Introduction - Harvard Universitypeople.math.harvard.edu/~yusheng/Papers/Sequences of rational ma… · SEQUENCES OF RATIONAL MAPS WITH INFINITE NON-MONOMIAL RESCALING LIMITS YUSHENG

SEQUENCES OF RATIONAL MAPS WITH INFINITE

NON-MONOMIAL RESCALING LIMITS

YUSHENG LUO

Abstract. In this paper, we will construct a sequence of rational mapsof degree 4 which has infinitely many non-trivial rescaling limits.

1. Introduction

Let {fn} be a sequence of rational maps of degree d. We are interested in

the case when fn diverges in RatdC, i.e., converges to ∂ RatdC ⊂ P2d+1C . Such a

sequence will be called a degenerating sequence. Following the terminologyby Shishikura and in [Kiw15], we will call a degenerating sequence of degree 1a moving frame. In some special moving frames Mn, M−1

n ◦fqn◦Mn converges

to non-trivial dynamics on P1C. Such moving frames are called a ’rescaling’

of the original degenerating sequence, and the non-trivial dynamical systemare called a ’rescaling limit’.

Rescaling limits has been studied by a lot of people when investigatingasymptotic and boundary behavior of rational maps, see [Sti93], [Eps00],[DeM07] and [Kiw15] for different applications of rescaling limits.

We will now give the precise definition and state our main result.

Definition 1.1. Let {fn} be a sequence of rational maps of degree ≥ 2. Wesay a moving frame {Mn} is a rescaling for {fn}, if there is an integer q ≥ 1and a degree d′ ≥ 2 and a rational map g, and a finite subset S ⊂ P1

C suchthat

M−1n ◦ f qn ◦Mn(z)→ g(z), as n→∞

uniformly on compact subsets of P1C − S. We say that g is a rescaling limit

for {fn}. The minimal q > 1 such that the above holds is called the periodof the rescaling {Mn}.Definition 1.2. Two sequence of frames {Mn} and {Ln} are said to beindependent if M−1

n ◦Ln →∞ in PSL(2,C). Two sequences of frames {Mn}and {Ln} are said to be equivalent if M−1

n ◦ Ln →M ∈ PSL(2,C).

Definition 1.3. Given a sequence {fn} and two rescalings {Mn} and {Ln}of period q, we say they are dynamically dependent if after passing to subse-quence, there exist 1 ≤ m ≤ q and finite sets S1, S2 and rational maps g1, g2

of degree at least 1 such that

L−1nk◦ fmnk

◦Mnk(z)→ g1(z)

Date: Jan. 22 2018, v1, UNFINISHED.

1

2 YUSHENG LUO

uniformly on compact subsets of P1C − S1 and

M−1nk◦ f q−mnk

◦ Lnk(z)→ g2(z)

uniformly on compact subsets of P1C − S2.

In [Kiw15], Kiwi established

Theorem 1.4. Let d ≥ 2, and consider {fn} ⊂ RatdC. Then there existsat most 2d− 2 pairwise dynamically independent rescalings whose rescalinglimits are not postcritically finite.

Previously, the only known examples where fn have infinitely many post-critically finite rescaling limits are those, with at most finitely many excep-tions, the rescaling limit is conjugate to z±k for some k ≥ 2. We will call arescaling limit that is not conjugate to z±k a non-monomial rescaling limit.In [Kiw15], Kiwi asks if there are only finitely many non-monomial rescal-ing limits. In this paper, we will show the answer is no for degeneratingsequences of rational maps:

Theorem 1.5. There exists a sequence of rational maps {fn} ⊂ Rat4C with

a non-monomial rescaling limit of period 3j for every j ∈ N.

Note that since these rescaling limits have different periods, they are nec-essarily dynamically independent. The construction can also be generalizedto any degree. We will focus on the case of degree 4 to keep the examplemore concrete.

1.1. The construction. We will use theory of Berkovich spaces to con-struct the example. Non-Archimedean dynamics naturally emerges whenwe study degenerate holomorphic families {ft} ⊂ P2d+1

C . Here, the familyis parameterized by the unit disk ∆, and we say the family is degenerate ifft ∈ ∂ RatdC if and only if t = 0. We can regard such a family as a rationalmap f with coefficients in the field of formal Laurent series C((t)). Here,we prefer to work with the field of Puiseux series L, the algebraic closureof the completion of C((t)). We will extend the action of f : P1

L −→ P1L

to an action on the Berkovich space P1,anL . If x ∈ P1

L is a fixed point of f ,we can naturally associate it with the tangent map (or the reduction map)Txf : P1

C −→ P1C. We call a periodic point x of period q a repelling periodic

point if the tangent map Txfq has degree ≥ 2. The rescaling limits of a

sequence are closely related to repelling periodic points on P1,anL .

We will first construct a sequence of rational maps in Rat4L:

Theorem 1.6. For every n, there exists a rational map fn ∈ Rat4L such

that for each 0 ≤ k ≤ n− 1, there is a repelling periodic point xk ∈ P1,anL of

period pk = 3k with tangent map Txkfpk not conjugate to z±j.

SEQUENCES OF RATIONAL MAPS WITH INFINITE NON-MONOMIAL RESCALING LIMITS3

The construction can be thought of as a ’renormalization of maps ontrees’. We start with the McMullen family

f(z) = z2 +t2

z2

This family has been studied intensely by Devaney et al., we refer thereaders to [Dev10] [CD16] for detailed exposition.

We will see that the dynamics of f in P1,anL contains a degree 2 full tent

map as a subsystem (see Section 4). This actually implies that there areinfinitely many monomial rescaling limit.

Inductively on n, we will define the sequence fn as a perturbation of f

fn(z) = z2 +t2

z2+ rn

where rn =∑n

j=0 ajtαj .

The exponents αn > 12 are increasing to a number α∞ < 1, so rn is

a divergent sequence in L. Hence, the sequence fn does not converge toa rational map in Rat4

L. By taking a diagonal sequence, we are able toconstruct a degenerating sequence of rational maps with infinitely manynon-trivial rescaling limit and establish the main theorem.

1.2. Related Results. In [AC16], using Shishikura tree and tree of spheres,Arfeux and Cui has constructed a sequence of rational maps with arbitrarynon-monomial rescaling limits.

2. Rational Maps on Berkovich Space

In this section, we will give a brief introduction on the Berkovich spacesand the dynamics of rational maps on it. We are going to summarize someof the properties that will be used in this paper, and refer the readers to[BR10] for more detailed exposition of this grand theory.

2.1. The field L. In this paper, we will be mainly focusing on the field L,field of formal Puiseux series. It is the algebraic closure of the completionof the field of formal Laurent series C((t)). An element in a ∈ L can berepresented by a formal series

a =∑j>0

ajtλj

where aj ∈ C, λj ∈ Q if aj does not vanish for sufficiently large j, thenλj →∞ as j →∞. The absolute value is given by

|a| = exp(−min{λj : aj 6= 0})

provided a 6= 0. The valuation group is |L×| = exp(Q).We will be using either bold symbols like a or a(t) to represent a point in

L.

4 YUSHENG LUO

Given a ∈ L, we call

B(a, r) := {z ∈ L : |z− a| ≤ r}

a closed ball, and

B(a, r)− := {z ∈ L : |z− a| < r}

an open ball.The valuation ring of L is DL = B(0, 1), and its maximal ideal is MLB(0, 1)−.

The residual field of L = DL/ML is canonically identified with C by the map

c 7→ c+ ML

Given a rational map f ∈ Ratd(L), the reduction map f is given by takingthe reduction on its coefficients. Roughly speaking, the reduction map isobtained from f by setting t = 0.

2.2. The Berkovich projective line P1,anL . The Berkovich projective line

P1,anL is, as a topological space, a compact, Hausdorff, arcwise connected

space which contains P1L as a dense subset. It has a tree structure, i.e., any

two points x, y can be connected via a unique arc [x, y].

We define the tangent direction v at x ∈ P1,anL as a connected component

of P1,anL − {x}. The tangent space at x is the set of tangent directions, and

write TxP1,anL . The tangent space can be identified with the projective space

of the residual field, in our case, P1C.

There are many different ways to view the points in P1,anL , one way to

view them is via the Berkovich’s classification. Berkovich’s classificationsays that there are exactly 4 types of points in P1,an

L :

(1) Type I: Points in P1L, which we will also call the classical points;

(2) Type II: Points corresponding to a closed disk B(a, r) with r ∈ |L×|;(3) Type III: Points corresponding to a closed disk B(a, r) with r /∈ |L×|;(4) Type IV: Points corresponding to a nested sequence {B(ai, ri)} with

empty intersection.

The point corresponding to B(0, 1) is called the Gauss point and it isdenoted by xg. The Berkovich hyperbolic space HL is defined by

HL := P1,anL − P1

L

Given two Type II or III points x, y corresponding to the balls B(a, r)and B(b, s) respectively. Let B(a, R) be the smallest ball containing bothB(a, r) and B(b, s), then R = max{r, s, |a − b|}. We define the distancefunction

d(x, y) = 2 logR− log r − log s

This distance formula is easily extended continuously to arbitrary pointsx, y ∈ HL.


2.3. Rational maps on P1,anL . Given a rational map f ∈ Ratd(L), f extends

continuous to a map on P1,anL . The extension respects compositions and

thus iterations of rational maps. One can define the local degree at a pointx ∈ P1,an

L , and each point has exactly d preimages counted multiplicity.In the case of a degree 1 map, i.e., M ∈ PSL(2,L), the map is an au-

tomorphism of P1,anL . Given x ∈ P1,an

L corresponding to a nested sequence{B(ai, ri)}, then M(x) is the point corresponding to the nested sequence{M(B(ai, ri))}

The following proposition is standard (see Proposition 3.3 in [Kiw15] orSection 9.1 in [BR10]):

Proposition 2.1. Let f ∈ RatdL, and M,L ∈ PSL(2,L), then f(M(xg)) =L(xg) if and only if the reduction of L−1 ◦ f ◦M has degree ≥ 1.

2.4. Notations and preferred coordinates. Note that in non-Archimedeanspace, every point in the a ball is its center. This creates some inconveniencein our presentation and computation. Hence we define the following:

Definition 2.2. Let x ∈ P1,anL be type 2 point, represented by a closed

ball B(a(t), r) in L, we call the point a(t) =∑ant

αn ∈ B(a(t), r) withmaxαn < − log r the preferred center of the closed disk B(a(t), r).

We call M ∈ PSL(2,L) the preferred moving frame for x if

M(z) = t− log rz + a(t).

Using the preferred moving frames, we can identify the tangent spaceTP1,an

L |x with P1C by pulling back the tangent space at x to xg. More pre-

cisely, letM(z) = tlz + a(t)

be the preferred moving frame at x, v ∈ C is identified with the tangentdirection associated to the point L(xg) where

L(z) = tl+εz + a(t) + vtl

for ε > 0. And ∞ is identified with the tangent direction associated to thepoint L(xg) where

L(z) = tl−εz + a(t)

for ε > 0.

Definition 2.3. Let x ∈ P1,anL be type 2 point, represented by a closed ball

B(a(t), r) in L, and v ∈ P1C∼= TP1,an

L |x with v 6= ∞, we call the the arc[x, y] =: γ(x, v, δ) the preferred arc if y = L(xg) with L(z)

L(z) = t− log r+δz + a(t) + vt− log r.

Let g ∈ Ratd(L), and y = g(x), we call the reduction map of

˜L−1 ◦ g ◦M(z) =: Tg|xthe preferred tangent map at x if M and L are preferred moving frames forx and y.

6 YUSHENG LUO

3. Tent Map

Let I = [0, 1], and

T : I −→ I

t 7→

{2t if t ∈ L2− 2t if t ∈ R

where L = [0, 12 ] and R = [1

2 , 1].

Given a point x ∈ I, its forward orbit is denoted by

(x0, x1, ...) := (x, T (x), ...)

If x is not eventually mapped to 0, then the itinerary of x is a sequenceI(x) = (A0, A1...) of L and R, where Ak = L or R if xk ∈ L or R. Theitinerary of x uniquely determines the point x. If x is a periodic point, thenI(x) is a periodic sequence. In this case, we will represent I(x) by the finitesequence of its periodic cycle.

As we will see in the next section, this tent map is closely related to thedynamics of f on P1,an

L . In our construction, we will be using the followingvery specific sequence of periodic points of T .

Let r0 = 2/3 be the unique fixed point of T . Let rn be the smallestperiodic point of period 2n larger than rn−1, and let r∞ := lim rn andδn = r∞− rn. We also denote O(rn, T ) as the periodic orbit of rn. Here aresome simple facts about this sequence which we will be using later. We leftthese for readers to check.

(1) rn satisfies the recursive formula

rn = rn−1 +2

22n + 1· (1− rn−1)

(2) The itinerary of rn is I(rn) = (I(rn−1), I(rn−1)∗) where I(rn−1)∗ isderived from I(rn−1) by changing the last letter to the opposite.

(3) Each component of I − ∪n−1k=0O(rk, T ) contains exactly 1 point in

O(rn, T ). In particular, rn = maxO(rn, T ).(4) Any two points in ∪nk=0O(rk, T ) are separated by at least δn.


(5) Let r ∈ O(rn, T ), then

r + δn <

{2− 3r if r < 1

2

5r − 2 if r > 12

i.e., for r ∈ O(rn, T ), we have

δn < |4r − 2|

Figure 3.1. A Plot of O(rk, T ) for k = 0, 1, 2, 3

4. A study of the map f(z) = z2 + t2

z2

In this section, we will study some of the properties of the dynamics of

f(z) = z2 + t2

z2on P1,an

L .

Lemma 4.1. Let f(z) = z2 + t2

z2and Mr(z) = trz. Consider

Φ : I = [0, 1] −→ P1,anL

r 7→Mr(xg)

then Φ is a conjugacy onto its image between the Tent map T and f |Φ(I).Moreover, the preferred tangent map

T f |Mr(xg)(z) =

{z2 if r < 1

21z2

if r > 12

Proof. If r < 1/2, consider

M−12r ◦ f ◦Mr(z) =

t2rz2 + t2−2r/z2

t2r

= z2 + t2−4r/z2

Since r < 1/2, the reduction of this map is z2. This proves the case forr < 1/2.

If r > 1/2, consider

M−12−2r ◦ f ◦Mr(z) =

t2rz2 + t2−2r/z2

t2−2r

= t4r−2z2 + 1/z2

8 YUSHENG LUO

Since r > 1/2, the reduction of this map is 1/z2. This proves the case forr < 1/2. �

Motivated by this, we will call the arc Φ(I) the Major Arc.As an immediate consequence of 4.1, we have

Corollary 4.2. Let f(z) = z2 + t2

z2, Mr(z) = trz, and rn be the sequence

defined in Section 3, then xn := Mrn(xg) is a periodic point of period 2n.Moreover, the preferred tangent map

T f |xn(z) = 1/z22n

Proof. This follows from the fact rn is periodic of period 2n under T , andO(rn, T ) contains odd under of points in (1/2, 1). �

A similar computation allows one to compute for the points not too faroff the Major Arc.

Lemma 4.3. Let f(z) = z2 + t2

z2and M(z) = tlz +

∑nk=0 akt

αk , with 0 <α0 < α1 < ... < αn and

αn < l <

{min{2− 3α0, 2α1 − α0} if α0 <

12

min{5α0 − 2, 2α1 − α0} if α0 >12

Then f(M(xg)) = L(xg) where

L(z) =

{tl+α0z + a2

0t2α0 +

∑nk=1 2a0akt

α0+αk if α0 <12

tl+2−3α0z + 1a20t2−2α0 −

∑nk=1

2aka30t2+αk−3α0 if α0 >

12

Moreover, the preferred tangent map

T f |M(xg)(z) =

{2a0z if α0 <

12

− 2a30z if α0 >

12

Proof. Consider the first case when α0 <12 , then

L−1(z) =z − a2

0t2α0 −

∑nk=1 2a0akt

α0+αk

tl+α0

We will compute the reduction of L−1 ◦ f ◦M. We will first break it into twosums

L−1 ◦ f ◦M = S1 + S2

where

S1(z) =M(z)2 − a2

0t2α0 −

∑nk=1 2a0akt

α0+αk

tl+α0

and

S2(z) =t2

tl+α0M(z)2


Note that the second sum

S2(z) =t2−l−α0

t2lz2 + 2∑n

k=0 aktαktlz + (

∑nk=0 akt

αk)2

=t2−l−3α0

a20 +O(tα1−α0)

= O(t2−l−3α0)

Since l < 2− 3α0, the reduction of S2 is 0.The first sum

S1(z) = t−(l+α0) · (t2lz2 + 2

n∑k=0

aktαktlz + (

n∑k=0

aktαk)2

− a20t

2α0 −n∑k=1

2a0aktα0+αk)

= O(tl−α0)z2 + 2a0z +O(tα1−α0)z +O(t2α1−l−α0)

Since α0 < l < 2α1 − α0, the reduction of S1 is S1(z) = 2a0z. Therefore,the reduction of L−1 ◦ f ◦M is 2a0z, proving the lemma in this case.

Now we consider the second case when α0 >12 , then

L−1(z) =z − 1

a20t2−2α0 +

∑nk=1

2aka30t2+αk−3α0

tl+2−3α0

We again break L−1 ◦ f ◦M into two sums

L−1 ◦ f ◦M = S1 + S2

where

S1(z) =M(z)2

tl+2−3α0

and

S2(z) =t2 − ( 1

a20t2−2α0 −

∑nk=1

2aka30t2+αk−3α0)M(z)2

tl+2−3α0M(z)2

The first sum

S1(z) =t2lz2 + 2

∑nk=0 akt

αktlz + (∑n

k=0 aktαk)2

tl+2−3α0

= O(t3α0+l−2)z2 +O(t4α0−2)z +O(t5α0−l−2)

Since l < 5α0 − 2 and α0 >12 , the reduction of S1 is 0.

10 YUSHENG LUO

The second sum

S2(z) =t2 − ( 1

a20t2−2α0 −

∑nk=1

2aka30t2+αk−3α0)M(z)2

tl+2−3α0M(z)2

=O(t2+2l−2α0)z2 − 2

a0t2+l−α0z +O(t2+l+α1−2α0)z +O(t2+2α1−2α0)

O(t2+3l−3α0)z2 +O(t2+2l−2α0)z + a20t

2+l−α0 +O(t2+l+α1−2α0)

=O(tl−α0)z2 − 2

a0z +O(tα1−α0)z +O(t2α1−α0−l)

O(t2l−2α0)z2 +O(tl−α0)z + a20 +O(tα1−α0)

Since α0 < l < 2α1−α0, the reduction of S2 is S2(z) = − 2a30z. Therefore,

the reduction of L−1 ◦ f ◦M is − 2a30z, proving the lemma in this case.

�

Remark 4.4. Let x be a point which is not too far off the Major Arc,(i.e., satisfies the condition of the Lemma 4.3), and let y be the point onthe Major Arc where [xg, x] branches off, and a ∈ C represents the tangentdirection of [y, x] at y. By Lemma 4.3, for any w ∈ (y, x], we have

T fw(z) = D(T fy)(a)z

where D represents the derivative.

Remark 4.5. Let x = M(xg) where M(z) = tlz +∑n

k=0 aktαk be a point

which is not too far off the Major Arc, (i.e., satisfies the condition of the

Lemma 4.3), and x′ = M′(xg) where M(z) = tαnz+∑n−1

k=0 aktαk be the last

’branch point’ on [xg, x]. By Lemma 4.3, we have that f isometrically mapsγ(x′, an, l − αn) to γ(f(x′), T f |x′(an), l − αn).

5. The Proof of Theorem 1.6

In this section we will prove Theorem 1.6.

Let f(z) = z2 + t2

z2. The idea is essentially to construct a sequence of

perturbations of f which contains more and more non-monomial rescalinglimits.

Throughout this section, we will use rn to denote the sequence in [0, 1]defined in Section 3, and xn to denote the point Mrn(xg) where

Mrn(z) = trnz

Given a type 2 point x ∈ P1,anL , we define

h(x) := − log r

where r is the radius of the disk corresponding to the x, and call it the heightof the point x. Note that h(x) = d(x, xg).

We will construct a sequence (an) ∈ C and the sequence of perturbationsof f

fn(z) = z2 +t2

z2+

n∑k=0

aktrk


Lemma 5.1. Let x ∈ P1,anL , then h(fn(x)) = h(f(x)). If h(f(x)) < rk, then

fn(x) = fk−1(x).

Moreover, the preferred tangent maps

T fn|x(z) = T f |x(z) + δ

where δ = ak if h(f(x)) = rk for k = 0, ..., n, and 0 otherwise.

Proof. This follows immediately as fn is a composition of the linear mapz 7→ z +

∑nk=0 akt

rk and f . �

In order to study the dynamics of fn on P1,anL , we define

Definition 5.2. Let x ∈ P1,anL , with preferred moving frame

M(xg) = tαmz +

m−1∑k=0

aktαk .

The preferred arc [x, y] = γ(x, v, l − αm) is called n-admissible if

(1) γ(x, v, l − αm) is not contained in the Major Arc,

(2) l <

{2− 3α0 if α0 <

12

5α0 − 2 if α0 >12

(3) l − α1 < α1 − α0

(4) h(fn((x, y))) ∩ {rk : k ≤ n} = ∅

The following lemma is an immediate consequence of Lemma 5.1, Defini-tion 5.8 and Remark 4.5.

Lemma 5.3. Let γ(x, v, ε) be an n-admissible arc, then fn maps γ(x, v, ε)isometrically to γ(fn(x), T fn|x(v), ε).

xg

x0

x∞

(a) T−1

xg

x0

x∞

(b) T0

xg

x0

x∞x1 f2

1 (x1)

f1(x1)

0

1

23

4

56

7

8

(c) T1

Figure 5.1. Some examples of the trees Tn with canonicalordering labeled on the edge

12 YUSHENG LUO

Given a (connected) finite tree T , and a ∈ T , we say a is a vertex of T ifa is either a branch point for T (i.e. T −{a} contains 3 or more component)or a is a tip point (i.e. T − {a} is still connected). We use V (T ) to denotethe set of vertices and B(T ) to denote the set of branch points. An edge Eis defined as a component T − V (T ) together with its two boundary points.If both end points are in B(T ), we call E an interior edge, otherwise, anend.

Given fn(z) = z2 + t2

z2+∑n

k=0 aktrk , let Tn be the minimal tree containing

all forward orbits of xg and xk’s, i.e.,

Tn = CV ({xg} ∪∞⋃j=0

O(xj , fn))

We will show

(Tn, fn(z) = z2 +t2

z2+

n∑k=0

aktrk)

satisfies the following properties:

(1) xk is periodic of period pk :=

{3k if k ≤ n+ 1

2k−n−1 · 3n+1 if k > n+ 1with

O(xk, fn) ⊂ Tnand maxx∈O(xk,fn) h(x) ≤ rk;

(2)

B(Tn) =n⋃k=0

O(xk, fn)

and each branch point has valence 3;(3) Let x ∈ O(xk, fn), the tangent map of the first return map permutes

the tangent space of Tn at x.(4) If E = [xn, x∞], then fn(E) is contained in the edge [xg, fn(xn)];

If E is an interior edge in [xg, x∞] which does not contain the

point t1/2xg, then fn(E) is a union of interior edges;

If E is an interior edge in [xg, x∞] which contains t1/2xg, then oneof the end point x ∈ O(xn, fn) and the other y ∈ O(xn−1, fn), andthere exists w,w′, x′ ∈ [x, y] such that fn([x,w]) = fn([w′, x′]) is anend of Tn, fn[x′, y] is an edge of Tn and fn((w,w′)) ∩ Tn = ∅;

If E is an interior edge (or respectively an end) not contained in[xg, x∞], then fn(E) is an interior edge (or respectively an end) ofTn and fn maps E isometrically to fn(E);

(5) Let E be an edge, then E is isometrically mapped to an edge I ⊂[xg, x∞] ∼= [0, rn] by f jn for some j ≥ 0, and

O(xk, fn) ∩ E = (f jn)−1(O(rk, T ) ∩ I)

Note that f−1 = z2 + t2

z2and T−1 := [xg, x∞] satisfies all the properties.

Hence the base case is proved.


By P1 and P2, there are exactly 3n+1 edges. These edges can be orderedin the following way: E0 = [xn, x∞] = γ(xn, v0, δn), Ej is the unique edge

of Tn containing γ(f jn(xn), T f jn|xn(v0), ε) for some ε > 0. We will call thisordering the canonical ordering of the edges of Tn.

We will call the unique edge of Tn containing the point t1/2xg the criticaledge, and the unique end of Tn in its image the minor edge. We also havethe following characterization of the minor edge:

Proposition 5.4. Let the edges of Tn be ordered as above, and let Em bethe minor edge of Tn, then h(Em) ≥ rn and

m = minj>1{j : Ej is an end of Tn}

= minj>0{j : There exists x ∈ Ej with h(x) > rn}

and for all m ≤ j < 3n+1, Ej is an end of Tn.

Proof. By P5, h(fm−1n (xn)) is mapped to rn by T , so h(fmn (xn)) = rn by

Lemma 5.1. Hence, by P4, we conclude that h(Em) ≥ rn.If Ej is an end, then Ej maps isometrically to an end Ej+1 unless j = 0

or 1, so the first equality holds as Em−1 is the critical edge. This also provesthat Ej is an end for all m ≤ j < 3n+1.

By P1, maxx∈O(x,k,fn) ≤ rk, so an edge with a point x of height h(x) > rnis necessarily an end of Tn. So the second equality follows from the first oneand h(Em) ≥ rn.

�

Assume that we have constructed Tn−1 and fn−1 satisfying all the prop-erties, and we let fn(z) = fn−1(z) + ant

rn . We now compute the preferredtangent map at xn:

Proposition 5.5. The preferred tangent map at xn is

T f3n

n |xn(z) =A

z22n+Ban

for some A,B ∈ C− {0}.

Proof. By P1, maxx∈O(xk,fn−1) h(x) ≤ rk, so by Lemma 5.1, fn−1 = fn onthe orbit of xk when k ≤ n, and the preferred tangent map

T fn|x(z) =

{T fn−1|x(z) if h(fn(x)) < rn

T fn−1|x(z) + an if h(fn(x)) = rn

Let j1 denote the smallest integer such that

h(f j1n (xn)) = rn,

Let E = [x, y] ⊂ Tn−1 be the end containing xjn, then E is the minor edge

of Tn−1. Let j2, j3 denote the smallest integer such that f j2n (x) = xn−1, and

14 YUSHENG LUO

f j3n (xn−1) = x. So the the first return map f3nn of xn can be broken into

xnfj1n (x)−−−−→ xj1n

fj2n (x)−−−−→ xj1+j2

nfj3n (x)−−−−→ xj1+j2+j3

nfj2n (x)−−−−→ xn

xn−1

xn

x∞

xj1+j2n

x = xj1n−1 = xj3n−1

xj1+j2+j3n xj1n

Note that by P3 and P5, T f3nn−1|xn(z) = C/z22

n

for some C. For j1 ≤ j <3n, xjn /∈ [xg, x∞], so T fn−1|xjn(z) = cz for some c. By our choice of j1 and

Lemma 5.1,

T f j1n |xn(z) = T f j1n−1|xn(z) + an =C1

z22n+ an.

Since for j1 ≤ j < 3n, T fn|xjn is of the form cz or cz + an,

T f j2n |xj1n (z) = C2z + C3an

for some C2 and C3.We claim that C2 + C3 6= 0. Note that 0 ≤ j < j2, xj1+j

n and xj are inthe same end of Tn−1 so by Remark 4.4 for 0 ≤ j < j2 the derivative of thetwo linear maps T fn|xj1+j

nand T fn|xj are the same. Since the map fn on

an end is an isometry, h(fn(xj1+jn )) = rn if and only if h(fn(xj)) = rn−1, so

T f j2n |x(z) = C2z + C3an−1. Since T f j3n |xn−1(z) = c

z22n−1 + an−1 for some c .

Therefore

T f3n−1

n |xn−1(z) = C2(c

z22n−1 + an−1) + C3an−1.

Since T f3n−1

n |xn−1(z) is not monomial, C2 + C3 6= 0.

Since T f j2n |x(z) = C2z + C3an−1, reversing the argument also shows that

T f j2n |xj1+j2+j3n

(z) = C2z + C3an

For 0 ≤ j < j3, the map T fn|xjn−1is not always a linear map. Let vj

represents the the tangent direction at xjn−1 of [xjn−1, xj1+j2+jn ], then by

Remark 4.4, the derivative associated to the direction vj

D(T fn|xjn−1)(vj) = DT fn|xj1+j2+j

n


By our choice of j3, h(xjn−1) < rn−1 for 0 < j < j3, hence h(xj1+j2+jn ) < rn

for 0 < j < j3, and h(xj1+j2+j3n ) = rn. So by the chain rule,

T f j3n |xj1+j2n

(z) = D(T f j3n |xn−1)(v0)z + an

To compute D(T f j3n |xn−1)(v0), we note that T f j2n |x(z) = C2z+C3an−1, so

D(T f j3n |xn−1)(v0) = D(T f3n−1

n |xn−1)(v0)/C2.

Since v0 is in the critical orbit of T f3n−1

n (z) of period 3, we know T f3n−1

n (z) =

c/z22n−1

+ v0 and T f3n−1

n (v0) = 0. Hence, we have

D(T f3n−1

n |xn−1)(v0)/C2 = 22n−1/C2

Therefore,

T f ln|xj+kn

(z) = 22n−1/C2z + an

and

T f3n

n |xn(z) = T fkn |xj+k+ln

◦ T f ln|xj+kn◦ T fkn |xjn ◦ T f jn|xn(z)

= C2(22n−1/C2(C2(C1/z

22n

+ an) + C3an) + an) + C3an

= 22n−1C1C2/z

22n

+ (1 + 22n−1)(C2 + C3)an

Hence B 6= 0. (Note that A 6= 0 follows from the fact f3nn (xn) = xn.)

�

Let an be chosen so that

A/(Ban)22n

+Ban = 0

then the tangent map T f22n

n |xn has exactly two critical points of multiplicity22n − 1 forms a periodic cycle of period 3.

We define Cn := CV ({xg} ∪ O(xn, fn)) as the convex hull of the Gausspoint and the orbit of xn and

Tn := Cn ∪⋃

(x,v)∈O((xn,v0),(fn,T fn))−(Cn,TCn)

γ(x, v, δn)

where δn = r∞ − rn.Let E = [x, y] = [x(E), y(E)] be an end of Tn−1, then by IH5, there is a

unique point w = w(E) ∈ E ∩ O(xn, fn−1). Hence we have

Tn−1 = Cn ∪⋃

E is an end of Tn−1

[w(E), y(E)]

Topologically, Tn can be thought of constructed from Tn−1 by attachingan end on each edge of Tn−1.

Proposition 5.6. (Tn,fn) satisfies all the properties.

16 YUSHENG LUO

Remark 5.7. Note that once we prove this proposition, then by P1, P4and P5, we immediately have that

Tn = CV ({xg} ∪∞⋃j=0

O(xj , fn))

Proof. To check P2, it is clear that the set of branch points isn⋃k=0

O(xk, fn)

and the valence at⋃n−1k=0 O(xk, fn) is 3. from our definition and IH2

So we only need to check that the valence at x ∈ O(xn, fn) is 3.We first note that [xg, x∞] ∈ Tn by our construction, and the valence at

xn is 3.Let x ∈ O(xn, fn), and assume x ∈ Ek in the canonical ordering of the

edges of Tn−1. If Ek is an interior edge, then by Proposition 5.4, h(Ej) ≤rn−1 < rn for all 0 < j ≤ k, so T fkn−1|xn(z) = T fkn |xn(z). Note that two of thetangent directions of Tn at xn are the tangent directions of Tn−1 at xn, hencewe have only attached 1 new end to Cn at x, so the valence is 3. Otherwise,

denote Ej = [aj , bj ] where aj ∈ O(xn−1, fn), then by P1, h([aj , xjn]) ≤ rn

for all j, where xjn = f jn(xn). Hence f jn|[xn−1,xn] = f jn−1|[xn−1,xn]. Hence thetangent direction (x, vx→xg) is contained in orbit of (xn, vxn→xg).Hence wehave only attached 2 ends to Cn at x, so the valence is 3.

To check P4, we need the following lemma.

Lemma 5.8. If E = [x, y] is an end of Tn, then it is an n-admissible oflength δn;

Proof. Note each point x ∈ O(xn, fn) has valence 3, so it cannot be a tippoint of Tn, so an end is of the form E = γ(x, v, δn) by construction.

We will prove using induction and assume that the ends in Tn−1 is n− 1-admissible.

Let J = [a, b] ⊂ Tn−1 be the edge or the end containing x, and x = M(xg)

where M(z) = tαmz +∑m−1

k=1 aktαk , then b = L(xg) where L(z) = tα

′mz +∑m−1

k=1 aktαk . By IH5, we know that α′m − αm ≥ min(mins,t∈∪nk=0O(rk,T ) |s−

t|, δn) = δn.For the first inequality, we note that if m = 0, then by IH5, α0 ∈ O(rn, T ),

so

α0 + δn <

{2− 3α0 if α0 <

12

5α0 − 2 if α0 >12

follows from the fact on Tent maps. If m > 0, then

αm + δn ≤ α′m <

{2− 3α0 if α0 <

12

5α0 − 2 if α0 >12

follows from the hypothesis J is n− 1-admissible.


For the second inequality, we note that if m = 0, it is trivially satisfied.If m = 1, then by IH5, α1 − α0 ≥ mins,t∈∪nk=0O(rk,T ) |s− t| > δn. Hence,

α1 + δn − α1 < α1 − α0.

If m > 1, then

αm + δn − α1 ≤ α′m − α1 < α1 − α0

follows from the hypothesis J is admissible.For the third one, we note that if m = 0, then α0 ∈ O(rn, T ) and

(α0, α0 + δn) ∩ {rk : k ≤ n} = ∅.

If m > 0, then since J is n − 1-admissible, and (αn, αn + δn) ⊂ (αn, α′n),

so the only possible intersection with {rk : k ≤ n} is rn. Let x′ ∈ Jwith h(x′) = rn. Note that J is necessarily an end in Tn−1, so by IH5,x′ ∈ O(xn, fn). Hence x′ = x as J ∩ O(xn, fn) contains exactly 1 point.Hence (αn, αn + δn) ∩ {rk : k ≤ n} = ∅.

Therefore, E is n-admissible.�

If E = [xn, x∞], then since maxx∈f(E) h(x) ≤ r0, so fn(E) = f(E). By IH5,fn(z) has the smallest height, so [xg, fn(z)] is an edge, and fn(E) ⊂ [xg, fn(z)].

If E is an edge which does not contain t1/2xg, then then fn is injectiveon E. Since the end points of E are in ∪nk=0O(xk, fn), so are their image.Hence, fn(E) is a union of interior edges.

If E is an edge containing 1/2, by IH5 and the fact that

∪nk=0O(xk, fn) = ∪nk=0O(xk, fn−1),

we have

∪nk=0O(xk, fn) ∩ [xg, x∞] = ∪nk=0O(rk, T ).

So one of the end points x ∈ O(xn, fn) and the other y ∈ O(xn−1, fn) followsimmediately from the itineraries of the rk’s for the Tent map. Now let x′

be the symmetric point of x to 1/2, w1 ∈ [x, y] be of distance δn away fromx, and w2 be its symmetric point, then fn(x) = fn(x′) and fn(w) = fn(w′).By Lemma 4.1, fn([x,w]) = fn([w′, x′]) = γ(fn(x), T fn|x(vx→w), δn). Sincevx→w is a critical direction, we conclude that fn([x,w]) = fn([w, x′]) is anend. Note that fn is injective on [x′, y] and fn(x′) = fn(x) ∈ O(xn, fn),fn(y) ∈ O(xn−1, fn), so fn([x′, y]) is an edge follows from IH4 and IH5. Notethat the height of fn(u) for u ∈ (w,w′) is strictly bigger than r∞. Hencefn((w,w′)) ∩ Tn = ∅.

If E is an edge which does not intersect [xg, x∞], then fn|E = fn−1|Eas maxx∈Cn h(x) ≤ rn by IH1. So fn is an isometry from E to fn(E),hence there is no branch points in fn(E) by IH4. Since both end points arein ∪nk=0O(xk, fn), so are the end points of the image. Hence, fn(E) is aninterior edge.

If E = [x, y] is an end which does not intersect [xg, x∞], then E =γ(x, v, δn) for some (x, v) is in the orbit of (xn, v0) under (fn, T fn). By

18 YUSHENG LUO

Lemma 5.8, E is n-admissible, so fn(E) = γ(fn(x), T fn|x(v), δn), and fnmaps isometrically from E to fn(E). Note from our construction, either

(1) There are two ends attached to x(2) There is only one end attached to x

In the first case, let I ⊂ Tn be the unique edge attached to x and J ⊂ Tn−1

be the end containing x. Note that I ⊂ J .If J is not in [xg, x∞], then by IH4 fn−1(J) is an end. Note fn|I = fn−1|I

as I ⊂ Cn. Therefore, γ(fn(x), T fn|x(v), δn) is an end of Tn as the tangentmap T fn|x is injective on the tangent plane of Tn.

If J = [xn−1, x∞], then fn−1|J = fn|J , and fn(J) ⊂ [xg, fn(xn−1)] ⊂ Cn.Therefore γ(fn(x), T fn|x(v), δn) is an end of Tn.

If J = [xg, fn(xn−1)], then fn−1|J = fn|J . Therefore, fn(J) ⊂ Cn, soγ(fn(x), T fn|x(v), δn) is an end of Tn.

In the second case, let I1, I2 ⊂ Tn be the two edges attached to x.If neither I1 and I2 contains 1/2, then since fn(I1) and fn(I2) are union

of interior edges, so γ(fn(x), T fn|x(v), δn) is an end of Tn as there must bean end attached to fn(x).

If I1 contains 1/2, then fn(I2) is an interior edge and there are two endsattached to fn(x), so γ(fn(x), T fn|x(v), δn) is an end of Tn.

To check P5, if k ≤ n, it follows from IH5 as O(xk, fn) = O(xk, fn−1).We will first prove each xk is periodic. Let k > n and x ∈ [xn, x∞] ∩

O(xk, fn) and let j be the smallest integer such that h(f jn(x)) > rn, then

f j−1n (x) = f j−1

n−1(x). Let I ⊂ Tn be the edge that contains f j−1n (x), then I

contains 1/2. Hence by P4, we know that f jn(x) is contained in the minoredge of Tn. Let E1 = [a1, b1] ⊂ Tn and E2 = [a2, b2] ⊂ Tn−1 be the end

containing f jn(x) and f jn−1(x) respectively. Since h(f jn(x)) = h(f jn−1(x)) by

Lemma 5.1, so d(a1, fjn(x)) + rn = d(a2, f

jn−1(x)) + rn−1. Let k1 and k2 be

such that fk1n (E1) = [xn, x∞] and fk2n−1(E2) = [xn−1, x∞]. Hence

fk1n (f jn(x)) = fk2n−1(f jn−1(x))

as the maps fk1n |E1 and fk2n−1|E2 are both isometries. Therefore, by IH5, theperiodic cycles of xk under fn intersecting [xn, x∞] equals to

{Mr(xg) : Mr(z) = trz where r ∈ O(rk, T ) ∩ [rn, r∞]}

If E is an end of Tn which is not in [xg, x∞], and j be such that f jn(E) =[xn, x∞]. By P4, the inverse map(fn)−1 is single valued on any end I other

than the minor edge. Hence (f jn)−1 is single valued on [xn, x∞], so

O(xk, fn) ∩ E = (f jn)−1(O(xk, fn) ∩ [xn, x∞]).

Now let Ej be an interior edge in the canonical ordering, then f in([xn, x∞])

does not intersect an end for all i ≤ j by Proposition 5.4. Hence f jn|[xn,x∞](z) =

f jn−1|[xn,x∞](z). The claim now follows from IH5.


To check P1, from P5 it is clear that each xk is periodic, O(xk, fn) ⊂ Tnand maxx∈O(xk,fn) h(x) ≤ rk, so we only need to check that the period is asdescribed. Note that for k ≤ n, this is clear as fn−1 agrees with fn. So weassume that k ≥ n+1. Note that there are exactly 3n additional ends (1 foreach iterate of xn) in Tn, and each end is eventually isometrically mappedto [xn, x∞]. Since there are 2k−n−1 points O(rk, T ) in [rn, r∞], by IH1, theperiod of xk for fn−1 is 2k−n+1 · 3n, hence the new period is

2k−n · 3n + 2k−n−1 · 3n = 2k−n−1 · 3n

To check P3, for k < n, this follows from IH3, and for k = n, this followsfrom our definition. We only need to check k > n. Note that in the proofof P5, we actually proved

T f2k−n−1·3nn |xk(z) = T f2k−n·3n

n−1 |xk(z).

Hence the result now follows from IH3.�

6. Construction of the Sequence

In this section, we will construct the sequence and prove the Theorem1.5.

In the previous section, we have constructed a sequence of fn ∈ Ratd(L)of the form

fn(z) = z2 +t2

z2+

n∑k=0

aktrk

such that Mrk(xg) = trk(xg) is a periodic point of period 3k, with rescalinglimit as a non-monomial rational map of degree 22n for all k ≤ n.

Let ρn be a sequence of positive numbers decreasing to 0, and consider

sn :=

u(n)∑k=0

akρrkn

We choose u(n) to be non decreasing sequence diverging to +∞ and sothat

|akρ(rk−rk−1)/2n | ≤ 1/2k

for all k ≤ u(n).This is possible as each ρn is decreasing to 0, and rk − rk−1 > 0.

20 YUSHENG LUO

Let Sm,n :=∑m

k=0 akρrkn , then for all sufficiently large n, (i.e., n satisfies

that m ≤ u(n)), we have

|sn − Sm,n|ρrmn

≤u(n)∑

k=m+1

|akρrk−rmn |

≤ ρ(rm+1−rm)/2n

u(n)∑k=m+1

|akρ(rk−rk−1)/2n |

= O(ρ(rm+1−rm)/2n )

We will now show

Proposition 6.1. The sequence

fn(z) = z2 +ρ2n

z2+ sn

has a non-monomial rescaling limit of period 3j for every j ∈ N.

Proof. Let fm(z) = z2+ t2

z2+∑m

k=0 aktrk , and xm = Mrm(xg) where Mrm(z) =

trmz be as in the previous section. Let xjm := f jm(xm) and Mj be the pre-

ferred moving frames of xjm. Hence the reduction of the map

M−1j+1 ◦ fm ◦Mj(z)

is a rational map of degree ≥ 1.Let Fm,n and Mj,n be the sequence obtained from fm and Mj by sub-

stituting t with the positive number ρn. Note this is well defined as thecoefficients in fm and Mj are finite sums of tα’s. More concretely,

Fm,n = z2 +ρ2n

z2+ Sm,n

A coefficient a in M−1j+1 ◦ fm ◦Mj(z) is a rational functions of finite sums

tα’s, therefore by substituting t by ρn, we conclude that the sequence con-verges to the reduction a as ρn → 0.

Therefore, the sequence

M−1j+1,n ◦ Fm,n ◦Mj,n(z)

converges algebraically to the rational map ˜Mj+1−1 ◦ fm ◦Mj(z).

Assume Mj(z) = tljz+aj , then we know lj ≤ rm, as maxx∈O(xm,fm) h(x) =

rm. Since |sn − Sm,n| = O(ρrm+(rm+1−rm)/2n ), so

|M−1j+1,n ◦ Fm,n ◦Mj,n(z)−M−1

j+1,n ◦ fn ◦Mj,n(z)| = |sn − Sm,n|/ρlj+1n

= O(ρrm+(rm+1−rm)/2−lj+1n )

Hence the sequenceM−1j+1,n ◦ fn ◦Mj,n(z)

converges algebraically to the rational map ˜Mj+1−1 ◦ fm ◦Mj(z) as well.


Hence for any m ∈ N, {Lm,n(z) = ρrmn z} is a rescaling of {fn} of period 3m

whose rescaling limit is a non-monomial degree 22m rational map. Note fordifferent m’s, the periods are not the same, so they cannot be dynamicallydependent.

This proves the result. �

References

[AC16] Matthier Arfeux and Guizhen Cui. Arbitrary large number of non trivial rescal-ing limits, 2016.

[BR10] Matthew Baker and Robert S. Rumely. Potential Theory and Dynamics on theBerkovich Projective Line. Mathematical surveys and monographs. AmericanMathematical Soc., 2010.

[CD16] Daniel Cuzzocreo and Robert L. Devaney. Simple mandelpinski necklaces forz2 +λ/z2. In Lluıs Alseda i Soler, Jim M. Cushing, Saber Elaydi, and Alberto A.Pinto, editors, Difference Equations, Discrete Dynamical Systems and Applica-tions, pages 63–72, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg.

[DeM07] Laura DeMarco. The moduli space of quadratic rational maps. Journal of theAmerican Mathematical Society, 20(2):321–355, 2007.

[Dev10] Robert L. Devaney. Singular Perturbations of Complex Analytic Dynamical Sys-tems, pages 13–29. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.

[Eps00] Adam Epstein. Bounded hyperbolic components of quadratic rational maps. Er-godic Theory and Dynamical Systems, 20(3):727–748, 2000.

[Kiw15] Jan Kiwi. Rescaling limits of complex rational maps. Duke Math. J.,164(7):1437–1470, 05 2015.

[Sti93] James Stimson. Degree two rational maps with a periodic critical point. PhDthesis, University of Liverpool, 1993.

Dept. of Mathematics, Harvard University, One Oxford Street, Cambridge,MA 02138 USA

E-mail address: [email protected]

Introduction - Harvard Universitypeople.math.harvard.edu/~yusheng/Papers/Sequences of rational ma… · SEQUENCES OF RATIONAL MAPS WITH INFINITE NON-MONOMIAL RESCALING LIMITS YUSHENG

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