di whose measure is arbitrarily close to the measure of the setfm(Qℓ(x)∩W i) (see [27] for details). One can now strengthen the inequality(11.7) (setting L = 1 in (11.6)) and prove that

∣∣∣∣∣νW 1(f−m(W 1

m))

νW 2(f−m(W 2m))

− 1

∣∣∣∣∣ ≤ K5∆(W 1,W 2) (11.8)

for all sufficiently large m. Here K5 > 0 is a constant which depends onlyon ℓ. It follows from (11.8) that the Jacobian Js(π)(y) for y ∈ Qℓ(x) ∩W 1

satisfies

|Js(π)(y) − 1| ≤ K5∆(W 1,W 2). (11.9)

Moreover, one can obtain the following formula for the Jacobian of theholonomy map:

Js(π)(y) =∞∏

k=0

Jac(dfk(y)f |Es(fk(y))

Jac(dfk(x)f |Es(fk(x)).

12. Absolute Continuity and Smooth Invariant Measures

One of the main manifestations of absolute continuity is the descriptionof the ergodic properties of diffeomorphisms preserving smooth measures. Inorder to achieve this goal we provide in this section a more detailed descrip-tion of absolute continuity property of local stable and unstable manifoldsfor diffeomorphisms preserving smooth measures.

Let f be a C1+α diffeomorphism of a smooth compact Riemannian man-ifold M without boundary. We assume that f preserves a smooth measureν, i.e., a measure which is equivalent to the Riemannian volume, and thatthis measure is hyperbolic, and thus the set R of nonuniformly hyperbolicpoints has full measure.

Let Rℓ be a regular set of positive measure. Fix a Lebesgue point x ∈ Rℓ

and consider the family of local stable manifolds

L(x) = V s(w) : w ∈ Rℓ ∩B(x, r)

where r = r(ℓ) is sufficiently small. We denote by νs(w) = νV s(w) for w ∈Rℓ ∩B(x, r), the Riemannian volume on V s(w) induced by the Riemannianmetric on M .

There exists a family of smooth submanifolds T s(w) for w ∈ B(x, r)with the following properties:

1. T s(w) is transverse to the family of local stable manifolds L(x) forevery w ∈ B(x, r);


2. T s(w1) ∩ T s(w2) = ∅ if w2 /∈ T s(w1) and T s(w1) = T s(w2) if w2 ∈T s(w1);

3.⋃

w∈B(x,r) Ts(w) ⊃ B(x, r).

4. T s(w) depends continuously on w ∈ B(x, r).

In other words, the collection of smooth submanifolds T s(w) generates apartition of the ball B(x, r). We denote this partition by T s. We alsodenote by µs(w) = µT s(w) the Riemannian volume on T s(w) induced by theRiemannian metric on M .

Consider now the partition ξs of the set Qℓ(x) (defined by (11.1)) intolocal stable manifolds V s(w) for w ∈ Rℓ ∩ B(x, r). It is clear that thispartition is measurable. Denote by νs(w) the conditional measure on V s(w)generated by the partition ξs and the measure ν. It is easy to see that thefactor space Qℓ(x)/ξs can be identified with the subset

As(w) =y ∈ T s(w) : there exists z ∈ R

ℓ ∩B(x, r)

such that y = T s(w) ∩ V s(z)

for every w ∈ B(x, r). We denote by νs the factor measure generated by thepartition ξs and the measure ν.

Theorem 12.1. The following statements hold:

1. the measures νs(w) and νs(w) are equivalent for ν-almost every w ∈Rℓ ∩B(x, r);

2. the factor measure νs is equivalent to the measure µs(w)|As(w) forν-almost every w ∈ Rℓ ∩B(x, r).

Proof. Consider the partition T s. Denote by µs(w) for w ∈ B(x, r)the system of conditional measures and by µs the factor measure generatedby the partition T s and the measure ν. One can see that the factor spaceB(x, r)/T s can be identified with a local stable manifold V s(w) for somew ∈ Rℓ ∩ B(x, r). Since the partition T s is smooth there exist measurablebounded functions g(w, z) and h(w) (where w ∈ Rℓ∩B(x, r) and z ∈ T s(w))such that

dµs(w)(z) = g(w, z) dµs(w)(z), dµs(w) = h(w) dνs(w).

Let B ⊂ Qℓ(x) be a Borel subset of positive ν-measure. We have

ν(B) =

∫

B(x,r)/T s

∫

T s(w)χB(w, z) dµs(w)(z) dµs(w)

=

∫

B(x,r)/T s

∫

T s(w)χB(w, z)g(w, z) dµs(w)(z) dµs(w)

=

∫

B(x,r)/T s

∫

T s(w0)χB(w, z)g(w, z)Js(πw0w)(y) dµs(w0)(y) dµ

s(w),

where χB(w, z) is the indicator function of the set B at the point z ∈ T s(w),πw0w is the holonomy map between the transversals T s(w0) and T s(w), z =


πw0w(y), and Js(πw0w)(y) is the Jacobian of the map πw0w at y. ApplyingFubini’s theorem we obtain

ν(B) =

∫

T s(w0)

∫

B(x,r)/T s

χB(w, z)g(w, z)Js(πw0w)(y) dµs(w) dµs(w0)(y)

=

∫

T s(w0)

∫

V s(w)χB(w, z)g(w, z)Js(πw0w)(y)h(w) dνs(w) dµs(w0)(y).

This implies the desired result.

As an immediate consequence of Theorem 12.1 we obtain the followingresults.

Theorem 12.2. For ν-almost every x ∈ R we have

νs(V s(x) \ R) = 0.

Let x ∈ Rℓ and W be a smooth submanifold which is transverse to thefamily of local smooth manifolds L(x). Let N be a set of zero Lebesguemeasure in W .

Theorem 12.3. We have ν(⋃V s(w)) = 0 where the union is taken over

all points w ∈ Rℓ ∩B(x, r) for which V s(w) ∩W ∈ N .

13. Ergodicity of Nonuniformly Hyperbolic Systems Preserving

Smooth Measures

In this section we move to the core of smooth ergodic theory and con-sider smooth dynamical systems preserving smooth hyperbolic invariantmeasures. A sufficiently complete description of their ergodic properties isone of the main manifestations of the above results on local instability (seeSections 9 and 10) and absolute continuity (see Sections 11 and 12). It turnsout that smooth hyperbolic invariant measures have an abundance of ergodicproperties which makes smooth ergodic theory a deep and well-developedpart of the general theory of smooth dynamical systems. Nevertheless, manyinteresting problems in this theory still remain open.

Let f be a C1+α diffeomorphism of a smooth compact Riemannian mani-foldM , and ν a Borel f -invariant measure onM . We assume that ν is equiv-alent to the Riemannian volume on M and that the set Λ of points withnonzero Lyapunov exponents (defined by (6.4)) has positive ν-measure.

The following statement is one of the main results of smooth ergodictheory. It describes the decomposition of the measure ν into ergodic com-ponents.

Theorem 13.1. There exist invariant sets Λ0, Λ1, . . . such that:

1.⋃

i≥0 Λi = Λ, and Λi ∩ Λj = ∅ whenever i 6= j;

2. ν(Λ0) = 0, and ν(Λi) > 0 for each i ≥ 1;3. f |Λi is ergodic for each i ≥ 1.

Proof. We begin with the following statement.


Lemma 13.2. Given an f-invariant Borel function ϕ, there exists a setN ⊂ M of zero measure such that if y ∈ Λ ∩ B(x, r) and z, w ∈ V s(y) \Nor z, w ∈ V u(y) \N then ϕ(z) = ϕ(w).

Proof of the lemma. Let z, w ∈ V s(y) and ψ be a continuous func-tion. By Birkhoff’s ergodic theorem the functions

ψ(x) = limn→∞

1

2n+ 1

n∑

k=−n

ψ(fkx),

ψ+(x) = limn→∞

1

n

n∑

k=1

ψ(fkx), and ψ−(x) = limn→∞

1

n

n∑

k=1

ψ(f−kx),

are defined for ν-almost every point x. We also have that ψ(x) = ψ+(x) =ψ−(x) outside a subset N ⊂M of zero measure.

Since ρ(fnz, fnw) → 0 as n→ ∞ (see Theorem 9.2) and ψ is continuous,we obtain

ψ(z) = ψ+(z) = ψ+(w) = ψ(w).

Notice that the continuous functions are dense in L1(M,ν) and hence, thefunctions of the form ψ are dense in the set of f -invariant Borel functions.The lemma follows.

Consider the regular sets

Λℓ =

x ∈ Λ : C(x, ε) ≤ ℓ,K(x, ε) ≥ 1

ℓ

,

where the functions C(x, ε) and K(x, ε) are given by Theorem 6.3. Letx ∈ Λℓ be a Lebesgue point of Λℓ, and for each r > 0 set

P ℓ(x, r) =⋃

y∈Λℓ∩B(x,r)

(V s(y) ∪ V u(y)).

Lemma 13.3. There exists r = r(ℓ) > 0 such that the map f is ergodicon the set

Q(x) =⋃

n∈Z

fn(P ℓ(x, r)). (13.1)

Proof of the lemma. Let ϕ be an f -invariant function, andN the setof zero measure constructed in Lemma 13.2. Choose 0 < r < minrℓ, δℓ (seeRemark 10.2). We also assume that the number r satisfies the requirementsin Section 12. By Theorem 12.1 there exists a point y ∈ (Λℓ ∩ B(x, r)) \Nsuch that

νsy(V

s(y) ∩N) = 0 and νuy (V u(y) ∩N) = 0,

where νsy and νu

y are, respectively, the measures induced on V s(y) and V u(y)by the Riemannian volume. Let Rs =

⋃V s(z) and Ru =

⋃V u(z), where

the unions are taken over all points z ∈ Λℓ∩B(x, rℓ) for which, respectively,V s(z) ∩ V u(y) ∈ N and V s(y) ∩ V u(z) ∈ N . By Theorem 12.3, we haveν(Rs) = ν(Ru) = 0.


Let z1, z2 ∈ P ℓ(x, r)\(Rs∪Ru∪N), and define a point wi ∈ Λℓ∩B(x, r)so that zi ∈ V s(wi) or zi ∈ V u(wi) for i = 1, 2. Depending on the locationof the points z1 and z2, there are four possible cases:

1) z1 ∈ V u(w1) and z2 ∈ V u(w2);2) z1 ∈ V s(w1) and z2 ∈ V u(w2);3) z1 ∈ V u(w1) and z2 ∈ V s(w2);4) z1 ∈ V s(w1) and z2 ∈ V s(w2).We will consider only the first two cases; the other two can be treated in

a similar fashion. Notice that in the first case the intersection V s(y)∩V u(wi)is nonempty and consists of a single point yi for i = 1, 2. It follows fromthe definition of the set Ru that y1, y2 6∈ N . Therefore, by Lemma 13.2,

ϕ(z1) = ϕ(y1) = ϕ(y2) = ϕ(z2).

In the second case, the intersection V s(wi)∩V u(y) is nonempty and consistsof a single point yi for i = 1, 2. It is easy to see that y1, y2 6∈ N . Sincey 6∈ N , Lemma 13.2 implies that

ϕ(z1) = ϕ(y1) = ϕ(y) = ϕ(y2) = ϕ(z2).


Lemma 13.4. We have ν(Q(x)) > 0 and Q(x) ⊂ Λ (mod 0).

Proof of the lemma. Since Q(x) ⊃ P ℓ(x, r) ⊃ Λℓ ∩B(x, r), we have

ν(Q(x)) ≥ ν(Λℓ ∩B(x, r)) > 0.

By Lemma 13.3 the map f |Q(x) is ergodic. Therefore

Q(x) = Q(x) ∩ Λ ⊂ Λ (mod 0)

and the lemma follows.

Since almost every point x ∈ Λ is a Lebesgue point of Λℓ for some ℓ, theinvariant sets Q(x) (defined by (13.1) for different values of ℓ) cover the setΛ (mod 0).

By Lemma 13.4, there is at most countable number of such sets. Wedenote them by Q1, Q2, . . .. We have ν(Qi) > 0 for each i ≥ 1, and the setΛ0 = Λ \⋃i≥1Qi has zero measure.

By Lemma 13.3, the map f |Qi is ergodic for each i ≥ 1. This yields

Qi ∩Qj = ∅ (mod 0), (13.2)

whenever i 6= j. For each n ≥ 1, let us set Λn = Qn \ ⋃n−1i=1 Qi. We have

Λi ∩ Λj = ∅ whenever i 6= j. It follows from (13.2) that ν(Qi) = ν(Λi) > 0for each i ≥ 1. This completes the proof of the theorem.

As an immediate consequence of Theorem 13.1 we obtain the followingresult. It is a generalization of Theorem 12.2 to the case when the set Λ haspositive but not necessarily full measure.


Theorem 13.5. For ν-almost every x ∈ Λ we have

νs(V s(x) \ Λ) = 0, νu(V u(x) \ Λ) = 0.

Theorem 13.1 is the first step in describing the ergodic properties ofsmooth hyperbolic invariant measures. The following Spectral Decomposi-tion Theorem provides a substantially deeper information.

We remind the reader the notion of Bernoulli automorphism. Let (X,µ)be a Lebesgue space, that is, a probability measure space with at most acountable number of atoms whose union Y ⊂ X is such that µ|(X \ Y ) ismetrically isomorphic to the Lebesgue measure on some interval [0, a]. Onecan naturally associate to (X,µ) a two-sided Bernoulli shift σ : XZ → XZ

defined by (σx)n = xn+1 for each n ∈ N, which preserves the convolutionµZ. A Bernoulli automorphism (T, ν) is an invertible (mod 0) measure pre-serving transformation which is metrically isomorphic to the Bernoulli shiftassociated to some Lebesgue space (X,µ).

We now state the Spectral Decomposition Theorem.

Theorem 13.6 (see [28]). For each i ≥ 1 the following properties hold:

1. Λi is a disjoint union of sets Λji , for j = 1, . . ., ni, which are cycli-

cally permuted by f , i.e., f(Λji ) = Λj+1

i for j = 1, . . ., ni − 1, andf(Λni

i ) = Λ1i ;

2. fni |Λji is a Bernoulli automorphism for each j.

We remark that the proof of Statement 2 requires the more sharp esti-mate (11.9) of the Jacobian of the holonomy map.

We shall now consider the case of a C1+α flow ϕt on a compact manifoldM . We assume that ν is a smooth measure which is ϕt-invariant. This meansthat ν(ϕtA) = ν(A) for any Borel set A ⊂M and t ∈ R. In a similar fashionto that in Sections 9 and 12 one can establish the existence of families oflocal stable and unstable manifolds (see Remark 10.13), and show that thesemanifolds possess the absolute continuity property (the precise formulationof the results is entirely analogous).

Following [29], we shall now formulate results concerning the ergodicdecomposition and the Bernoulli property for flows. Note that the Lyapunovexponent along the flow direction is zero. We assume that all other valuesof the Lyapunov exponent are nonzero for ν-almost every point. We alsoassume that ν vanishes on the set of fixed points of ϕt.

Theorem 13.7. There exist invariant sets Λ0, Λ1, . . . such that:

1.⋃

i≥1 Λi = Λ, and Λi ∩ Λj = ∅ whenever i 6= j;

2. ν(Λ0) = 0, and ν(Λi) > 0 for each i ≥ 1;3. ϕt|Λi is ergodic for each i ≥ 1.

Recall that a Bernoulli flow is a flow ϕt such that every transformationϕt is a Bernoulli automorphism.


Theorem 13.8. For each i ≥ 1, one of the following exclusive alterna-tives holds:

1. ϕt|Λi is a Bernoulli flow;2. ϕt|Λi is isomorphic to a constant-time suspension over a Bernoulli

automorphism.

14. Local Ergodicity

In the case of an Anosov diffeomorphism f preserving a smooth invariantmeasure on a connected smooth Riemannian manifold one can strengthen theSpectral Decomposition Theorem 13.6 by showing that f is indeed, ergodic.

For a general C1+α diffeomorphism preserving a smooth hyperbolic mea-sure one should expect that the number of open ergodic component is count-able (not finite). Dolgopyat, Hu, and Pesin constructed such an example inAppendix B. It is therefore, an interesting problem in smooth ergodic the-orem to find additional conditions which would guarantee that the ergodiccomponents of positive measure are open (mod 0). This is known as thelocal ergodicity problem.

The main obstacles for local ergodicity are the following:

1. the stable and unstable foliations are measurable but not necessarilycontinuous;

2. the unstable leaves may not expand under the action of fn (we re-mind the reader that they were defined as being exponentially con-tracting under f−n); the same is true for stable leaves with respectto the action of f−n;

3. the stable and unstable distributions are measurable but not neces-sarily continuous.

There are two different ways to obtain sufficient conditions for localergodicity. One was suggested by Pesin in [28] and the other one by Katokand Burns in [13]. Each of them is based on requirements which eliminateone or more of the above mentioned obstacles.

We first describe the approach developed in [28]. Roughly speaking itrequires that the stable (or unstable) foliations are locally continuous.

Given a subset X ⊂ M , we call a partition ξ of X a C1 continuouslamination of X if there exist continuous functions δ : X → (0,∞) andq : X → (0,∞) and an integer k > 0 such that for each x ∈ X

1. there exists a smooth immersed k-dimensional manifold W (x) con-taining x for which ξ(x) = W (x)∩X where ξ(x) is the element of thepartition ξ containing x; the manifold W (x) is called the (global) leafof the lamination at x; the connected component of the intersectionW (x)∩B(x, δ(x)) that contains x is called the local leaf at x and isdenoted by V (x);

2. there exists a continuous map ϕ : B(x, q(x)) → C1(D,M) (whereD ⊂ Rk is the unit ball) such that for every y ∈ X ∩B(x, q(x)) themanifold V (y) is the image of the map ϕ(y) : D →M .


For every x ∈ X and y ∈ B(x, q(x)) we set U(y) = ϕ(y)(D) and we callit the local leaf of the lamination at y. Note that U(y) = V (y) for y ∈ X.

A C1 continuous lamination of the whole manifold M is called a C1

continuous foliation on M . The stable (as well as unstable) foliation of anAnosov diffeomorphism of a compact smooth connected Riemannian mani-fold M is a C1 continuous foliation of M (see [12]).

Theorem 14.1. Let f be a C1+α diffeomorphism of a compact smoothRiemannian manifold preserving a smooth measure ν and Λ the nonuni-formly hyperbolic set for f (see Section 7). Assume that ν(Λ) > 0 and thatthere exists a C1 continuous lamination W of Λ such that W (x) = W s(x)for every x ∈ Λ (where W s(x) is the global stable manifold at x; see Sec-tion 10). Then every ergodic component of f of positive measure is open(mod 0) in Λ (with respect to the induced topology).

Proof. We need the following statement which is an immediate conse-quence of Theorem 12.2.

Lemma 14.2. There exists a set N ⊂M of zero measure such that

νsx(V s(x) \ Λ) = νu

x (V u(x) \ Λ) = 0

for every x ∈ Λ \N .

Let Q ⊂ Λ be an f -invariant set with ν(Q) > 0. We assume that f |Qis ergodic, and we will show that Q is open (mod 0). By Lemma 13.3,there exists a density point x of the set Λℓ such that Q = Q(x) (mod 0)(see (13.1)). By Lemma 14.2, νu

x -almost every point y ∈ V u(x) belongs toΛ. Let BU (y, r) be the ball in U(y) (with respect to the intrinsic metric)centered at y of radius r. For a νu

x -measurable set Y ⊂ V u(x) we denote

R(x, r, Y ) =⋃

y∈Y

BU (y, r).

We also set

R(r) = R(x, r, V u(x)), R(r) = R(x, r, V u(x) ∩ Λ)

and given m ∈ N,Rm(r) = R(x, r, V u(x) ∩ Λm).

Clearly,

Rm(r) ⊂ R(r) ⊂ R(r)

for every m ∈ N.Since W is a C1 continuous lamination one can find δ0 > 0 such that

BU (y, δ0) ⊂ U(y) for any y ∈ V u(x). By Theorem 9.2 and Remark 10.2,there exists rm > 0 such that BU (y, rm) ⊂ V s(y) for any y ∈ V u(x) ∩ Λm.

Fix r ∈ (0, rm]. Given y ∈ Rm(r/2), we denote by ni(y) the successivereturn times of the positive semi-trajectory of y to the set Rm(r/2). We also

denote by zi ∈ V u(x)∩Λm a point for which fni(y)(y) ∈ BU (zi, r/2) for i ∈ N.In view of the Poincare recurrence theorem, one can find a subset N ⊂ M


of zero ν-measure for which the sequence ni(y) is well-defined provided thaty ∈ Rm(r/2) \N .

Lemma 14.3. For ν-almost every y ∈ Rm(r/2) we have

W s(y) =

∞⋃

i=1

f−ni(y)(BU (zi, r)). (14.1)

Proof of the lemma. Let y ∈ Rm(r/2) \ N and z ∈ W s(y). By

Theorem 10.12 we have ρ(fni(y)y, fni(y)z) ≤ r/2 for all sufficiently large i.Therefore,

fni(y)(z) ∈ BU (fni(y)(y), r/2) ⊂ BU (zi, r)

and the lemma follows.

Denote by ξm(δ0) the partition of the set Rm(δ0) into the sets BU (y, δ0).

Lemma 14.4. The partition ξm(δ0/2) is measurable and has the followingproperties:

1. the conditional measure on the element BU (y, δ0/2) of this partitionis absolutely continuous with respect to the measure νs

y;2. the factor measure on the factor space Rm(δ0/2)/ξm(δ0/2) is abso-

lutely continuous with respect to the measure νux |V u(x) ∩ Λm.

Proof of the lemma. Choose r = minδ0/100, rm. By Lemma 14.3,for almost every point w ∈ V u(x) ∩ Λm one can find a point y(w) ∈BU (w, r/2) ⊂ Rm(r/2) for which (14.1) holds. Moreover, the point y(w)can be chosen in such a way that the map

w ∈ V u(x) ∩ Λm 7→ y(w) ∈ Rm(r/2)

is measurable. For each n ∈ N, set

Rn =⋃

w∈V u(x)∩Λm

⋃

ni(y)≤n

(f−ni(y(w))(BU (zi, r)) ∩Rm(3δ0/4)

).

Observe that

Rm

(3

4δ0

)=⋃

n∈N

Rn.

Given ε > 0, there exists p > 0 and a set Y ⊂ V u(x) ∩ Λm such that

νux ((V u(x) ∩ Λm) \ Y ) ≤ ε and R(δ0/2, Y ) ⊂

⋃

n≤p

Rn. (14.2)

It follows from Theorem 12.1 that the partition ξm(δ0/2)|Rn satisfies State-ments 1 and 2 of the lemma for each n > 0. Since ε is arbitrary the desiredresult follows.

We proceed with the proof of the theorem. Denote by ξ(δ0) the partition

of the set R(δ0) into the sets BU (y, δ0) and by ξ(δ0) the partition of the set

R(δ0) into the sets BU (y, δ0). Since W is a C1 continuous lamination, the


factor space R(δ0)/ξ(δ0) can be identified with V u(x) and the factor space

R(δ0)/ξ(δ0) with V u(x) ∩ Λ.

Letting m→ ∞ in Lemma 14.4, we conclude that the partition ξ(δ0/2)also satisfies Statements 1 and 2 of the lemma. By Lemma 13.3, we have

Q ⊃ R(δ0/2). This implies that Q =⋃

n∈Zfn(R(δ0/2)).

Note that the set R(δ0/2) is open. We will show that the set

A = Λ ∩ (R(δ0/2) \ R(δ0/2))) (14.3)

has zero measure. Assuming the contrary we have that the set

Am = Λm ∩ (R(δ0/2) \ R(δ0/2))) (14.4)

has positive measure for all sufficiently large m. Therefore, for ν-almostevery z ∈ Am we obtain that νu

z (V u(z) ∩ Am) > 0. Consider the setR(z, δ0/2, A

m). Clearly,

R(z, δ0/2, Am) ⊂ R(δ0/2) \ R(δ0/2).

Let Bm = R(z, δ0/2, Am) ∩ V u(x). This set is not empty. It follows from

(14.4) that Bℓ ⊂ V u(x) \ Λ and thus νux (Bℓ) = 0 (see Lemma 14.2). More-

over, repeating arguments in the proof of Lemma 14.4 (see 14.2) one canshow that the holonomy map π that moves V u(z) ∩ Am onto Bm is abso-lutely continuous. Hence, νu

x (Bm) > 0. This contradiction implies 14.3 andcompletes the proof of the theorem.

Theorem 14.1 provides a way to establish ergodicity of the map f |Λ.Recall that a map f is called topologically transitive if given two nonemptyopen sets there exists a positive integer n such that fnA ∩B 6= ∅.

Theorem 14.5. Assume that the conditions of Theorem 14.1 hold andthat Λ is an open (mod 0) set (in particular is a set of full measure). Then:

1. every ergodic component of f of positive measure lying in Λ is open(mod 0);

2. if f |Λ is topologically transitive then f |Λ is ergodic.

Proof. The first statement follows from Theorem 14.1. Assume thatthe map f |Λ is topologically transitive. Let C, D be two distinct ergodiccomponents of positive measure. Since ν(Λ) = 1 we have that C,D ⊂ Λ(mod 0). Moreover, ν(fnC ∩D) = 0 for any integer n. By Theorem 14.1,the sets C and D are open (mod 0) and since the map f is topologicallytransitive, we have fmC∩D 6= ∅ for some m. Furthermore, the set fmC∩Dis also open (mod 0) and since ν is equivalent to the Riemannian volume,we conclude that ν(fmC ∩ D) > 0. This contradiction implies that f isergodic.

For a general diffeomorphism preserving a smooth hyperbolic measure,one should not expect the unstable (and stable) foliation to be locally con-tinuous. In order to explain why this can happen consider a local unstablemanifold V u(x) passing through a point x ∈ Λ. For a sufficiently large


ℓ, the set V u(x) ∩ Λℓ has positive Riemannian volume (as a subset of thesmooth manifold V u(x)) but is, in general, a Cantor-like set. When thelocal manifold is moved forward a given time n one should expect a suffi-ciently small neighborhood of the set V u(x) ∩ Λℓ to expand. Other piecesof the local manifold (corresponding to bigger values of ℓ) will also expandbut with smaller rates. This implies that the global leaf W u(x) (defined by(10.5), see Remark 10.11) may bend “uncontrollably” — this phenomenonhas not yet been observed in any example but is thought to be “real” (andeven “typical” in some sense). As a result the map x 7→ ϕx in the definitionof local continuity may not be indeed continuous. Furthermore, the globalmanifold W u(x) may turn out to be “bounded”, i.e., it may not admit anembedding of an arbitrarily large ball in Rk (where k = dimW u(x)). Thisphenomenon, again, has not yet been observed.

The local continuity of the global foliation often comes up in the followingsetting. Using some additional information on the system one can builda locally continuous invariant foliation whose leaves contain leaves of theunstable foliation. This along may not yet guarantee the local continuity ofthe unstable foliation. However, often one may observe that local unstableleaves expand in a “controllable” way when they are moved forward. Thissituation occurs, for example, for geodesic flows on compact Riemannianmanifolds of nonpositive curvature (see Section 16).

We now state a formal criterion for local continuity.

Theorem 14.6 (see [28]). Let f be a C1+α diffeomorphism of a compactsmooth Riemannian manifold preserving a smooth hyperbolic measure ν andΛ the nonuniformly hyperbolic set for f of full measure. Let also W be a C1

continuous lamination of Λ with the following properties:

1. W (x) ⊃ V s(x) for every x ∈ Λ;2. there exists a number δ0 > 0 and a measurable function n(x) on Λ

such that for almost every x ∈ Λ and any n ≥ n(x),

f−n(V s(x)) ⊃ BU (f−n(x), δ0).

Then every ergodic component of f of positive measure is open (mod 0).

Proof. Let x be a Lebesgue point of the set Λℓ for some sufficientlylarge ℓ > 0. Set A(r) = Λℓ ∩B(x, r) where B(x, r) is the ball in M centeredat x of radius r. Applying Lemma 14.3 to the set A(r) for sufficiently small rand using the conditions of the theorem we find that W s(y) ⊃ BU (y, δ0) foralmost every y ∈ A(r). One can now obtain the desired result by repeatingarguments in the proof of Theorem 14.1.

For one-dimensional foliations the second condition of Theorem 14.6holds automatically and hence, can be omitted.

Theorem 14.7 (see [28]). Let W be a C1 continuous one-dimensionallamination of Λ, satisfying the following property: W (x) ⊃ V s(x) for every


x ∈ Λ. Then every ergodic component of f of positive measure is open(mod 0). Moreover, W s(x) = W (x) for almost every x ∈ Λ.

Proof. Fix ℓ > 1 sufficiently large. For almost every point x ∈ Λℓ, theintersection A(x) = V s(x)∩Λℓ has positive Lebesgue measure in V s(x). Forevery y ∈ A(x) let s(y) be the distance between x and y measured alongV s(x). Then there exists a differentiable curve γ : [0, s(y)] → V s(x) withγ(0) = x and γ(s(y)) = y, satisfying

ρW (f−nx)(f−nx, f−ny) =

∫ s(y)

0‖dγ(t)f

−nγ′(t)‖ dt

≥∫ s(y)

0‖dfnγ(t)f

nγ′(t)‖−1 dt ≥ ℓ−1e−εnλ−n ≥ δ0

for sufficiently large n (see Section 7), where δ0 is a positive constant. There-fore, the second condition of Theorem 14.6 holds and the desired resultfollows.

Remark 14.8. Theorems 14.6 and 14.7 can be extended to the case whenthe set Λ is open (mod 0) and has positive (not necessarily full) measure.They can also be extended (with trivial modifications) to dynamical systemswith continuous time.

We now describe the approach in [13] to study the local ergodicity. Acontinuous function Q : TM → R is called an infinitesimal eventually strictLyapunov function for f over a set U ⊂M if:

1. for each x ∈ U the function Qx = Q|TxM is homogeneous of degreeone, and takes both positive and negative values;

2. there exist continuous distributions Dsx ⊂ Cs(x) and Du

x ⊂ Cu(x)such that TxM = Ds

x ⊕Dux for all x ∈ U , where

Cs(x) = Q−1((−∞, 0)) ∪ 0 and Cu(x) = Q−1((0,∞)) ∪ 0are called the stable and unstable cones of Qx;

3. if x ∈ U , n ∈ N, fn(x) ∈ U , and v ∈ TxM then

Qfn(x)(dxfnv) ≥ Qx(v);

4. for ν-almost every x ∈ U there exists k = k(x), ℓ = ℓ(x) ∈ N suchthat fk(x) ∈ U , f−ℓ(x) ∈ U , and if v ∈ TxM \ 0 then

Qfk(x)(dxfkv) > Qx(v) and Qf−ℓ(x)(dxf

−ℓv) < Qx(v).

A function Q is called an infinitesimal eventually uniform Lyapunov functionfor f over a set U ⊂ M if it satisfies Conditions 1–3 and the followingcondition: there exists ε > 0 such that for ν-almost every x ∈ M one canfind k = k(x), ℓ = ℓ(x) ∈ N for which fk(x) ∈ U , f−ℓ(x) ∈ U , and ifv ∈ TxM \ 0 then

Qfk(x)(dxfkv) > Qx(v) + ε‖v‖ and Qf−ℓ(x)(dxf

−ℓv) < Qx(v) − ε‖v‖.


The following result of Katok and Burns gives a criterion of local ergodicityin terms of infinitesimal Lyapunov functions.

Theorem 14.9. The following properties hold:

1. If f possesses an infinitesimal eventually strict Lyapunov functionQ over an open set U ⊂M , then almost every ergodic component off on the set

⋃n∈Z

fnU is open (mod 0).2. If f possesses an infinitesimal eventually uniform Lyapunov functionQ over an open set U ⊂M , then every connected component of theset

⋃n∈Z

fnU belongs to one ergodic component of f . Moreover, ifU is connected then f |U is a Bernoulli transformation.

Sketch of the proof. When Q is an infinitesimal eventually strictLyapunov function, given a compact set K ⊂ U one can use the uniformcontinuity of x 7→ Qx on the set K, and Condition 3 to show that the sizeof the stable and unstable manifolds on K is uniformly bounded away fromzero. Furthermore, using Condition 4 one can show that for ν-almost everypoint z ∈M there exist θ = θ(z) > 0 and a neighborhood N of z such thatfor ν-almost every x ∈ N and y ∈ V u(x) ∩N the tangent space TyV

u(x) isin the θ-interior of Cu(y). A similar statement holds for stable manifolds.

Together with Condition 2 this implies that the stable and unstablemanifolds have almost everywhere a “uniform” product structure; namely,for almost every x ∈ U there exist a neighborhood N(x) of x and δ > 0 suchthat:

1. V s(y) and V u(y) have size at least δ for almost every y ∈ N(x);2. V s(y) ∩ V u(z) 6= ∅ for ν × ν-almost every (y, z) ∈ N(x) ×N(x).

The proof of Statement 1 follows now from a similar argument to that inthe proof of Lemma 13.3.

When Q is an infinitesimal eventually uniform Lyapunov function, thefunction θ(z) is uniformly bounded away from zero. This can be used toestablish that for every x (and not only almost every x) there exists a neigh-borhood N(x) of x and δ > 0 with Properties 1 and 2. A similar argumentnow yields the first claim in Statement 2. The last claim is an immediateconsequence of Theorem 13.6.

In conclusion we will state a few interesting open problems relevant tothe above discussion. In what follows M is a compact smooth Riemannianmanifold, f is a Cr diffeomorphism of M (r ≥ 2) preserving the Riemannianvolume, and Λ is a nonuniformly hyperbolic set for f .

Problem 1. Construct a diffeomorphism f for which:

1. the set Λ has positive but not full measure;2. the set Λ is not open (mod 0);3. the set Λ has full measure but some (or all) ergodic components of

positive measure are not open (mod 0).

Problem 2. Assume that dimM = 2.


1. Is it true that every ergodic component of positive measure is open(mod 0)?

2. Is it true that there can be only finitely many ergodic componentsof positive measure?

15. The Entropy Formula

One of the main ideas of Smooth Ergodic Theory is that sufficient insta-bility of trajectories yields rich ergodic properties of the system. The entropyformula is in a sense a “quantitative manifestation” of this idea and is yetanother pearl of Smooth Ergodic Theory. It expresses the Kolmogorov–Sinaientropy hν(f) of a diffeomorphism, preserving a smooth hyperbolic measure,in terms of the values of the Lyapunov exponent.

In [33], Ruelle established an upper bound for the metric entropy of a dif-feomorphism, preserving an arbitrary Borel probability measure, in terms ofthe Lyapunov exponents. Independently, the same upper bound was foundby Margulis (unpublished) in the case of volume-preserving diffeomorphisms.

We now briefly recall the relevant notions from the entropy theory. Let(X,B, µ) be a Lebesgue measure space. This means that µ is a Lebesguemeasure on X with σ-algebra B. A finite or countable family ξ ⊂ B is calleda measurable partition of X if µ(

⋃C∈ξ C) = µ(X), and µ(C ∩ D) = 0 for

every C, D ∈ ξ such that C 6= D. The entropy of the measurable partitionξ (with respect to µ) is given by

Hµ(ξ) = −∑

C∈ξ

µ(C) log µ(C),

with the convention that 0 log 0 = 0. Given two measurable partitions ξ andζ we also define the conditional entropy of ξ with respect to ζ by

Hµ(ξ|ζ) = −∑

C∈ξ

∑

D∈ζ

µ(C ∩D) logµ(C ∩D)

µ(D).

Given two measurable partitions ξ and η of X we shall write ξ ⊂ η iffor every D ∈ η there exists C ∈ ξ such that D ⊂ C (mod 0). If ξ, η,and ζ are measurable partitions with ξ ⊂ η we have Hµ(ξ) ≤ Hµ(η) andHµ(ζ|ξ) ≥ Hµ(ζ|η).

Consider now a measurable function T : X → X. We define the entropyof T with respect to partition ξ by the limit

hµ(T, ξ) = limn→∞

1

nHµ

(n−1∨

k=0

T−kξ

), (15.1)

which always exists. It can be shown that

hµ(T, ξ) = infn

1

nHµ

(n−1∨

k=0

T−kξ

)= lim

n→∞Hµ

(ξ|

n∨

k=1

T−kξ

). (15.2)


When ξ− =∨∞

k=0 T−kξ is a partition we have hµ(T, ξ) = Hµ(ξ|ξ−). We

finally define the entropy of T with respect to µ by

hµ(T ) = supξhµ(T, ξ),

where the supremum is taken over all measurable partitions ξ with finite en-tropy (which indeed, coincides with the supremum over all finite measurablepartitions). We have hµ(T, ξ) ≤ Hµ(ξ) (by (15.2)) and hµ(Tm) = mhµ(T )for every m ∈ N.

When T is invertible with measurable inverse one can replace T−kξ byT kξ in the formulas (15.1) and (15.2). In particular, hµ(T−1, ξ) = hµ(T, ξ)and hµ(T−1) = hµ(T ). Therefore, hµ(Tm) = |m|hµ(T ) for every m ∈ Z.

Let f be a C1 diffeomorphism of a smooth compact Riemannian man-ifold and ν an invariant Borel probability measure on M . Recall that theLyapunov spectrum of the exponent χ(x, ·) is (see Section 6)

Spχ(x) = (χi(x), ki(x)) : 1 ≤ i ≤ s(x).Theorem 15.1 (Margulis–Ruelle Inequality). We have

hν(f) ≤∫

M

∑

i:χi(x)>0

ki(x)χi(x) dν(x).

Proof. By decomposing ν into its ergodic components we may assumewithout loss of generality that ν is ergodic. Then ki(x) = ki and χi(x) =χi are constant ν-almost everywhere for each i. Fix m > 0. Since M iscompact, there exists tm > 0 such that for every 0 < t ≤ tm, y ∈ M , andx ∈ B(y, t), we have

1

2dxf

m(exp−1

x B(y, t))⊂ exp−1

fmx fmB(y, t) ⊂ 2dxf

m(exp−1

x B(y, t)),

(15.3)where for a set A ⊂ TzM and z ∈M , we write αA = αv : v ∈ A.

Lemma 15.2. Given ε > 0, there exists a partition ξ of M satisfying thefollowing conditions:

1. diam ξ ≤ tm/10 and hν(fm, ξ) ≥ hν(f

m) − ε;2. for every element C ∈ ξ there exist balls B(x, r) and B(x, r′), such

that r < 2r′ ≤ tm/20 and B(x, r′) ⊂ C ⊂ B(x, r);3. there exists 0 < r < tm/20 such that if C ∈ ξ then C ⊂ B(y, r) for

some y ∈M , and if x ∈ C then

1

2dxf

m(exp−1

x B(y, r))⊂ exp−1

fmx fmC ⊂ 2dxf

m(exp−1

x B(y, r)).

Proof of the lemma. Given α > 0, consider a maximal α-separatedset Γ, i.e., a finite set of points for which d(x, y) > α whenever x, y ∈ Γ.For each x ∈ Γ define

DΓ(x) = y ∈M : d(y, x) ≤ d(y, z) for all z ∈ Γ \ x.


Obviously, B(x, α/2) ⊂ DΓ(x) ⊂ B(x, α). Note that the sets DΓ(x) corre-sponding to different points x ∈ Γ intersect only along their boundaries, i.e.,at a finite number of submanifolds of codimension greater than zero. Sinceν is a Borel measure, if necessary we can move the boundaries slightly sothat they have zero measure. Thus, we obtain a partition ξ with diam ξ ≤ α.Moreover, we can choose a partition ξ such that

hν(fm, ξ) > hν(f

m) − ε and diam ξ < tm/10.

This implies Statements 1 and 2. Statement 3 follows from (15.3).

We proceed with the proof of the theorem. We have

hν(fm, ξ) = lim

k→∞Hν(ξ|fmξ ∨ · · · ∨ fkmξ)

≤ Hν(ξ|fmξ) =∑

D∈fmξ

ν(D)H(ξ|D)

≤∑

D∈fmξ

ν(D) log cardC ∈ ξ : C ∩D 6= ∅,

(15.4)

where H(ξ|D) is the entropy of ξ with respect to the conditional measureon D induced by ν. Our goal is to obtain a uniform exponential estimate forthe number of those elements C ∈ ξ which have nonempty intersection witha given element D ∈ fmξ, and then an exponential bound for the numberof those sets D ∈ fmξ which contain regular points.

Lemma 15.3. There exists a constant K1 > 0 such that if D ∈ fmξ,then

cardC ∈ ξ : D ∩ C = ∅ ≤ K1 sup‖dxf‖mn : x ∈Mwhere n = dimM .

Proof. By the Mean Value Theorem,

diam(fmC) ≤ sup‖dxf‖m : x ∈MdiamC.

Thus, if C ∩ D 6= ∅ then C is contained in the 4r′-neighborhood of D.Therefore,

diamC ≤ (sup‖dxf‖m : x ∈M + 2)4r′

and hence, the volume of C is bounded from above by

Krn sup‖dxf‖mn : x ∈M,whereK > 0 is a constant. On the other hand, by Property 2 of the partitionξ, the set C contains a ball B(x, r′), and hence, the volume of C is at leastK ′(r′)n, where K ′ is a positive constant. This implies the desired result.

Fix ε > 0 and let Rm be the set of forward regular points x ∈M whichsatisfy the following condition: if k > m and v ∈ TxM , then

ek(χ(x,v)−ε)‖v‖ ≤ ‖dxfkv‖ ≤ ek(χ(x,v)+ε)‖v‖.


Lemma 15.4. If D ∈ fmξ has nonempty intersection with Rm then thereexists a constant K2 > 0 such that

cardC ∈ ξ : D ∩ C 6= ∅ ≤ K2eεm

∏

i:χi>0

em(χi+ε)ki .

Proof. Let C ′ ∈ ξ be such that C ′ ∩ Rm 6= ∅ and fmC ′ = D. Pick

a point x ∈ C ′ ∩ f−m(Rm) and let B = B(x, 2 diamC ′). The set B0 =

dxfm(exp−1

x B) ⊂ TfmxM is an ellipsoid and D ⊂ B0 = expfm(x)(B0). If aset C ∈ ξ has nonempty intersection with B0 then it lies in the set

B1 = y ∈M : d(y,B0) < diam ξ.Therefore,

cardC ∈ ξ : D ∩ C 6= ∅ ≤ vol(B1)(diam ξ)−n,

where vol(B1) denotes the volume of B1. Up to a bounded factor, vol(B1)

is bounded by the product of the lengths of the axes of the ellipsoid B0.Those of them that correspond to nonpositive exponents are at most subex-ponentially large. The remaining ones are of size at most em(χi+ε), up to abounded factor, for all sufficiently large m. Thus,

vol(B1) ≤ Kemε(diamB)n∏

i:χi>0

em(χi+ε)ki

≤ Kemε(2 diam ξ)n∏

i:χi>0

em(χi+ε)ki ,

for some constant K > 0. The lemma follows.

By Lemmas 15.3 and 15.4 and (15.4), we obtain

mhν(f) − ε = hν(fm) − ε ≤ hν(f

m, ξ)

≤∑

D∩Rm 6=∅

ν(D)

logK2 + εm+m

∑

i:χi>0

(χi + ε)ki

+∑

D∩Rm=∅

ν(D)(logK1 + nm log sup‖dxf‖ : x ∈M)

≤ logK2 + εm+m∑

i:χi>0

(χi + ε)ki

+ (logK1 + nm log sup‖dxf‖ : x ∈M)ν(M \Rm).

By the Multiplicative Ergodic Theorem 6.2, we have⋃

m≥0

Rm = M (mod 0)

for all sufficiently small ε. It follows that

hν(f) ≤ ε+∑

i:χi>0

(χi + ε)ki.


Letting ε→ 0 we obtain the desired upper bound.

An important consequence of the Margulis–Ruelle Inequality is that anyC1 diffeomorphism with positive topological entropy has an invariant mea-sure with at least one positive and one negative Lyapunov exponent. Inparticular, a surface diffeomorphism with positive topological entropy al-ways possesses a hyperbolic invariant measure.

Notice that for an arbitrary invariant measure the Margulis–Ruelle In-equality may be strict. The reader can examine a diffeomorphism with ahyperbolic fixed point and an atomic measure concentrated at this point.

We shall now prove the lower bound and hence, the entropy formula. Itwas established by Pesin in [28].

Theorem 15.5 (Entropy Formula). If f is of class C1+α and ν is ahyperbolic smooth invariant measure on M , then

hν(f) =

∫

M

∑

i:χi(x)>0

ki(x)χi(x) dν(x). (15.5)

Proof. We only need to show that

hν(f) ≥∫

M

∑

i:χi(x)>0

ki(x)χi(x) dν(x),

or equivalently (by replacing f by f−1 and using Theorem 6.2) that

hν(f) ≥ −∫

M

∑

i:χi(x)<0

ki(x)χi(x) dν(x). (15.6)

Consider the set

Γ = x ∈M : χ+(x, v) < 0 for some v ∈ TxM.If ν(Γ) = 0 the desired result follows immediately from (15.6). We therefore

assume that ν(Γ) > 0. Consider the subset Γ ⊂ Γ of regular points. For

each x ∈ Γ set

Tn(x) = Jac(dxfn|Es(x)) and g(x) =

∏

i:χi(x)<0

eki(x)χi(x).

Fix ε > 0 and for each n ∈ N consider the invariant sets

Γn = x ∈ Γ : (1 + ε)−n < g(x) ≤ (1 + ε)−n+1.We shall evaluate the entropy of the restriction f |Γn with respect to the

measure νn = ν|Γn.

Lemma 15.6. Given ε > 0, there exists a Borel positive function L(x)

such that for any x ∈ Γ and n ∈ N,

Tn(x) ≤ L(x)g(x)neεn.

Proof of the lemma. This is an immediate consequence of Theo-rems 6.1 and 5.2.


Fix n ∈ N. For each ℓ ∈ N we define the measurable sets

Γn,ℓ = x ∈ Γn ∩ Rℓ : L(x) ≤ ℓ,

and also

Γℓ =⋃

x∈Γn,ℓ

V s(x) and Γ =⋃

n∈Z

fn(Γℓ).

Note that Γ is f -invariant. Fix β > 0. By choosing ℓ sufficiently large, we

obtain νn(Γn,ℓ) > 1 − β. Since Γn,ℓ ⊂ Γℓ ⊂ Γ ⊂ Γn (mod 0) we also have

νn(Γ \ Γℓ) ≤ β and νn(Γn \ Γ) ≤ β. (15.7)

Consider now any finite measurable partition ξ of M such that each elementξ(x) of ξ is homeomorphic to a ball, has piecewise smooth boundary, andhas diameter at most rℓ (see Lemma 15.2).

We define a partition η of Γ composed of Γ \ Γℓ, and of the elements

V s(y) ∩ ξ(x) for each y ∈ ξ(x) ∩ Γn,ℓ.

Given x ∈ Γℓ and a sufficiently small r ≤ rℓ (see Section 11), we set

Bη(x, r) = y ∈ η(x) : ρ(y, x) < r.

Lemma 15.7. There exists qℓ > 0 and a set Aℓ ⊂ Γℓ with νn(Γℓ \Aℓ) ≤ βsuch that η−(x) ⊃ Bη(x, qℓ) for every x ∈ Aℓ.

Proof of the lemma. For each δ > 0 set

∂ξ =⋃

y∈M

∂(ξ(y)) and ∂ξδ = y : ρ(y, ∂ξ) ≤ δ.

One can easily show that there exists a constant C1 > 0 such that

νn(∂ξδ) ≤ C1δ. (15.8)

Let

Dq = x ∈ Γℓ : Bη(x, q) \ η−(x) 6= ∅.If x ∈ Dq then there exist m ∈ N and y ∈ Bη(x, q) such that y 6∈ (f−mη)(x).Hence, ∂ξ ∩ fm(Bη(x, q)) 6= ∅. Therefore, by Theorem 9.1 if x ∈ Dq thenfm(x) ∈ ∂ξ(C2λmeεmq) for some constant C2 = C2(ℓ) > 0. Thus, in view of(15.8),

νn(Dq) ≤ C1

∞∑

m=0

C2λmeεmq ≤ C3q

for some constant C3 = C3(ℓ) > 0. The lemma follows by setting qℓ = βC3−1

and Aℓ = Γℓ \Dqℓ.

For each x ∈ Rℓ let νx be the measure induced by the Riemannianvolume on V s(x) as a smooth submanifold in M (see Section 12).

The following statement is a crucial step in the proof of the lower boundand exploits the absolute continuity property of the measure ν in an essentialway.


Lemma 15.8. There exists a constant C5 = C5(ℓ) ≥ 1 such that if x ∈Aℓ ∩ V s(y) and y ∈ Γℓ,n then

C5−1 ≤ dν−x

dνy≤ C5.

Proof of the lemma. The statement follows immediately from The-orem 12.1 and Lemma 15.7.

We also need the following statement.

Lemma 15.9. There exists a constant C4 = C4(ℓ) > 0 such that for every

x ∈ Γℓ ∩ V s(y) with y ∈ Γℓ,n and for every m ∈ N we have

νfmy(fm(η(x))) ≤ C4Tm(y).

Proof of the lemma. We have

νfmy(fm(η(x))) =

∫

η(x)Tm(z) dνy(z).

A similar argument to that in the proof of Lemma 11.8 shows that thereexists a constant C ′ > 0 depending only on ℓ such that∣∣∣∣

Tm(z)

Tm(y)− 1

∣∣∣∣ ≤ C ′

for every z ∈ η(x) and all sufficiently large m ∈ N. The desired resultfollows.

We now complete the proof of the theorem. Write f = f |Γ. For everym ∈ N we have

hνn(f |Γn) ≥ hνn(f) =1

mhνn(fm) ≥ 1

mHνn(fmη|η−), (15.9)

where η− =∨∞

k=0 f−kη. To estimate the last expression we find a lower

bound for

Hνn(fmη|η−(x)) =

∫

η−(x)− log ν−x (η−(x) ∩ (fmη)(y)) dν−x (y),

where ν−x is the conditional measure on the element η−(x) of the parti-tion η−.

Note that if x ∈ Aℓ then

νx(Bη(x, qℓ)) ≥ C6qℓdim M

for some constant C6 = C6(ℓ) > 0. It follows from Lemmas 15.6, 15.9and 15.8 that for x ∈ Aℓ and n ∈ N

Hνn(fmη|η−(x)) ≥ −C5 log(C5νx(η−(x) ∩ (fmη)(x))

)νx(Bη(x, qℓ))

≥ −C5 log(C5C4Tm(x))νx(Bη(x, qℓ))

≥ −C5 log(C5C4L(x)g(x)meεm)C6qℓdim M

≥ −C5 log(C5C4ℓ(1 + ε)(−n+1)meεm)C6qℓdim M .


Choosing qℓ sufficiently small we may assume that C5C6qdim Mℓ < 1. There-

fore,

Hνn(fmη|η−(x)) ≥ −m(−n+ 1) log(1 + ε) −mε− C7

≥ −m log g(x) −m(log(1 + ε) + ε) − C7

for some constant C7 > 0 depending only on ℓ. Integrating this inequalityover the elements of η− we obtain

1

mHνn(fmη|η−) ≥

∫

Aℓ

(− log g(x) − (log(1 + ε) + ε) − C7

m

)dνn(x).

Given δ > 0, we can choose numbers m and ℓ sufficiently large and ε suffi-ciently small such that (in view of (15.7), (15.9), and Lemma 15.7)

hνn(f |Γn) ≥ − 1

ν(Γn)

∫

Γn

∑

i:χi(x)<0

ki(x)χi(x) dν(x) − δ.

Summing up all these inequalities over n (note that the sets Γn are disjointand invariant) we obtain

hν(f) ≥∞∑

n=1

ν(Γn)hνn(f |Γn)

≥∞∑

n=1

−

∫

Γn

∑

i:χi(x)<0

ki(x)χi(x) dν(x) − δν(Γn)

=

∫

M

∑

i:χi(x)<0

ki(x)χi(x) dν(x) − δ.

Since δ is arbitrary the desired result follows.

Remark 15.10. We note that the above proof also establishes the en-tropy formula (15.5) when ν is not a smooth measure, i.e., a measure whichis equivalent to the Riemannian volume on M but is just absolutely contin-uous.

In the case of Anosov diffeomorphisms, Sinai established in [37] a state-ment equivalent to the entropy formula.

We now describe a more general class of measures for which the entropyformula holds.

Consider the measurable partition ξs of the set Qℓ(x) (see (11.1)) intolocal stable manifolds for w ∈ Rℓ ∩ B(x, r). We denote by νs(w) the asso-ciated conditional measure on V s(w) generated by the partition ξs and themeasure ν (see Section 12). In a similar fashion the measurable partition ξu

into local unstable manifolds on Rℓ∩B(x, r) generates conditional measuresνu(w) on V u(w), w ∈ Rℓ ∩ B(x, r). The invariant measure ν is called anSRB-measure (after Sinai, Ruelle, and Bowen) if f has at least one nonzeroexponent ν-almost everywhere, and has absolutely continuous conditional


measures on stable manifolds (or unstable manifolds), that is, the measureνs(w) (or the measure νu(w)) is absolutely continuous with respect to theRiemannian volume on V s(w) (or on V u(w)) for ν-almost every point w. ByTheorem 12.1, a smooth measure ν (or more generally a measure ν abso-lutely continuous with respect to the Riemannian volume) invariant undera C1+α diffeomorphism is an SRB-measure.

Ledrappier and Strelcyn extended the entropy formula to SRB-measuresinvariant under a C1+α diffeomorphism [16].

The following result is due to Ledrappier and Young and provides acharacterization of SRB-measures in terms of the entropy formula.

Theorem 15.11 ([17]). For a Borel measure ν invariant under a C2

diffeomorphism, the entropy formula (15.5) holds if and only if ν is an SRB-measure.

16. Ergodic Properties of Geodesic Flows on Compact Surfaces

of Nonpositive Curvature

As we saw in Section 8 the geodesic flow gt on a compact surface M ofnonpositive curvature and of genus greater than 1 is nonuniformly hyperbolicon the set ∆ of positive Liouville measure which is defined by (8.4). Sincethe Liouville measure is invariant under the geodesic flow the results ofSection 13 apply and show that the ergodic components are of positiveLiouville measure (see Theorem 13.1). In this section we show that, indeed,the geodesic flow on ∆ is ergodic.

First, we state a remarkable result by Eberlein [8].

Proposition 16.1. The geodesic flow gt on M is topologically transitive.

We now wish to show that every ergodic component of gt|∆ is open(mod 0). To achieve this we construct C1 continuous laminations of SM ,W− and W+, such that W s(x) = W−(x) and W u(x) = W+(x) for almostevery x ∈ ∆ (W− and W+ are known as the stable and unstable horocy-cle foliations; see below). We then apply Theorems 14.5 and 14.7 (moreprecisely, their analog for flows; see Remark 14.8) to derive that the flowgt|∆ is ergodic. In order to proceed in this direction we need some basicinformation on manifolds of nonpositive curvature.

We denote by H the universal Riemannian cover of M , i.e., a simplyconnected complete Riemannian manifold for which M = H/Γ where Γ isa discrete subgroup of the group of isometries of H, isomorphic to π1(M).According to the Hadamard–Cartan theorem, any two points x, y ∈ H arejoined by a single geodesic which we denote by γxy. For any x ∈ H, theexponential map expx : Rp → H is a diffeomorphism. Hence, the map

ϕx(y) = expx

(y

1 − ‖y‖

)

is a homeomorphism of the open unit ball B onto H.


Two geodesics γ1(t) and γ2(t) in H are called asymptotic for t > 0 if

supt>0

ρ(γ1(t), γ2(t)) <∞,

where ρ is the distance in H induced by the Riemannian metric. Given apoint x ∈ H there is a unique geodesic starting at x which is asymptoticfor t > 0 to a given geodesic. The asymptoticity is an equivalence relation,and the equivalence class γ(∞) corresponding to a geodesic γ is called apoint at infinity. The set of these classes is denoted by H(∞) and is calledthe ideal boundary of H. One can extend the topology of the space H toH = H ∪H(∞) so that H becomes a compact metric space.

The map ϕx can be extended to a homeomorphism (still denoted by ϕx)of the closed ball B = B ∪ Sp−1 (where Sp−1 is the (p − 1)-dimensionalsphere in Rp) onto H by the equality

ϕx(y) = γy(+∞), y ∈ Sp−1.

In particular, ϕx maps Sp−1 homeomorphically onto H(∞).For any two distinct points x and y on the ideal boundary there is a geo-

desic which joins them. This geodesic is uniquely defined if the Riemannianmetric is of strictly negative curvature (i.e., if the inequality (8.1) is strict).Otherwise, there may exist a pair of distinct points x, y ∈ H(∞) which canbe joined by more than one geodesic. More precisely, there exists a geodesi-cally complete embedding into H of an infinite strip of zero curvature whichconsists of geodesics joining x and y. Moreover, any two geodesics on theuniversal cover which are asymptotic both for t > 0 and for t < 0 (i.e., theycan be joined to distinct points on the ideal boundary) bound a flat strip.This statement is known as the flat strip theorem (see [29, 6]).

The fundamental group π1(M) of the manifold M acts on the universalcover H by isometries. This action can be extended to the ideal boundaryH(∞). Namely, if p = γv(+∞) ∈ H(∞) and ζ ∈ π1(M), then ζ(p) is theequivalent class of geodesics which are asymptotic to the geodesic ζ(γv(t)).

We now describe the horocycle foliations for the geodesic flow.

Proposition 16.2 (see [8, 29]). The distributions E− and E+ are inte-grable. Their integral manifolds form C1 continuous foliations of SM whichare invariant under the flow gt.

We denote these foliations by W− and W+. The foliations W− and W+

can be lifted from the manifold SM to the manifold SH. We denote theselifts by W− and W+ respectively. The set L(x, p) = πW−(v) is called thelimit cycle (horocycle) centered at p = γv(+∞) ∈ H(∞) passing throughthe point x = πv.

The proof of Proposition 16.2 is based upon constructing the limit cycleL(x, p) (p = γv(+∞), x = πv) as a limit of circles S1(γv(t), t) as t → +∞(notice that these circles pass through x and their centers approach p).The convergence means that for any open ball B(x,R) in H centered atx of radius R the family of smooth curves B(x,R) ∩ S1(γv(t), t) converges


uniformly to the smooth curve B(x,R) ∩ L(x, p) as t → +∞ in the C1

topology (see [29] for details). More precisely, one can prove the followingresult.

Proposition 16.3. The following properties hold:

1. For any x ∈ H and p ∈ H(∞) there exists a unique limit cycleL(x, p) centered at p and passing through x; this limit cycle is alimit in the C1 topology of circles S1(γ(t), t) as t → +∞ where γ isthe unique geodesic joining x and p.

2. The leaf W+(v) is the framing of the limit circle L(x, p) (x = πv andp = γv(+∞)) by orthonormal vectors which have the same directionas the vector v (i.e., they are “inside” the limit sphere). The leafW−(v) is the framing of the limit circle L(x, p) (x = πv and p =γv(−∞) = γ−v(+∞)) by orthonormal vectors which have the samedirection as the vector v (i.e., they are “outside” the limit sphere).

3. For every ζ ∈ π1(M), we have

ζ(L(x, p)) = L(ζ(x), ζ(p)),

dvζW−(v) = W−(dvζv), dvζW

+(v) = W+(dvζv).

4. For every v, w ∈ SH, for which γv(+∞) = γw(+∞) = p, thegeodesic γw(t) intersects the limit circle L(π(v), p) at some point.

We now state our main result.

Theorem 16.4. Let M be a compact surface of nonpositive curvatureand of genus greater than 1. Then the following properties hold:

1. the set ∆ defined by (8.4) has positive Liouville measure, is open(mod 0), and is everywhere dense;

2. the geodesic flow gt|∆ is ergodic.

Proof. By Theorem 8.6, µ(∆) > 0 where µ is the Liouville measure.We shall show that the set ∆ is open (mod 0). Note that given v ∈ ∆ thereis a number t such that the curvature Kx of M at the point x = γv(t) isstrictly negative. Therefore there is a ball B(x, r) in M centered at x ofradius r such that Ky < 0 for every y ∈ B(x, r). It follows that there is aball D(v, q) in SM centered at v of radius q such that γw(t) ∈ B(x, r) forevery w ∈ D(v, q). Moreover, for almost every w ∈ D(v, q) (with respect tothe Liouville measure) there exists a sequence of numbers tn → ∞ such thatγw(tn) ∈ B(x, r). It follows that w ∈ ∆.

Note that since the geodesic flow is topologically transitive and the set ∆is open (mod 0) this set is everywhere dense. This proves the first statement.

For µ-almost every v ∈ ∆, consider one-dimensional local stable and

unstable manifolds V s(v) and V u(v). We denote by V s(v) and V u(v) their

lifts to SH (with v being a lift of v). Given w ∈ V s(v), we have

ρ(πgtv, πgtw) → 0 as t→ ∞. (16.1)


It follows that the geodesics γv(t) and γw(t) are asymptotic and hence,

γv(+∞) = γw(+∞). We wish to show that w ∈ W+(v). Assuming thecontrary consider the limit circle L(πv, γv(+∞)). Let z be the point ofintersection of the geodesic γw(t) and this limit circle (such a point existsby Proposition 16.3). We have

ρ(πgtv, πgtw) ≥ ρ(πw, z) > 0

which contradicts to (16.1). It follows that V s(v) ⊂ W−(v) for every v ∈ ∆

and every lift v of v. Arguing similarly, one can show that V u(v) ⊂ W−(v)for every v ∈ ∆ and every lift v of v. By Proposition 16.2 and Theorems 14.5and 14.7 (more precisely, by their analog for flows; see Remark 14.8), weconclude that gt|∆ is ergodic.

References

[1] D. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature,Proc. Steklov Inst. Math. 90 (1969), 1–235.

[2] D. Anosov and Ya. Sinai, Certain smooth ergodic systems, Russian Math. Surveys 22

(1967), no. 5, 103–167.[3] L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy

and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.[4] W. Ballmann, Lectures on spaces of nonpositive curvature, with an appendix by

M. Brin, DMV Seminar 25, Birkhauser Verlag, Basel, 1995.[5] A. Blohin, Smooth ergodic flows on surfaces, Trans. Moscow Math. Soc. 27 (1972),

117–134.[6] K. Burns and A. Katok, Manifolds with nonpositive curvature, Ergodic Theory Dy-

nam. Systems 5 (1985), no. 2, 307–317.[7] D. Bylov, R. Vinograd, D. Grobman and V. Nemyckii, Theory of Lyapunov exponents

and its application to problems of stability, Izdat. “Nauka”, Moscow, 1966, in Russian.[8] P. Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95

(1972), 492–510.[9] P. Eberlein, When is a geodesic flow of Anosov type? I, J. Differential Geom. 8

(1973), 437–463; II, J. Differential Geom. 8 (1973), 565–577.[10] A. Fathi, M. Herman and J.-C. Yoccoz, A proof of Pesin’s stable manifold theorem,

Geometric Dynamics (Rio de Janeiro, 1981), Jacob Palis, ed., Lecture Notes in Math.1007, Springer Verlag, 1983, pp. 177–215.

[11] H. Federer, Geometric measure theory, Springer Verlag, 1969.[12] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math. 583,

1977.[13] A. Katok and K. Burns, Infinitesimal Lyapunov functions, invariant cone families and

stochastic properties of smooth dynamical systems, Ergodic Theory Dynam. Systems14 (1994), 757–785.

[14] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,Cambridge University Press, 1995.

[15] A. Katok and J.-M. Strelcyn, Invariant manifolds, entropy and billiards; smooth maps

with singularities, with the collaboration of Francois Ledrappier and Feliks Przytycki,Lecture Notes in Math. 1222, Springer Verlag, 1986.

[16] F. Ledrappier and J.-M. Strelcyn, A proof of the estimate from below in Pesin’s

entropy formula, Ergodic Theory Dynam. Systems 2 (1982), 203–219.


[17] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Character-

ization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985),509–539.

[18] P.-D. Liu and M. Qian, Smooth ergodic theory of random dynamical systems, LectureNotes in Math. 1606, Springer Verlag, 1995.

[19] A. Lyapunov, The general problem of the stability of motion, Taylor & Francis, 1992.[20] I. Malkin, A theorem on stability via the first approximation, Dokladi, Akademii Nauk

USSR 76 (1951), no. 6, 783–784.[21] R. Mane, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und

ihrer Grenzgebiete 3. Folge·Band 8, Springer Verlag, 1987.[22] R. Mane, Lyapunov exponents and stable manifolds for compact transformations, Geo-

metric Dynamics (Rio de Janeiro, 1981), Jacob Palis, ed., Lecture Notes in Math.1007, Springer Verlag, 1983, pp. 522–577.

[23] J. Milnor, Fubini foiled: Katok’s paradoxical example in measure theory, Math. Intel-ligencer 19 (1997), 30–32.

[24] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for

dynamical systems Trans. Moscow Math. Soc. 19 (1968), 197–221.[25] O. Perron, Die Ordnungszahlen linearer Differentialgleichungssyteme, Math. Zs. 31

(1930), 748–766.[26] Ya. Pesin, An example of a nonergodic flow with nonzero characteristic exponents,

Func. Anal. and its Appl. 8 (1974), no. 3, 263–264.[27] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic

exponents, Math. USSR-Izv.40 (1976), no. 6, 1261–1305.[28] Ya. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Russian

Math. Surveys 32 (1977), no. 4, 55–114.[29] Ya. Pesin, Geodesic flows on closed Riemannian manifolds without focal points, Math.

USSR-Izv. 11 (1977), no. 6, 1195–1228.

[30] C. Pugh, The C1+α hypothesis in Pesin theory, Inst. Hautes Etudes Sci. Publ. Math.

59 (1984), 143–161.[31] C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972), 1–23.[32] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), no. 1,

1–54.[33] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat.

9 (1978), no. 1, 83–87.

[34] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes EtudesSci. Publ. Math. 50 (1979), 27–58.

[35] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. ofMath. (2) 115 (1982), no. 2, 243–290.

[36] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,Invent. Math. 139 (2000), 495–508.

[37] Ya. Sinai, Dynamical systems with countably-multiple Lebesgue spectrum II, Amer.Math. Soc. Trans. (2) 68 (1966), 34–88.

Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lis-

boa, Portugal

E-mail address: [email protected]

URL: http://www.math.ist.utl.pt/~barreira/

Department of Mathematics, The Pennsylvania State University, Univer-

sity Park, PA 16802, U.S.A.


URL: http://www.math.psu.edu/pesin/

99

Appendix A. Holder Continuity of Invariant Distributions

M. Brin

Although the stable and unstable subspaces of a nonuniformly hyperbolicdynamical system f depend continuously on the point inside an appropriateset (see Section 7 above), in general, they are not differentiable functionsof the point even if f is real analytic. This happens because the stableand unstable subspaces depend on the infinite future and past, respectively.We prove here that the stable and unstable subspaces depend Holder con-tinuously on the point. We consider only the stable subspaces, the Holdercontinuity of the unstable subspaces follows by reversing the time.

For the uniformly hyperbolic case, the Holder continuity was establishedby Anosov [A]. The main idea of the argument below is the same as in [BK].

A k-dimensional distribution E on a subset Λ of a differentiable manifoldM is a family of k-dimensional subspaces E(x) ⊂ TxM , x ∈ Λ. A Riemann-ian metric on M naturally induces distances in TM and in the space ofk-dimensional subspaces in TM . The Holder continuity of a distribution Ecan be defined using these distances. However, by the Whitney EmbeddingTheorem [H], every manifold M can be embedded in RN with a sufficientlylarge N . If M is compact, the Riemannian metric on M is equivalent to thedistance ‖x− y‖ induced by the embedding. The Holder exponent does notchange if the Riemannian metric is changed for an equivalent smooth met-ric, the Holder constant does change. Therefore we assume in Theorem A.3,without loss of generality, that the manifold is embedded in RN .

For a subspace A ⊂ RN and a vector v ∈ RN , set

dist(v,A) = minw∈A

‖v − w‖.

i.e., dist(v,A) is the length of the difference between v and its orthogonalprojection to A. For subspaces A, B in RN , define

dist(A,B) = max

max

v∈A,‖v‖=1dist(v,B), max

w∈B,‖w‖=1dist(w,A)

.

A k-dimensional distribution E defined on a subset Λ ⊂ RN is Holdercontinuous with Holder exponent α ∈ (0, 1] and Holder constant L > 0 ifthere is ε0 > 0 such that

dist(E(x), E(y)) ≤ L · ‖x− y‖α

for all x, y ∈ Λ with ‖x− y‖ ≤ ε0.The following two lemmas can be used to prove the Holder continuity of

invariant distributions for a variety of dynamical systems.

Lemma A.1. Let Ak and Bk, k = 0, 1, . . ., be two sequences of realN ×N matrices such that for some ∆ ∈ (0, 1) and a > 1,

‖Ak −Bk‖ ≤ ∆ak for k = 0, 1, 2, . . ..

100 M. BRIN

Suppose that there are subspaces EA, EB ⊂ RN and positive λ < µ andC ′ > 1 so that λ < a and for each positive integer k,

‖Akv‖ ≤ C ′λk‖v‖ if v ∈ EA; ‖Akw‖ ≥ C ′−1µk‖w‖ if w ⊥ EA;

‖Bkv‖ ≤ C ′λk‖v‖ if v ∈ EB; ‖Bkw‖ ≥ C ′−1µk‖w‖ if w ⊥ EB.

Then dist(EA, EB) ≤ 3C ′2 µλ∆

log µ−log λlog a−log λ .

Proof. Set

QkA = v ∈ RN : ‖Akv‖ ≤ 2C ′λk‖v‖

and

QkB = v ∈ RN : ‖Bkv‖ ≤ 2C ′λk‖v‖.

For v ∈ RN , write v = vλ + v⊥, where vλ ∈ EA and v⊥ ⊥ EA. If v ∈ QkA,

then

‖Akv‖ = ‖Ak(vλ + v⊥)‖ ≥ ‖Akv

⊥‖ − ‖Akvλ‖ ≥ C ′−1

µk‖v⊥‖ − C ′λk‖vλ‖,

and hence ‖v⊥‖ ≤ C ′µ−k(‖Akv‖ + C ′λk‖vλ‖) ≤ 3C ′2(λ/µ)k‖v‖. Therefore

dist(v,EA) ≤ 3C ′2(λ

µ

)k

‖v‖. (A.1)

Set γ = λ/a < 1. There is a unique nonnegative integer k such that γk+1 <∆ ≤ γk. Let w ∈ EB. Then

‖Akw‖ ≤ ‖Bkw‖ + ‖Ak −Bk‖ · ‖w‖ ≤ C ′λk‖w‖ + ∆ak‖w‖≤ (C ′λk + (γa)k)‖w‖ ≤ 2C ′λk‖w‖.

It follows that w ∈ QkA and hence EB ⊂ Qk

A. By symmetry, EA ⊂ QkB. By

(A.1) and by the choice of k,

dist(EA, EB) ≤ 3C ′2(λ

µ

)k

≤ 3C ′2µ

λ∆

log µ−log λlog a−log λ

Lemma A.2. Let f : M → M be a C1+β map of a compact, m-dimen-

sional, C2 submanifold M ⊂ RN . Then for every a > (maxz∈M ‖dzf‖)1+β

there is C1 > 1 such that for each n ∈ N and all x, y ∈M

‖dxfn − dyf

n‖ ≤ C1an · ‖x− y‖β .

Proof. Let C2 be such that ‖dxf − dyf‖ ≤ C2‖x − y‖β and set a1 =maxz∈M ‖dzf‖ ≥ 1. Observe that ‖fn(x) − fn(y)‖ ≤ an

1‖x − y‖ for all x,y ∈M . Fix a > a1. Then the lemma holds true for n = 1 and any C1 ≥ C2.

HOLDER CONTINUITY OF INVARIANT DISTRIBUTIONS 101

For the inductive step we have

‖dxfn+1 − dyf

n+1‖ ≤‖dfn(x)f‖ · ‖dxfn − dyf

n‖+ ‖dfn(x)f − dfn(y)f‖ · ‖dyf

n‖≤a1C1a

n‖x− y‖β + C2

(an

1‖x− y‖)βan

1

≤C1an+1‖x− y‖β

(a1

a+C2

C1

(a1+β1 )n

an+1

).

If a > a1+β1 , then there is C1 ≥ C2 for which the last factor in parentheses

is less than 1.

The subspaces E1, E2 ⊂ RN are θ-transverse if ‖v1−v2‖ ≥ θ for all unitvectors v1 ∈ E1 and v2 ∈ E2.

Theorem A.3. Let M be a compact, m-dimensional, C2 submanifold ofRN , m < N , and f : M → M a C1+β map, β ∈ (0, 1). Suppose that thereexist a subset Λ ⊂ M and real numbers 0 < λ < µ, C > 0, θ > 0 such thatfor each x ∈ Λ there are θ-transverse subspaces Es(x), Eu(x) ⊂ TxM withthe following properties: TxM = Es(x) ⊕ Eu(x), ‖dxf

kvs‖ ≤ Cλk‖vs‖ and‖dxf

kvu‖ ≥ C−1µk‖vu‖ for all vs ∈ Es(x), vu ∈ Eu(x), and all positiveintegers k.

Then for every a > (maxz∈M ‖dzf‖)1+β, the distribution Es is Holdercontinuous with exponent α = log µ−log λ

log a−log λβ.

Proof. For x ∈M , let E⊥(x) denote the orthogonal complement to thetangent plane TxM in RN . Since E⊥ is a smooth distribution, it is sufficient

to prove the Holder continuity of Es = Es ⊕ E⊥.Since Es(x) and Eu(x) are θ-transverse and of complementary dimen-

sions in TxM , there is C > 1 such that ‖dxfkw‖ ≥ C−1µk‖w‖ for all x ∈ Λ

and w ⊥ Es(x).For x, y ∈ Λ and a positive integer k, let Ak and Bk be N ×N matrices

such thatAkv = dxfkv if v ∈ TxM , Akw = 0 if w ⊥ TxM andBkv = dyf

kv if

v ∈ TyM , Bkw = 0 if w ⊥ TyM . By Lemma A.2, ‖Ak−Bk‖ ≤ C1ak‖x−y‖β .

Now the theorem follows from Lemma A.1 with ∆ = C1‖x − y‖β , EA =

Es(x), EB = Es(y), and C ′ = max(C, C).

References

[A] D. Anosov, Tangential fields of transversal foliations in U-systems, Math. Notes 2

(1967), no. 5, 818–823.[BK] M. Brin and Yu. Kifer, Dynamics of Markov chains and stable manifolds for random

diffeomorphisms, Ergodic Theory Dynam. Systems 7 (1987), 351–374.[H] M. Hirsch, Differential topology, Springer Verlag, 1994.

Department of Mathematics, University of Maryland, College Park, MD

20742-4015, U.S.A.


URL: http://www.math.umd.edu/~mib/

102

Appendix B. An Example of a Smooth Hyperbolic Measure with

Countably Many Ergodic Components

D. Dolgopyat, H. Hu and Ya. Pesin1

B.1. Introduction. We construct an example of a diffeomorphismwith nonzero Lyapunov exponents with respect to a smooth invariant mea-sure which has countably many ergodic components. More precisely we willprove the following result.

Theorem B.1. There exists a C∞ diffeomorphism f of the three dimen-sional torus T3 such that

1. f preserves the Riemannian volume µ on T3;2. µ is a hyperbolic measure;3. f has countably many ergodic components which are open (mod 0).

B.2. Construction of the Diffeomorphism f . Let A : T2 → T2 bea linear hyperbolic automorphism. Passing if necessary to a power of A wemay assume that A has at least two fixed points p and p′. Consider the mapF = A× Id of the three dimensional torus T3 = T2 × S1. We will perturb Fto obtain the desired map f .

Consider a countable collection of intervals In∞n=1 on the circle S1,where

I2n = [(n+ 2)−1, (n+ 1)−1], I2n−1 = [1 − (n+ 1)−1, 1 − (n+ 2)−1].

Clearly,⋃∞

n=1 In = (0, 1) and int In are pairwise disjoint.By Proposition B.2 below, for each n one can construct a C∞ volume

preserving ergodic diffeomorphism fn : T2×[0, 1] → T2×[0, 1] which satisfies:

1. ‖F − fn‖Cn ≤ e−n2;

2. for all 0 ≤ m <∞, Dmfn|T2 × z = DmF |T2 × z for z = 0 or 1;3. fn has nonzero Lyapunov exponents µ-almost everywhere.

Let Ln : In → [0, 1] be the affine map and πn = (Id, Ln) : T2 × In →T2 × [0, 1]. We define the map f by setting f |T2 × In = π−1

n fn πn for alln and f |T2 × 0 = F |T2 × 0. Note that for every n > 0 and 0 ≤ m ≤ n

1Key words and phrases. Hyperbolic measure, Lyapunov exponents, stable ergodicity,accessibility.

D. Dolgopyat was partially supported by the Sloan Foundation. H. Hu was partiallysupported by the National Science Foundation grant #DMS-9970646. Ya. Pesin was par-tially supported by the National Science Foundation grant #DMS-9704564 and by theNATO grant CRG970161. D. Dolgopyat and Ya. Pesin wish to thank the Isaac NewtonInstitute for Mathematical Sciences (Cambridge, UK) for hospitality and support dur-ing their stay there in June-July 2000. H. Hu wishes to thank the Centre de PhysiqueTheorique (Marseille-Luminy, France) for hospitality and support during his stay there inMay-July 2000.

AN EXAMPLE OF A SMOOTH HYPERBOLIC MEASURE 103

we have

‖DmF |T2 × In − π−1n Dmfn πn‖Cn ≤ ‖π−1

n (DmF −Dmfn) πn‖Cn

≤ e−n2 · (n+ 1)n → 0

as n→ ∞. It follows that f is C∞ on M and has the required properties.

B.3. Main Proposition. The goal of this section is to prove the fol-lowing statement. Set I = [0, 1].

Proposition B.2. For any k ≥ 2 and δ > 0, there exists a map g ofthe three dimensional manifold M = T2 × I such that:

1. g is a C∞ volume preserving diffeomorphism of M ;2. ‖F − g‖Ck ≤ δ;3. for all 0 ≤ m < ∞, Dmg|T2 × z = DmF |T2 × z for z = 0

and 1;4. g is ergodic with respect to the Riemannian volume and has nonzero

Lyapunov exponents almost everywhere.

Before giving the formal proof let us outline the main idea. The re-sult will be achieved in two steps. First applying an argument of [SW] weconstruct a perturbation map which has nonzero average central exponent∫M χc(x) dµ(x) 6= 0, where χc(x) denotes the Lyapunov exponent of x along

the neutral subspace Ec(x). We then further perturb this diffeomorphismmodifying an approach in [NT] to ensure that it has the accessibility prop-erty and therefore, is ergodic (see Section B.4 for details).

We believe that this approach works in a more general setting. Namely,we conjecture that the following statement holds.

Conjecture. Consider a one parameter family gε with g0 = F. Thenfor sufficiently small ε, gε satisfies the conditions of Proposition B.2 ex-cept for a positive codimension submanifold in the space of one parameterfamilies.

Proof of Proposition B.2. Consider the linear hyperbolic map Aof the torus T2. We may assume that its eigenvalues are η and η−1, whereη > 1. Let p and p′ be fixed points of A. Choose a number ε0 > 0 suchthat d(p, p′) ≥ 3ε0. Consider the local stable and unstable one-dimensionalmanifolds for A at points p and p′ of “size” ε0 and denote them respectivelyby V s(p), V u(p), V s(p′), and V u(p′).

Let us choose the smallest positive number n1 such that the intersectionA−n1(V s(p′)) ∩ V u(p) ∩ B(p, ε0) consists of a single point which we denoteby q1 (here B(p, ε0) is the ball in T2 of radius ε0 centered at p). Simi-larly, we choose the smallest positive number n2 such that the intersectionAn2(V u(p′)) ∩ V s(p) ∩ B(p, ε0) consists of a single point which we denoteby q2.

Given a sufficiently small number ε ∈ (0, ε0),

ε ≤ 1

2mind(p, q1), d(p, q2),

104 D. DOLGOPYAT, H. HU AND YA. PESIN

p

B(p, ε)

B(p, ε0)A−ℓ−1(q1)

p′

q1

q2

V u(p)

V s(p)

V u(p′)

V s(p′)

Figure B.1

there is ℓ ≥ 2 such that (see Figure B.1)

A−ℓ(q1) 6∈ B(p, ε), A−ℓ−1(q1) ∈ B(p, ε). (B.1)

We now choose ε′ ∈ (0, ε) such that A−ℓ−1(q1) ∈ B(p, ε′).Finally, we assume ε to be so small that for some q ∈ T2 we have

B(p, ε) ∩ (A−n1(V s(p′)) ∪An2(V u(p′))) = ∅,

Ai(B(q, ε)) ∩B(q, ε) = ∅, Ai(B(q, ε)) ∩B(p, ε) = ∅for i = 1, . . ., N , where N > 0 will be determined later, and ε = ε(N).

Set Ω1 = B(p, ε0)×I and Ω2 = Buc(q, ε0)×Bs(q, ε0), where q = (q, 1/2)and Buc(q, ε0) ⊂ V u(q) × I and Bs(q, ε0) ⊂ V s(q) are balls of radius ε0about q.

After this preliminary consideration we describe the construction of themap g.

Consider the coordinate system in Ω1 originated at (p, 0) ∈ M with x,y, and z-axes to be unstable, stable, and neutral directions respectively forthe map F . If a point w = (x, y, z) ∈ Ω1 and F (w) ∈ Ω1 then F (w) =(ηx, η−1y, z).

Choose a C∞ function ξ : I → R+ satisfying:

1. ξ(z) > 0 on (0, 1);

2. ξ(i)(0) = ξ(i)(1) = 0 for i = 0, 1, . . .,k;


3. ‖ξ‖Ck ≤ δ.

We also choose two C∞ functions ϕ = ϕ(x) and ψ = ψ(y) which are definedon the interval (−ε0, ε0) and satisfy

4. ϕ(x) = ϕ0 if x ∈ (−ε′, ε′) and ψ(y) = ψ0 if y ∈ (−ε′, ε′), where ϕ0

and ψ0 are positive constants;5. ϕ(x) = 0 if |x| ≥ ε; ψ(y) ≥ 0 for any y and ψ(y) = 0 if |y| ≥ ε;6. ‖ϕ‖Ck ≤ δ, ‖ψ‖Ck ≤ δ;

7.∫ ±ε0 ϕ(s) ds = 0.

We now define the vector field X on Ω1 by

X(x, y, z) =

(−ψ(y)ξ′(z)

∫ x

0ϕ(s)ds, 0, ψ(y)ξ(z)ϕ(x)

).

It is easy to check that X is a divergence free vector field supported on(−ε, ε) × (−ε, ε) × I.

We define the map ht on Ω1 to be the time t map of the flow generatedby X and we set ht = Id on the complement of Ω1. It is easy to see thatht is a C∞ volume preserving diffeomorphism of M which preserves the ycoordinate (the stable direction for the map F ).

Consider now the coordinate system in Ω2 originated at (q, 1/2) with x,y, and z-axes to be unstable, stable, and neutral directions respectively. Wethen switch to the cylindrical coordinate system (r, θ, y), where x = r cos θ,y = y, and z = r sin θ.

Consider a C∞ function ρ : (−ε0, ε0) → R+ satisfying:

8. ρ(r) > 0 if 0.2ε′ ≤ r ≤ 0.9ε and ρ(r) = 0 if r ≤ 0.1ε′ or r ≥ ε;9. ‖ρ‖Ck ≤ δ.

We define now the map hτ on Ω2 by

hτ (r, θ, y) = (r, θ + τψ(y)ρ(r), y). (B.2)

and we set hτ = Id on M \Ω2. It is easy to see that for every τ the map hτ

is a C∞ volume preserving diffeomorphism of M .

Let us set g = gtτ = ht F hτ . For all sufficiently small t > 0 and τ , themap gtτ is Ck close to F and hence, is a partially hyperbolic (in the narrowsense) C∞ diffeomorphism of M . It preserves the Riemannian volume in Mand is ergodic by Proposition B.3. It remains to show that gtτ has nonzeroLyapunov exponents almost everywhere.

Denote by Estτ (w), Eu

tτ (w), and Ectτ (w) the stable, unstable, and neutral

subspaces at a point w ∈ M for the map gtτ . It suffices to show that foralmost everywhere point w ∈M and every vector v ∈ Ec

τ (w), the Lyapunovexponent χ(w, v) 6= 0.

Set κtτ (w) = Dgtτ |Eutτ (w), w ∈ M . By Proposition B.6, for all suffi-

ciently small τ > 0, ∫

Mlog κ0τ (w) dw < log η.


The subspace Eutτ (w) depends continuously on t and τ (for a fixed w; for

details see the paper by Burns, Pugh, Shub, and Wilkinson in this volume)and hence, so does κtτ . It follows that for all sufficiently small τ > 0, thereis t > 0 such that ∫

Mlog κtτ (w) dw < log η.

Denote by χstτ (w), χu

tτ (w), and χctτ (w) the Lyapunov exponents of gtτ at

the point w ∈M in the stable, unstable, and neutral directions respectively(since these directions are one-dimensional the Lyapunov exponents do notdepend on the vector). By the ergodicity of gtτ , we have that for almostevery w ∈M ,

χutτ (w) = lim

n→∞

1

nlog

n−1∏

i=0

κtτ (gitτ (w)).

By the Birkhoff ergodic theorem, we get

χutτ (w) =

∫

Mlog κtτ (w) dw < log η.

Since Estτ (w) = Es

00(w) = EsF (w) for every t and τ , we conclude that

χstτ (w) = − log η for almost every w ∈ M . Since gtτ is volume preserv-

ing,

χstτ (w) + χu

tτ (w) + χctτ (w) = 0

for almost every w ∈M . It follows that χctτ (w) 6= 0 for almost every w ∈M

and hence, gtτ has nonzero Lyapunov exponents almost everywhere. Thiscompletes the proof of the proposition.

B.4. Ergodicity of the Map gtτ .

Proposition B.3. For every sufficiently small t > 0 and τ ≥ 0 the mapgtτ is ergodic.

Proof. Consider a partially hyperbolic (in the narrow sense) diffeomor-phism f of a compact Riemannian manifold M preserving the Riemannianvolume. Two points x, y ∈ M are called accessible (with respect to f) ifthey can be joined by a piecewise differentiable piecewise nonsingular pathwhich consists of segments tangent to either Eu or Es. The diffeomorphismf satisfies the essential accessibility property if almost any two points in M(with respect to the Riemannian volume) are accessible. We will show thatthe map gtτ has the essential accessibility property. The ergodicity of themap will then follow from the result by Pugh and Shub (see [PS]; see alsothe paper by Burns, Pugh, Shub, and Wilkinson in this volume).

Given a point w ∈M , denote by A(w) the set of points q ∈M such thatw and q are accessible. Set Ip = p × (0, 1).

Lemma B.4. For every z ∈ (0, 1),

A(p, z) ⊃ Ip. (B.3)


Proof of Lemma B.4. We use the coordinate system (x, y, z) in Ω1

described above. Since the map ht preserves the center leaf Ip, we have that

ht(0, 0, z) = (h(1)t (0, 0, z), h

(2)t (0, 0, z), h

(3)t (0, 0, z)) = (0, 0, h

(3)t (0, 0, z))

for z ∈ (0, 1). It suffices to show that for every z ∈ (0, 1),

A(p, z) ⊃ (p, a) : a ∈ [(h−ℓt )(3)(p, z), z], (B.4)

where ℓ is chosen by (B.1). In fact, since accessibility is a transitive relationand h−n

t (p, z) → (p, 0) for any z ∈ (0, 1), (B.4) implies that A(p, z) ⊃(p, a) : a ∈ (0, z]. Since this holds true for all z ∈ (0, 1) and accessibilityis a reflexive relation, we obtain (B.3).

Now we proceed with the proof of (B.4).Let q1 ∈ V u

tτ (p) and q2 ∈ V stτ (p) be two points constructed in Section B.3.

The intersection V stτ (q1)∩V u

tτ (q2) is not empty and consists of a single pointq3. We will prove that for any z0 ∈ (0, 1), there exist zi ∈ (0, 1), i = 1, 2, 3, 4such that

(q1, z1) ∈ V utτ ((p, z0)), (q3, z3) ∈ V s

tτ ((q1, z1)),

(q2, z2) ∈ V utτ ((q3, z3)), (p, z4) ∈ V s

tτ ((q2, z2))

andz4 ≤ (h−ℓ

t )(3)(p, z0). (B.5)

See Figure B.2. This means that (p, z4) ∈ A(p, z0). By continuity, weconclude that

(p, a) : a ∈ [z4, z0] ⊂ A(p, z0)

and (B.4) follows.Since gtτ preserves the xz-plane, we have that V uc

tτ ((p, z0)) = V ucF ((p, z0)).

Hence, there is a unique z1 ∈ (0, 1) such that (q1, z1) ∈ V utτ ((p, z0)). Notice

thatg−ntτ (p, z0) = (p, h−n

t ((p, z0)), g−ntτ (q1, z1) = (A−nq1, z1)

for n ≤ ℓ. This is true because the points A−nq1, n = 0, 1, . . ., ℓ lie outsidethe ε-neighborhood of Ip, where the perturbation map ht = Id. Similarly,since the points A−nq1, n > ℓ lie inside the ε′-neighborhood of Ip, and thethird component of ht depends only on the z-coordinate, we have

g−ntτ (q1, z1) = (A−nq1, h

−n+ℓt z1).

Since d(g−ntτ ((p, z0)), g

−ntτ ((q1, z1))) → 0 as n→ ∞, we have

d(h−nt ((p, z0)), h

−n+ℓt ((p, z1))) → 0

as n→ ∞. It follows that z1 = (h−ℓt )(3)((p, z0)).

By the construction of the map ht (that is ht = Id outside Ω1) the setsA−n1V s

tτ (p′) and An2V u

tτ (p′) are pieces of horizontal lines. This means that

z2 = z3 = z1.Since the third component of ht is nondecreasing from (q2, z2) to (p, z4)

along V stτ (p), we conclude that z4 ≤ z3 = z1 = (h−ℓ

t )3(p, z0) and thus (B.5)holds.


Ip

(p, z0)

(q1, z1)

(q3, z3)

(q2, z2)(p, z4)

(p, 0)

Figure B.2. Bold lines are stable manifolds, dotted — un-stable ones

The essential accessibility property follows from Lemma B.4 and thefollowing statement.

Lemma B.5 (see [NT]). Assume that any two points in Ip are accessible.Then the map gtτ satisfies the essential accessibility property.


Proof of Lemma B.5. It is easy to see that for any two points x, y ∈M which do not lie on the boundary of M one can find points x′, y′ ∈ Ipsuch that the pairs (x, x′) and (y, y′) are accessible. By Lemma B.4 thepoints x′, y′ are accessible. Since accessibility is a transitive relation theresult follows.

This completes the proof of the proposition.

B.5. Hyperbolicity of the Map g0τ . In this section we show that forall sufficiently small τ , the map g0τ has nonzero average Lyapunov exponentin the central direction. Since this map is ergodic this implies that g0τ hasnonzero Lyapunov exponents almost everywhere.

Proposition B.6. For any sufficiently small τ > 0,∫

Mlog κ0τ (w)dw < log η.

Proof. Our approach is an elaboration of an argument in [SW].For any w ∈M , we introduce the coordinate system in TwM associated

with the splitting EuF (w)⊕Es

F (w)⊕EcF (w). Given τ ≥ 0 and w ∈M , there

exists a unique number ατ (w) such that the vector vτ (w) = (1, 0, ατ (w))t

lies in Eu0τ (w) (where t denotes the transpose). Since the map hτ preserves

the y coordinate, by the definition of the function ατ (w), one can write thevector Dg0τ (w)vτ (w) in the form

Dg0τ (w)vτ (w) = (κτ (w), 0, κτ (w)ατ (gt0(w)))t (B.6)

for some κτ (w) > 1. Since the expanding rate of Dg0τ (w) along its unstabledirection is κ0τ (w) we obtain that

κ0τ (w) = κτ (w)

√1 + ατ (g0τ (w))2√

1 + ατ (w)2.

Since Eu0τ (w) is close to Eu

00(w) the function ατ (w) is uniformly bounded.Using the fact that the map g0τ preserves the Riemannian volume we findthat

Lτ =

∫

Mlog κ0τ (w) dw =

∫

Mlog κτ (w) dw. (B.7)

Consider the map hτ . Since it preserves the y-coordinate using (B.2), wecan write that

hτ (x, y, z) = (r cosσ, y, r sinσ),

where σ = σ(τ, r, θ, y) = θ + τψ(y)ρ(r). Therefore, the differential

Dhτ : EuF (w) ⊕ Ec

F (w) → EuF (g0τ (w)) ⊕ Ec

F (g0τ (w))


can be written in the matrix form

Dhτ (w) =

(A(τ, w) B(τ, w)C(τ, w) D(τ, w)

)

=

(rx cosσ − rσx sinσ ry cosσ − rσy sinσrx sinσ + rσx cosσ ry sinσ + rσy cosσ

),

where

rx =∂r

∂x=x

r= cos θ, rz =

∂r

∂z=y

r= sin θ,

σx =∂σ

∂x=

−zr2

+z

rτ ρr(y, r) =

sin θ

r+ τ ρr(y, r) cos θ,

σz =∂σ

∂z=

x

r2+x

rτ ρr(y, r) =

cos θ

r+ τ ρr(y, r) sin θ,

and ρ(y, r) = ψ(y)ρ(r). It is easy to check that

A = A(τ, w) = 1 − τrρr sin θ cos θ − τ2ρ2

2− τ2rρρr cos2 θ +O(τ3),

B = B(τ, w) = −τ ρ− τrρr sin2 θ − τ2rρρr sin θ cos θ +O(τ3),

C = C(τ, w) = τ ρ+ τrρr cos2 θ − τ2rρρr sin θ cos θ +O(τ3),

D = D(τ, w) = 1 + τrρr sin θ cos θ − τ2ρ2

2− τ2rρρr sin2 θ +O(τ3).

(B.8)

By Lemma B.7 below, we have

Lτ = log η −∫

Mlog(D(τ, w) − ηB(τ, w)ατ (g0τ (w)))dw.

By Lemma B.8, we have

dLτ

dτ

∣∣∣τ=0

= 0,d2Lτ

dτ2

∣∣∣τ=0

< 0.

So we can choose τ so small that Lτ 6= log η.

Lemma B.7.

Lτ = log η −∫

Mlog(D(τ, w) − ηB(τ, w)ατ (g0τ (w)))dw.

Proof of Lemma B.7. Since g0τ = h0 F hτ = F hτ , we have that

Dτ (w) = Dg0τ (w)|Eu0τ (w) ⊕ Ec

0τ (w) =

(ηA(τ, w) ηB(τ, w)C(τ, w) D(τ, w)

).

By (B.6),

Dτ (w)

(1

ατ (w)

)=

(ηA(τ, w) + ηB(τ, w)ατ (w)C(τ, w) +D(τ, w)ατ (w)

)

=

(κτ (w)

κτ (w)ατ (g0τ (w))

).

(B.9)


Since hτ is volume preserving, AD −BC = 1 and therefore,

A+Bα =1

D+B

D(C +Dα).

Comparing the components in (B.9), we obtain

κτ (w) =η(A(τ, w) +B(τ, w)ατ (w))

=η

(1

D(τ, w)+B(τ, w)

D(τ, w)(C(τ, w) +D(τ, w)ατ (w))

)

=η

(1

D(τ, w)+B(τ, w)

D(τ, w)(κτ (w)ατ (g0τ (w)))

).

Solving for κτ (w), we get

κτ (w) =η

D(τ, w) − ηB(τ, w)ατ (g0τ (w)).

The desired result follows from (B.7).

Lemma B.8.dLτ

dτ

∣∣∣τ=0

= 0,d2Lτ

dτ2

∣∣∣τ=0

< 0. (B.10)

Proof of Lemma B.8. In order to simplify notations we set D′τ = ∂D

∂τ ,

B′τ = ∂B

∂τ , C ′τ = ∂C

∂τ , D′′ττ = ∂2D

∂τ2 , and B′′ττ = ∂2B

∂τ2 . Since the function ατ (w)is differentiable over τ (see the paper by Burns, Pugh, Shub, and Wilkinsonin this volume) by Lemma B.7, we find

dLτ

dτ= −

∫

M

D′τ − ηB′

τα(g0τ (w)) − ηB ∂ατ (w)∂τ (g0τ (w))

D(τ, w) − ηB(τ, w)ατ (w)(g0τ (w))dw

and therefore,

d2Lτ

dτ2=

∫

M

(D′

τ − ηB′τα(g0τ (w)) − ηB(τ, w)∂ατ (w)

∂τ (g0τ (w))

D(τ, w) − ηB(τ, w)αs(g0τ (w))

)2

dw

−∫

M

E(τ, w)

D(τ, w) − ηB(τ, w)ατ (g0τ (w))dw,

where

E(τ, w) =D′′ττ − ηB′′

ττα(g0τ (w))

− ηB(τ, w)∂2ατ (w)

∂τ2(g0τ (w)) − 2ηB′

τ

∂ατ (w)

∂τ(g0τ (w)).

Note that for all w 6∈ Ω2,

A(τ, w) = D(τ, w) = 1, C(τ, w) = B(τ, w) = 0

and for all w ∈M ,

A(0, w) = D(0, w) = 1, C(0, w) = B(0, w) = 0, α0(w) = 0.


It follows thatdLτ

dτ

∣∣∣τ=0

=

∫

Ω2

D′τ dw, (B.11)

and also that

d2Lτ

dτ2

∣∣∣τ=0

=

∫

Ω2

[(D′

τ )2 −D′′

ττ + 2ηB′τ

∂ατ (w)

∂τ(g0τ (w))

]

τ=0

dw. (B.12)

By (B.8), we obtain that

D′τ (0, w) = rρr(r) sin θ cos θ

and hence, ∫

Ω2

D′τdw = 0.

Therefore, (B.11) implies the equality in (B.10).We now proceed with the inequality in (B.10). Applying Lemma B.9

below we obtain that

∂α

∂τ(g0τ (w))

∣∣∣τ=0

=C ′

τ (0, w)

η+

∞∑

n=1

C ′τ (0, g

−n00 (w))

ηn+1.

It follows that

2ηB′τ (0, w)

∂α

∂τ(g0τ (w))

∣∣∣τ=0

=2B′τ (0, w)C ′

τ (0, w)

+ 2B′τ (0, w)

∞∑

n=1

C ′τ (0, g

−n00 (w))

ηn.

First, we evaluate the term

F(w) = D′τ (0, w)2 −D′′

ττ (0, w) + 2B′τ (0, w)C ′

τ (0, w).

Using (B.8), we find that

F(w) =(rρr sin θ cos θ)2 + (ρ2 + 2rρρr sin2 θ)

− 2(ρ+ rρr sin2 θ)(ρ+ rρr cos2 θ)

= − ρ2 − (rρr sin θ cos θ)2 − 2rρρr cos2 θ.

(B.13)

Recall that Ω2 = Buc(q, ε0) ×Bs(q, ε0) and ρ(r) = 0 if r ≥ ε. We have∫

Ω2

2rρρr cos2 θ dw =

∫ ε0

−ε0

dy

∫ 2π

02 cos2 θ dθ

∫ ε

0r2ρρr dr. (B.14)

Since 0 = ρ(0) = ρ(ε) (by the definition of the function ρ), we find that∫ ε

0r2ρρr dr =

1

2r2ρ2

∣∣∣ε

0−∫ ε

0rρ2 dr = −

∫ ε

0rρ2 dr.

We also have that ∫ 2π

02 cos2 θ dθ =

∫ 2π

0dθ. (B.15)


It follows from (B.14)–(B.15) that

−∫

Ω2

2rρρr cos2 θ dw =

∫

Ω2

rρ2 dw ≤ ε

∫

Ω2

ρ2 dw. (B.16)

Arguing similarly one can show that

−∫

Ω2

(rρr sin θ cos θ)2 dw = −1

8

∫

Ω2

(rρr)2 dw (B.17)

Thus we conclude using (B.13), (B.16), and (B.17) that∫

Ω2

F(w) dw ≤ −(1 − ε)

∫

Ω2

ρ2r dw − 1

8

∫

Ω2

(rρr)2 dw < 0. (B.18)

We now evaluate the remaining term

G(w) =∞∑

n=1

1

ηi

∫

Ω2

2B′τ (0, w)C ′

τ (0, g−n00 (w)) dw.

Since the map g00 = F preserves the Riemannian volume we obtain that∫

Ω2

2B′τ (0, w)C ′

τ (0, g−n00 (w)) dw ≤

∫

Ω2

B′τ (0, w)2 dw +

∫

Ω2

C ′τ (0, g

−n00 (w))2 dw

=

∫

Ω2

B′τ (0, w)2 dw +

∫

Ω2

C ′τ (0, w)2 dw

Applying (B.8), we find that∫

Ω2

B′τ (0, w)2 dw +

∫

Ω2

C ′τ (0, w)2 dw

=

∫

Ω2

(ρ+ rρr sin2 θ)2 dw +

∫

Ω2

(ρ+ rρr cos2 θ)2 dw

≤ 4

(∫

Ω2

ρ2 dw +

∫

Ω2

r2ρ2r dw

).

It follows that for sufficiently large N > 0 (which does not depend on ε)

∞∑

i=N

1

ηi

∫

Ω2

2B′τ (0, w)C ′

τ (0, g−i00 (w)) dw ≤ 1

10

(∫

Ω2

ρ2 dw +

∫

Ω2

r2ρ2r dw

).

(B.19)Note that if g−n

00 Ω2 ∩ Ω2 = ∅, then B′τ (0, w)C ′

τ (0, g−n00 (w)) = 0 for all w.

Hence, ∫

Ω2

2B′τ (0, w)C ′

τ (0, g−n00 (w)) dw = 0.

We may choose the point q and a small ε such that g−n00 Ω2 ∩Ω2 = F−nΩ2 ∩

Ω2 = ∅ for all n = 1, 2, . . ., N . It follows from (B.12), (B.18), and (B.19)


that

d2Lτ

dτ2

∣∣∣τ=0

=

∫

Ω2

F(w) dw +

∫

Ω2

G(w) dw

≤ −(

9

10− ε

)∫

Ω2

ρ2 dw − 1

40

∫

Ω2

r2ρ2r dw < 0.

The desired result follows.

Lemma B.9.

∂α

∂τ(g0τ (w))

∣∣∣τ=0

=∞∑

n=0

C ′τ (0, g

−n00 (w))

ηn+1.

Proof of Lemma B.9. Define

R(τ, w, α) =C(τ, w) +D(τ, w)α

η(A(τ, w) +B(τ, w)α).

It follows from (B.6) that

ατ (g0τ (w)) = R(τ, w, ατ (w)). (B.20)

By (B.6) and (B.8), we have

∂R

∂τ

∣∣∣τ=0

=(C ′

τ +D′τα)(A+Bα) + (C +Dα)(A′

τ +B′τα)

η(A+Bα)2

∣∣∣τ=0

=C ′

τ (0, w)

η.

Since AD −BC = 1,

∂R

∂α

∣∣∣τ=0

=AD −BC

η(A+Bα)2

∣∣∣τ=0

=1

η.

It follows from (B.20) that

∂α

∂τ(g0τ (w))

∣∣∣τ=0

=C ′

τ (0, w)

η+

1

η· ∂α∂τ

(w)∣∣∣τ=0

.

Since this inequality holds for any w, replacing w with g−10τ (w) we obtain

∂α

∂τ(w)∣∣∣τ=0

=C ′

τ (0, g−10τ (w))

η+

1

η· ∂α∂τ

(g−10τ (w))

∣∣∣τ=0

.

The result follows by induction.

References

[NT] V. Nitica and A. Torok, An open and dense set of stably ergodic diffeomorphisms in

a neighborhood of a nonergodic one, Topology, to appear.[PS] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur.

Math. Soc. 2 (2000), no. 1, 1–52.[SW] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,

Invent. Math. 139 (2000), 495–508.





URL: http://www.math.psu.edu/dolgop/




URL: http://www.math.psu.edu/hu/




URL: http://www.math.psu.edu/pesin/

Barreira_pspm Lectures on Lyapunov Exponents Pesin

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