Equidistribution and Brownian motion on the Sierpi?ski gasket

EQUIDISTRIBUTION AND BROWNIAN

MOTION ON THE SIERPINSKI GASKET

Peter J. Grabner and Robert F. Tichy

Dedicated to Prof. Edmund Hlawka on the occasion of his 80th birthday

Abstract. We introduce several concepts of discrepancy for sequences on the Sier-pinski gasket. Furthermore a law of iterated logarithm for the discrepancy of tra-

jectories of Brownian motion is proved. The main tools for this result are regularity

properties of the heat kernel on the Sierpinski gasket. Some of the results can begeneralized to arbitrary nested fractals in the sense of T. Lindstrøm.

1. Introduction

As a starting point we consider the Sierpinski gasket, a well known planar fractalset introduced by W. Sierpinski [Si]. Let A0 be a closed equilateral triangle of unit

sides e1, e2, e3 with vertices P1(12 ,

√32 ), P2(0, 0), P3(1, 0). Let A1 be the set obtained

by deleting the open equilateral triangle whose vertices are the midpoints of theedges of A0. Thus A1 consists of 3 equilateral triangles with side 1

2 . Repeatingthis procedure we obtain successively A2, A3, . . . . An consists of 3n equilateraltriangles of side 2−n, which are called elementary triangles of level n. Furthermorewe denote the set of all vertices of An by Vn and the boundary of An by En. ThusFn = (Vn, En) is defining a finite graph.

Definition 1. The set G =⋂∞

n=0An is called the (bounded) Sierpinski gasket.

Remark 1. Any point p ∈ G can be represented by the triple (k1, k2, k3) withk1 + k2 + k3 = 2, where

ki = ki(p) =∞∑

ℓ=1

ε(i)ℓ

2ℓ, ε

(i)ℓ = 0 or 1

and ε(1)ℓ + ε

(2)ℓ + ε

(3)ℓ = 2 for all ℓ ≥ 1. Note that (1− ki)

√32 is just the distance of

p to the side ei.

1991 Mathematics Subject Classification. (Primary) 60B99 (Secondary) 11K06.Key words and phrases. diffusion processes, fractals, discrepancy, uniform distribution.

The authors are supported by the Austrian Science Foundation project Nr. P10223-PHY and

by the Austrian-Italian scientific cooperation program

Typeset by AMS-TEX

1

2 PETER J. GRABNER AND ROBERT F. TICHY

Figure 1

By standard techniques as discribed in [Fa] it is easy to see that the Sierpinski

gasket has Hausdorff dimension α = log 3log 2 and finite positive Hausdorff measure.

Let µ denote the (normalized) Hausdorff measure of dimension α on G.

In a series of papers methods and results from classical potential theory in theEuclidean space were extended to the Sierpinski gasket, the Sierpinski carpet, wherethe whole potential theory is developed in a series of papers (cf. [BB] for furtherreferences) or more generally to so called nested fractals. We want to mention herethe fundamental paper of M.T. Barlow and E.A. Perkins [BP], where a systematic

theory of Brownian motion on the Sierpinski gasket is developed. Let F be theinfinite graph defined by F =

⋃

k 2kFk.

EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPINSKI GASKET 3

Definition 2. The topological closure of

∞⋃

k=0

2−kF

is called the infinite Sierpinski gasket G.

The Brownian motion is introduced as a suitable limit process of discrete randomwalks on the vertices of F : Let Yk be the random walk on F with transititionprobabilities 1

4 from each vertex to its neighbour. In order to give a proper definitionof the limiting process one has to rescale the time according to the eigenvalues ofthe transition matrix. Thus we consider the processes

X(n)(t) = 2−nY[5nt] for t ≥ 0, n ∈ N.

In [BP, Theorem 2.8] it is shown, that the processes X(n) converge weakly to a

processX , where X is a continuous, non-constant, G-valued, strong Markov processstarting at O. This process X = Xt can be considered as the Brownian motionon G. The Laplacian is obtained as the infinitesimal generator of the semi-groupdescribing this process. Among other very interesting results the authors obtainregularity properties of the heat kernel.

Lindstrøm [Li] studies Brownian motion on compact nested fractals. In thecase of the finite (=compact) Sierpinski gasket G a trajectory of Brownian motion

is obtained from a trajectory on G just by factorizing G modulo the equivalencerelation ρ, which identifies all the translates of G whose union is G. Notice that ρidentifies the points P1, P2 and P3.

Another way to obtain the Brownian motion on G is to consider the limit processof random walks on the vertices of An, where the transition probabilities in eachpoint is 1

4 to any neighbouring point, and the transition probabilities for leaving

the vertices of A0 is 12(this is in direct correspondence to the equivalence relation

constructed above). This actually is the approach of Lindstrøm. The Laplacianis again defined as the infinitesimal generator of this process. The main results ofLindstrøm are concerned with the asymptotic behaviour of the eigenvalues of theLaplacian.

A completely different approach is due to a Japanese school. In [Ki] Kigamiconsiders finite difference operators, so called harmonic differences on Fn. TheLaplacian then is defined as a certain limit of these operators. The key idea is touse the step by step construction of the graphs to investigate the “evolution” of theeigenvalues (cf. [Sh]). Kigami starts with a detailed investigation of harmonic func-tions on the Sierpinski gasket and its N -dimensional generalizations. Fukushimaand Shima [FS] use this approach in order to develop a precise spectral analysis. Asurvey on this developments can be found in the monographs [DK] and [EI].

In classical papers on uniformly distributed sequences and functions the distri-bution behaviour of the trajectories of the Brownian motion on the p-dimensionaltorus Rp/Zp were analyzed (cf. [St], where the one-dimensional case is considered).W. Fleischer [Fl] has considered the p-dimensional case, and later on in [BDT] thisproblem could be settled in the case of Brownian motion on Riemannian manifolds.In [BDT] a law of iterated logarithm for the discrepancy is proved by applying a


general technique due to W. Philipp [Ph] and bounds for the eigenvalues of theLaplace-Beltrami-operator on the manifold.

In section 2 we develop the basic properties of uniformly distributed sequenceson the Sierpinski gasket. We define a natural metric and introduce various conceptsof discrepancy and obtain inequalities comparing these discrepancies. We discussspecial sequences including irregularities of distribution. Furthermore we proveexplicit formulæ for the Hausdorff measure of certain triangles contained in theSierpinski gasket. For related subsets of G this measure was computed in [Gr] asan application of summation formulæ for special q-multiplicative arithmetic func-tions. This is a consequence of the digital description of the gasket, which we havepresented above. In section 3 we conclude by proving a law of iterated logarithm forthe discrepancy of the trajectories of the Brownian motion on the compact gasketG using a method of Blumlinger [Bl].

2. Uniform distribution on the gasket

2.1. The geodesic metric on G.G is a compact space the topology of which is induced by the following metric

d. Any two points a and b in G are contained in elementary triangles of level k,∆k(a), ∆k(b), respectively. Let ak, bk be the lower left vertices of ∆k(a), ∆k(b)respectively, and observe that ak and bk are vertices of the finite graph Fk. We set

d(a, b) = limk→∞

2−kdk(ak, bk),

where dk is the minimal length of a chain connecting ak and bk. Obviously d(a, b)is the geodesic distance of a and b, i.e. the length of the shortest continuous curvein G connecting a and b. This distance has already been used in [BP].

Proposition 1. Let a and b be two points in G given by their digital representation

a = (ε(i)ℓ ), b = (δ

(i)ℓ ), i = 1, 2, 3, ℓ = 1, 2, . . . . Let L be the first index such that the

triples (ε(i)L ) and (δ

(i)L ) are distinct and define the indices i and j by ε

(i)L = 0 and

δ(j)L = 0. Then the distance of a and b is given by

∞∑

ℓ=L

2−ℓ(

ε(j)ℓ + δ

(i)ℓ − 1

)

.

(The formula does not depend on different representations of the same points.)

Proof. Assume first that a and b are contained in one elementary triangle of level1. Blowing up this triangle by a factor 2 yields d(a, b) = 1

2d(a, b), where a and

b are the homothetic images of a and b. This procedure can be continued as faras these iterated homothetic images of a and b lie in two different elementarytriangles of level 1. This happens after L − 1 iterations. In this case we haved(a, b) = 1

2d(a, Pj) +12d(b, Pi). Thus we only have to compute the distances of

a given point to one of the points P1, P2, P3. Observing that d(p, Pm) = km(p)(m = 1, 2, 3 see Remark 1) and d(p, Pm) = 2km(p) − 1 for km(p) ≥ 1

2 , we obtaind(a, b) = kj(a) + ki(b) − 1. Inserting the digital representations of Remark 1 weobtain the desired result. �


Remark 2. Let p ∈ G be a point different from Pi, i = 1, 2, 3 and let ε > 0 besufficiently small. Then the ε-ball

B(p, ε) = {x ∈ G | d(x, p) < ε}

consists of two congruent equilateral triangles (intersected with G) with one commonvertex. Obviously, B(Pi, ε) consists of one triangle. Thus the metric d induces thetopology of the gasket.

Since G is a compact metric space the general theory of uniform distribution (cf.[KN]) can be applied. A sequence (xn) of points in G is called uniformly distributed(with respect to the Hausdorff measure µ) if

(2.1) limN→∞

1

N

N∑

n=1

f(xn) =

∫

G

f(x)dµ(x)

holds for all continuous functions f on G. By [KN, Theorem 1.2, p. 175] (xn) isuniformly distributed if and only if (2.1) is satisfied for all functions f = χM , whereM is a Borel set with negligable boundary. In order to describe the distributionbehaviour more precisely we introduce several concepts of discrepancy.

2.2. Several Notions of Discrepancy.

Let D be some system of Borel sets A, such that the boundary of A is a null set.Then the discrepancy of a sequence (xn) with respect to D is defined by

(2.2) DN (xn) = DDN (xn) = sup

A∈D

∣

∣

∣

∣

∣

1

N

N∑

n=1

χA(xn)− µ(A)

∣

∣

∣

∣

∣

,

where χA is the characteristic function of the set A. Of special interest are discrep-ancy systems D which are “nice” from a topological or geometric point of view.

The first system we want to consider is the system B of all balls B(p, ε) withp ∈ G and ε > 0. We will call the corresponding discrepancy ball discrepancy.

We next introduce the gasket discrepancy: Let G be the system of all sets whichare intersections of G with triangles the sides of which are parallel to the sides ofA0 and whose vertices are elements of G and define DG

N as in (2.2). Furthermorewe consider the star discrepancy DS

N , which is defined via the discrepancy system Sconsisting of triangles of G that have one side in the boundary of A0 (see Figure 2).Finally we introduce the elementary discrepancy DE

N . In this case the supremumin (2.2) is extended over all elementary triangles.

In the following we establish some easy relations between these four types ofdiscrepancy. Let y ∈ G be arbitrary and let ∆1(y), ∆2(y) and ∆3(y) be threetriangles as defined in Figure 2. Note that one side of ∆i is a part of the side ei of


the equilateral triangle A0 (i = 1, 2, 3).

∆

∆ ∆

1

3 2

1

3 2

1

2 3

(y)

(y) (y)

e

e e

P

P P

y

Figure 2

We introduce three discrepancy functions of the sequence (xn):

(2.3) D(i)N (xn, y) =

1

N

N∑

n=1

χ∆i(y)(xn)− µ(∆i(y)), for i = 1, 2, 3.

Furthermore we define the corresponding discrepancies of a given sequence (xn)

(2.4) D(i)N (xn) = sup

y∈G

∣

∣

∣D

(i)N (xn, y)

∣

∣

∣.

The measures µ(∆i(p)) can be computed explicitely using the digital representationof Remark 1. This is a generalization of a result given in [Gr].

Proposition 2. Let p be a point in G given by its representation (k1, k2, k3) as inRemark 1

ki =

∞∑

ℓ=1

ε(i)ℓ

2ℓ.

Then the Hausdorff measure of ∆1(p) is given by

∞∑

ℓ=2

3−ℓ

ℓ−1∑

n=1

ε(2)n ε(3)n

((

1 + ε(2)n+1

)

· · ·(

1 + ε(2)ℓ−1

)

ε(2)ℓ +

(

1 + ε(3)n+1

)

· · ·(

1 + ε(3)ℓ−1

)

ε(3)ℓ

)

.

Proof. We note first that ∆1(p) = {q = (m1, m2, m2) | m2 ≤ k2, m3 ≤ k3}. In orderto compute the Hausdorff measure of this set we consider its finite approximations


by elementary triangles and count their numbers. Let ΦN be the digital functiongiven by

ΦN

(

ε(2)1 , . . . , ε

(2)N , ε

(3)1 , . . . , ε

(3)N

)

= #

{

(

δ(2)1 , . . . , δ

(2)N , δ

(3)1 , . . . , δ

(3)N

)

|

(

δ(i)1 , . . . , δ

(i)N

)

≤(

ε(i)1 , . . . , ε

(i)N

)

for i = 2, 3 and δ(2)n + δ(3)n > 0 for n ≤ N

}

.

An easy observation shows the following recurrence relation(2.15)

ΦN

(

ε(2)1 , . . . , ε

(2)N , ε

(3)1 , . . . , ε

(3)N

)

= ΦN−1

(

ε(2)2 , . . . , ε

(2)N , ε

(3)2 , . . . , ε

(3)N

)

+

ε(2)1 ε

(3)1

(

ΦN−1

(

ε(2)2 , . . . , ε

(2)N , 1, . . . , 1

)

+ ΦN−1

(

1, . . . , 1, ε(3)2 , . . . , ε

(3)N

))

.

By inserting special values we get

ΦN (1, . . . , 1, ε1, . . . , εN ) = (1 + ε1)ΦN−1 (1, . . . , 1, ε2, . . . , εN )+

ε1ΦN−1(1, . . . , 1, 1 . . . , 1).

Inserting ΦN (1, . . . , 1, 1 . . . , 1) = 3N into this equation yields

Φ(1, . . . , 1, ε1, . . . , εN ) = (1 + ε1) · · · (1 + εN ) +

N∑

k=1

(1 + ε1) · · · (1 + εk−1)εk3N−k.

Inserting this into (2.15) yields the following explicit formula

ΦN

(

ε(2)1 , . . . , ε

(2)N , ε

(3)1 , . . . , ε

(3)N

)

=

N∑

n=1

ε(2)n ε(3)n

(

(

1 + ε(2)n+1

)

· · ·(

1 + ε(2)N

)

+

N−n∑

k=1

(

1 + ε(2)n+1

)

· · ·(

1 + ε(2)n+k−1

)

ε(2)n+k3

N−n−k +(

1 + ε(3)n+1

)

· · ·(

1 + ε(3)N

)

+

N−n∑

k=1

(

1 + ε(3)n+1

)

· · ·(

1 + ε(3)n+k−1

)

ε(3)n+k3

N−n−k

)

.

We use this formula and the fact that

µ(∆1(p)) = limN→∞

3−NΦN

(

ε(2)1 , . . . , ε

(2)N , ε

(3)1 , . . . , ε

(3)N

)

to obtain the desired result. �

We note that any triangle with sides parallel to the sides of A0 can be representedas set-theoretic sum or difference of at most six triangles of types ∆1,∆2,∆3. Thuswe have

(2.5) DSN (xn) ≤ DG

N (xn) ≤ 6DSN (xn).


In order to compare the elementary and the gasket discrepancy we have to esti-mate how many elementary triangles are necessary to approximate a given gaskettriangle T contained in G. For this purpose we define Pn as the union of all elemen-tary triangles of level n, which are contained in T . Observe now, that Pn+1 \ Pn

consists of elementary triangles of level n + 1 which are contained in elementarytriangles of level n which intersect the boundary of T . The number of elementarytriangles of level n, which intersect the boundary of T is at most 3 · 2n. Thus Tcan be approximated by a union of 6 · 2n elementary triangles of level ≤ n with anerror of at most 6 · ( 2

3)n. From this we derive the inequality

DGN ≤ 6 · 2m

(

DEN + 3−m

)

for any m. By inserting m = [log31

DEN] we get

(2.6) DEN ≤ DG

N ≤ 24(

DEN

)

α−1α ,

where the left inequality is obvious.Finally we compare the ball discrepany with the elementary discrepancy. For

this purpose we observe that any elementary triangle ∆ of sidelength 2−k can beexhausted by balls by the following procedure: Take the midpoint p of an edge of ∆and consider the ball B(p, 2−k−1). Then ∆ \B(p, 2−k−1) is an elementary triangleof sidelength 2−k−1 and we can iterate the procedure. Thus we need K balls toapproximate an elementary triangle with accuracy µ(∆)3−K . This yields

DEN ≤ KDB

N + 3−K

for any integer K ≥ 0. We set K = [log31

DBN

] to obtain

(2.7) DEN ≤ DB

N

(

log31

DBN

+ 3

)

.

In order to obtain an inequality in the opposite direction we have to exhaust a givenball by elementary triangles. The procedure to do this is quite the same as in theproof of (2.6) and yields

(2.8) DBN ≤ 72

(

DEN

)

α−1α .

Remark 3. For any triangle ∆ ∈ G and arbitrary k ∈ N there exist two triangles∆′ and ∆′′ with vertices in Vk such that ∆′ ⊆ ∆ ⊆ ∆′′ and µ(∆′′ \∆′) = O(( 23)

k).Since Vk contains only finitely many points and µ(∆) is a continuous function ofthe vertices of ∆, compactness and uniform continuity immediately yield

limN→∞

DGN (xn) = 0

if and only if (xn) is uniformly distributed in G. The inequalities (2.5–8) imply thatthis holds for all the notions of discrepany discussed above.


2.3. Lp-Discrepancy.

We introduce the Lp-discrepancy of (xn) for arbitrary p ≥ 1:

(2.9) L(p)N =

(

∫ 1

0

∫

G

∣

∣

∣

∣

∣

1

N

N∑

n=1

χB(y,r)(xn)− µ(B(y, r)

∣

∣

∣

∣

∣

p

dµ(y) dr

)

1p

.

Obviously L(p)N ≤ DB

N (xn). In order to prove an opposite inequality we sketch aprocedure used in [Ti] to derive a general inequality of this type on compact metricspaces endowed with a Borel probability measure. This is a more general versionof an inequality between the usual discrepancy on [0, 1)s and the correspondingLp-discrepancy proved in [NTT].

Theorem 1. Let (X, d) be a compact metric space and λ and ζ be two Borelprobability measures on X, where λ satisfies the following additional conditions

|λ(B(x, r1))− λ(B(x, r2))| ≤ L1|r2 − r1|β

|λ(B(x1, r))− λ(B(x2, r))| ≤ L2d(x1, x2)β

λ(B(x, r) ≥ L0rs.

Then the discrepancy function D(y, r) = ζ(B(y, r)) − λ(B(y, r)) satisfies the fol-lowing inequality

∫

X

∫ ϑ

0

ϕ (|D(y, r)|) dr dλ(y) ≥ c‖D‖s+1β

∞ ϕ

(

1

6‖D‖∞

)

for any increasing function ϕ on [0, 1], where c is a positive constant only dependingon X,L0, L1, L2, β and s. ϑ denotes the diameter of X.

Corollary 1. Let L(p)N denote the Lp-discrepancy defined in (2.9). Then the fol-

lowing inequality holds

L(p)N ≥ c

(

DBN

)

α+1(α−1)p

+1

for a suitable positive constant c depending only on p. Thus limN→∞

L(p)N (xn) = 0 is

equivalent with the uniform distribution of the sequence (xn).

Sketch Proof of Theorem 1. Let D = ‖D‖∞. Then for any ε > 0 there exists a pair(x0, r) ∈ X×R

+ such that |D(x0, r)| > D−ε. We set ϑ(x0) = supy∈X d(x0, y) andshow that

m(a) := supr∈[a,ϑ(x0)−a]

|D(x0, r)| ≥1

3D − 2

3L1a

β

for arbitrary 0 < a < 12ϑ(x0). We choose

a = min

(

(

1

2ϑ(x0)

)1β

,

(

4L1 + 6L1 + L2

2β

)− 1β

,1

2ϑ(x0)

)

D1β

and take an r0 ∈ [a, ϑ(x0)− a] such that

|D(x0, r)| ≥1

3D − 2

3L1a

β − ε


for arbitrary ε > 0.For D(x0, r0) > 0 we have B(y, r) ⊇ B(x0, r0) for every y ∈ B(x0,

a4) and every

r ∈ [r0 +a2 , r0 + a] =: Ia. Thus we have (using the monotonicity of ζ(B(y, r)))

ζ(B(y, r))− λ(B(y, r)) ≥ ζ(B(x0, r0))− λ(B(y, r0))− L1(r − r0)β ≥

ζ(B(x0, r0))− λ(B(x0, r0))− L1(r − r0)β − L2

(a

4

)β

≥

D(x0, r0)− (L1 + L2)(a

4

)β

,

and by the choice of x0, r0 and a we derive

(2.10) |D(y, r)| ≥ 1

6D.

For D(x0, r0) < 0 we have B(y, r) ⊇ B(x0, r0) for every y ∈ B(x0,a4 ) and every

r ∈ [r0 − a2, r0

a4] =: Ia. Thus we have |D(y, r)| ≥ 1

6D.

Combining (2.10) and the last condition on the measure λ yields

∫

X

∫ ϑ

0

ϕ (|D(y, r)|) dr dλ(y) ≥∫

B(x0,a4 )

∫

Ia

ϕ

(

1

6D

)

dr dλ(y) ≥

L0

(a

4

)s+1

ϕ

(

1

6D

)

,

which (by the choice of a) gives the desired result. �

Proof of Corollary 1. In order to prove the Corollary we notice that

ζ(E) =1

N

N∑

n=1

χE(xn)

is a Borel measure, and λ = µ satisfies the conditions of Theorem 1 with β = α− 1and s = α and some suitable constants L0, L1, L2. �

2.4. Special Sequences and Irregularities of Distribution.

Obviously we have

(2.11)1

N≤ DN (xn) ≤ 1

for the four discrepancy systems under consideration in section 2.2. We note herethat in the case of the elementary discrepancy it is possible to find sequences (xn)such that NDE

N (xn) is bounded. Such sequences can be compared with the well-known net-sequences in the unit cube. We remark here that these net-sequenceshave recently been used for various applications in quasi Monte Carlo methods(cf. [Ni]). For the other two notions of discrepancy there is the phenomenon ofirregularities of distribution.

In the following we want to describe a gasket-analogon of the well-known vander Corput sequence γ = (γn). For this purpose we note that the digital expansiondescribed in Remark 1 can also be given as follows: let δℓ ∈ {0, 1, 2} be the index i


mod 3 such that ε(i)ℓ = 0. Then every point in the gasket can be encoded (not nec-

essarily uniquely) as an infinite triadic string (cf. [Cu]). For defining the sequenceγ we expand every integer n in triadic expansion

n =L∑

ℓ=0

δℓ+1(n)3ℓ

and define γn as the point encoded by (δ1, δ2, . . . , δL, 0∞).

Remark 4. Note that any elementary triangle of level k corresponds to a residueclass mod 3k. Thus the elementary discrepancy is O( 1

N). By (2.5), (2.6) and

(2.8) we immediately derive

(2.12) DSN (γ), DG

N(γ), DBN(γ) = O

(

1

Nα−1α

)

.

By a standard technique due to W. Philipp [Ph] the average rate of growth ofthe discrepancy of an arbitrary sequence xn ∈ G can be determined.

Proposition 3. The following law of iterated logarithm holds for D = G,S,B

(2.13) lim supN→∞

DDN (xn)

√N√

2 log logN=

1

2

for almost all (with respect to the infinite product measure generated by µ) sequenceson G.

Remark 5. For the elementary discrepancy a similar law of the iterated logarithm

can be shown by much simpler arguments; the constant 12 has to be replaced by

√23 .

The example of van der Corput sequence shows that there is a gap between

the lower bound 1N

and the upper bound O(

1

Nα−1α

)

. A simple application of

W. Schmidt’s theorem on irregularities of distribution [Sch] yields

Proposition 4. Let xn be a sequence in G. Then

DSN (xn) ≥ c

logN

N

holds for infinitely many N (where c > 0 denotes an absolute constant).

Proof. For the sequence xn we consider the sequence k1(xn) ∈ [0, 1] (cf. Remark 1).Note that for uniformly distributed xn, k1(xn) has the asymptotic distributionfunction F (x) = µ(∆2(p)), where p is given by k1(p) = x and k3(p) = 1, seeProposition 2 and [Gr]. Clearly this function is continuous and strictly increasing.Applying Schmidt’s lower bound to the sequence F−1(k1(xn)) yields

DSN (xn) ≥ D∗

N

(

F−1 (k1(xn)))

≥ 1

66 log 4

logN

N,

where D∗N denotes the usual star-discrepancy in the unit interval. �


Remark 6. Clearly, this is a very weak bound, since we have used only very specialgasket triangles to derive this inequality. It remains as an interesting open problemto improve this lower bound. Since we have no natural group structure it seems tobe very hard to apply Beck’s Fourier transform approach to the gasket.

Concluding this section we present a probabilistic approach for constructing aset ΓN of N points in G with small discrepancy.

Theorem 2. For every positive integer N > 1 there exists a point set ΓN consistingof N points such that

DBN (ΓN ) ≤ cN

12α−1 (logN)

12 ,

where c > 0 is an absolute constant.

Proof. In order to prove this theorem we use Beck’s probabilistic approach [BC].We define N sets Q1, . . . , QN as follows: let k be the uniquely determined integersuch that 3k−1 < N ≤ 3k and take Q1, . . . , QN1

as elementary triangles of levelk, QN1+1, . . . , QN1+N2

as the union of two elementary triangles of level k andQN1+N2+1, . . . , QN1+N2+N3

as the union of three elementary triangles of level k,whereN1+N2+N3 = N andN1+2N2+3N3 = 3k. We chooseN3 = max(3k−2N, 0),N2 = 3k −N − 2N3 and N1 = 2N − 3k +N3 (these values are all non-negative).

Let Z1, . . . , ZN be random variables such that Zn is uniformly distributed onQn (with respect to µ), n = 1, . . . , N . We observe that

Bℓ ={

B(x, r) | x ∈ Vℓ, r =m

2ℓ, m = 0, . . . , 2ℓ

}

has the property that for any ball B(x, r) there exist two balls B′, B′′ ∈ Bℓ suchthat B′ ⊆ B ⊆ B′′ and µ(B′′ \ B′) ≪ ( 23 )

ℓ. Furthermore #Bℓ ≤ 2 · 6ℓ. Now wecompute the expected value of the random variables Xn(S) = χS(Zn) for a ball

S ∈ Bℓ for some ℓ, which will be chosen later. Clearly EXn(S) =µ(S∩Qn)µ(Qn)

. Thus

we obtain that Xn(S) ≡ EXn(S) if S ∩ Qn = ∅ or S ∩ Qn = Qn. As in the proofof (2.6) we have

(2.14) #{n | ∅ 6= S ∩Qn 6= Qn} ≪ N log3 2,

where the implied constant is absolute. By [BC, Lemma 8.2] we derive

Prob

(∣

∣

∣

∣

∣

N∑

n=1

(Xn(S)− EXn(S))

∣

∣

∣

∣

∣

≥ γ

)

≤ 2 exp(

−Cγ2N− log3 2)

.

Setting γ = C′N12 log3 2−1

√logN and ℓ = [ 2 log 3−log 2

2(log 3−log 2) log 3 logN ], a suitable choice

of C′ yields

Prob

(∣

∣

∣

∣

∣

1

N

N∑

n=1

χS(Zn)− µ(S)

∣

∣

∣

∣

∣

≥ C′N12 log3 2−1

√

logN, for some S ∈ Bℓ

)

≤ 1

2.

Thus there exists a point set ΓN satisfying the bound given in Theorem 2. �


Remark 7. The main ingredient of the proof is the approximation of the discrep-ancy system by a finite system. Thus an analogous theorem can be proved for DS

and DG .

2.5. Uniform Distribution of Curves.

We recall here the definition of discrepancy for continuous functions x(t) on thegasket:

(2.15) DT (x(t)) = DDT (x(t)) = sup

A∈D

∣

∣

∣

∣

∣

1

T

∫ T

0

χA(x(t))dt− µ(A)

∣

∣

∣

∣

∣

.

If we interpret x(t) as the motion of a particle on the gasket, the discrepancy canbe considered as the deviation of the mean with respect to time and the spatialmean. The motion is called equidistributed if limT→∞DT (x(t)) = 0. The theoryof uniform distribution for continuous functions was developed in a series of papersby Hlawka [Hl], Kuipers et al. (cf. [KN]). We consider all discrepancy systems Dintroduced above. An application of a general result in [DT] yields

Proposition 5. Let D be one discrepancy systems considered in section 2.2 andx(t) a continuous function R

+0 → G with finite arclength s(T ) and lim

T→∞s(T ) = ∞.

Then there exists a constant c(D) > 0 such that

DDT (x(t)) ≥ c(D)

(

1

s(T )

)α

α−1

for T ≥ T0.

Remark 8. Note that the arclength of a continuous function x(t) is defined by

s(T ) = sup

N−1∑

n=0

d(x(tn), x(tn+1)),

where the supremum is extended over all partitions 0 = t0 < t1 < . . . < tN = T .

Remark 9. The proof is verbally the same as the proof of Theorem 1 in [DT]. Weonly note, that in the technical condition (2.2) in [DT] it is not necessary to takeballs B(x, r) and B(y, R) with the same center x = y.

Remark 10. Obviously the general inequalities (2.5–8) remain valid for the con-tinuous versions of the different kinds of discrepancies.

3. A uniform law of iterated logarithm for Brownian motion on G

Our aim is the generalization of the law of iterated logarithm (2.13) to thetrajectories of Brownian motion. For Brownian motion on manifolds similar resultscan be found in [BDT] and [Bl].

In the introduction we have defined Brownian motion on G as a limit process of adiscrete random walk. Barlow and Perkins [BP] have shown for the corresponding

process on the infinite gasket G that this is a symmetric Markov process withjointly continuous transition densities p(t, x, y) on [0,∞) × G × G. Furthermore

t 7→ p(t, x, y) is C∞, and for all t > 0, x, y ∈ G the following estimate holds

(3.1)c1t

log 3log 5 exp

(

−c2d(x, y)log 5

log 5−log 2 t−log 2

log 5−log 2

)

≤ p(t, x, y) ≤

c3tlog 3log 5 exp

(

−c4d(x, y)log 5


log 5−log 2

)


with suitable positive constants c1, c2, c3, c4.As described in the introduction the Brownian motion on G by factorizing G

modulo a suitable equivalence relation ρ. Thus we can decompose G as⋃

k TkG,where the Tk are translations, which originate from the definition of ρ. Since

p(t, x, y) =∑

k

p(t, x, Tky),

we have to combine the estimates (3.1) in order to give upper and lower bounds.

It follows from [Gr] that the number of copies of G in G whose points p have adistance ℓ ≤ d(p, 0) ≤ ℓ+ 1 is 2s(ℓ)+1, where s(ℓ) denotes the binary sum-of-digitsfunction. Therefore we get the bounds

(3.2)

γt = 2c1tlog 3log 5

∞∑

ℓ=0

2s(ℓ) exp(

−c2(ℓ+ 1)log 5


log 5−log 2

)

≤

p(t, x, y) ≤ γ′t = 2c3tlog 3log 5

∞∑

ℓ=0

2s(ℓ) exp(

−c4ℓlog 5


log 5−log 2

)

.

We want to prove that the functions γt and γ′t are bounded from above and below

by positive constants for t ≥ 1. In order to prove this and to estimate γt from belowwe apply partial summation to the first sum in (3.2). This yields

γt = 2c1tlog 3log 5

∞∑

ℓ=1

ℓ−1∑

k=0

2s(k)×

×(

exp(

−c2ℓlog 5


log 5−log 2

)

− exp(

−c2(ℓ+ 1)log 5


log 5−log 2

))

.

From [Ha], [FGKPT] and [Gr] we know that

1

2N log2 3 ≤

N−1∑

n=0

2s(n) ≤ N log2 3,

which implies(3.3)

γt ≥ c1tlog 3log 5

∞∑

ℓ=1

ℓlog2 3×

×(

exp(

−c2ℓlog 5


log 5−log 2

)

− exp(

−c2(ℓ+ 1)log 5


log 5−log 2

))

≥

c1tlog 3log 5

∞∑

ℓ=1

ℓlog2 3−1 exp(

−c2ℓlog 5


log 5−log 2

)

.

The last sum has been studied by Ramanujan (cf. [Be]). For estimating this sumwe apply the Mellin transform to the sum

f(u) =

∞∑

n=1

nlog2 3−1 exp(

−c2nlog 5

log 5−log 2 u)

,


which yields

f∗(s) =

∫ ∞

0

f(u)us−1 du = ζ

(

log 5

log 5− log 2s− log2 3 + 1

)

Γ(s).

By the well-known correspondence between the singularities of the transform and

the asymptotic behaviour of the function we obtain f(u) ∼ c5ulog 5(log 5−log 2)

log 2 log 5 with

some explicit constant c5. Inserting this into (3.3) (u = t−log 2

log 5−log 2 ) and applyingthe same procedure to γ′t we derive

(3.4) 0 < c6 ≤ p(t, x, y) ≤ c7 for t ≥ 1.

Now we introduce a gasket analogon of the classical Wiener measure. Let Cw bethe set of all continuous curves in G starting in a given point w ∈ G. Then for fixed0 < t1 < t2 < . . . < tn and a Borel set E ⊆ Gn the corresponding Wiener measureis given by

(3.5)

µw ({x ∈ Cw | (x(t1), x(t2), . . . , x(tn)) ∈ E}) =∫

E

p(tn − tn−1, xn, xn−1) · · ·p(t2 − t1, x2, x1)p(t1, x1, w) dµ(x1) · · ·dµ(xn).

Proposition 6. Let P (t, x, S) =∫

Sp(t, x, y)dµ(y) be the transition probabilities of

Brownian motion on G. Then

(3.6) |P (t, x, S)− µ(S)| ≤ Ae−at

(3.7) ‖P (t, x, dy)− dµ(y)‖ ≤ 2Ae−at,

with positive constants A, a, independent of x and the measurable set S. (‖.‖ denotesthe uniform norm with respect to x.)

Proof. From [Do, p. 197] it follows that there exists a measure νt such that

(3.8)∣

∣P (nt, x, S)− νt(S)∣

∣ ≤ (1− γt)n−1

for t > 0. Applying the Chapman-Kolmogorov equation we obtain

(3.9)

∫

G

P (nt, y, S)P (ns, x, dy) = P (n(s+ t), x, S).

For n → ∞, P (nt, x, S) converges to νt(S) uniformly in x ∈ G by (3.8). Thusthe integral in (3.9) converges to νt(S), whereas the right hand side converges toνs+t(S). Hence ν = νt is independent of t.

Next we identify ν as the Hausdorff measure µ. As p(t, x, y) is bounded byan absolute constant ν is absolutely continuous with respect to µ. Let f be thedensity of ν with respect to µ. By the above arguments f is essentially bounded.Let q(t, x, y) be the transition density with respect to the measure ν, i.e. q(t, x, y) =p(t, x, y)f(y). We want to show that f ≡ 1 and proceed indirectly, assuming thatf is non-constant. Let now C be the essential supremum of f and set

Aε = {x | f(x) < C − ε} .


For ε > 0 we have

q(t, x, y0)− q(t, y0, x) = p(t, x, y0)(f(y0)− f(x)) > 0 for x ∈ Aε and y0 ∈ ACε .

Next we choose ε so small that µ(Aε) > 0. Thus we obtain∫

ACε

(q(t, x, y0)− q(t, y0, x))dν(x) ≤ 0

and∫

Aε

(q(t, x, y0)− q(t, y0, x))dν(x) > 0.

Since µ(ACε ) → 0 for ε→ 0 we have

∫

G

(q(t, x, y0)− q(t, y0, x))dν(x) > 0.

On the other hand the integral on the left hand side is 0, which is a contradiction.Thus f ≡ 1 and the two measures µ and ν are equal.

It follows from (3.4) and (3.8) that

|P (t, x, S)− µ(S)| ≤ e−at+a

with a = − log(1 − γ1). Setting A = ea yields (3.6). The estimate (3.7) is animmediate consequence of the Hahn decomposition theorem. �

Let I be a time interval and FI the σ-algebra generated by events in I. A processis called ϕ-mixing if

(3.10) |P (E2 | E1)− P (E2)| ≤ ϕ(t),

and ψ-mixing if

(3.11) |P (E2 | E1)− P (E2)| ≤ P (E2)ψ(t)

for events E1, E2, with E1 being F[0,s]-measurable, E2 being F[s+t,∞]-measurableand ϕ(t) → 0 , ψ(t) → 0 as t→ ∞.

Proposition 7. The Brownian motion on G has the ψ-mixing property with ψ(t) =Ke−at for t ≥ 1. It satisfies the ϕ-mixing property for t ≥ 0 with ϕ(t) = K ′e−at.

Proof. By the Chapman - Kolmogorov equation and (3.7) we obtain

(3.12) |p(s+ t, x, y)− 1| ≤ 2γ′sAe−at.

Using the Markov property, a simple computation yields for t ≥ 1

|P (E2 | E1)− P (E2)| ≤ P (E2)Ke−at,

where K =4γ′

1/2A

infx,y∈G p(1,x,y) . Since |P (E2 | E1) − P (E2)| ≤ 1, the second assertion

follows immediately. �

Next we approach our main result, a uniform law of the iterated logarithm. Itfollows from a general result of W. Philipp [Ph] and is a gasket analogon of Theorem4 in [Bl]. As usual we will use the notation E(f) =

∫

Gf(x)dµ(x).


Theorem 4. Let θ be a positive integer and let Aθ be a family of real - valueduniformly bounded measurable functions on G such that 1 ≤ #Aθ ≤ eθk1 with someconstant k1. Let A be the set of all functions on G, uniformly bounded by 1 andhaving the following approximation property:

For all f ∈ A there exists a sequence of functions hθ, hθ ∈ A such that for allpositive integers L

L∑

θ=1

hθ ≤ f ≤L∑

θ=1

hθ

‖L∑

θ=1

(hθ − hθ‖1 ≤ e−k2L,

where k2 is a positive constant.Let w be a given point in G. Then for arbitrary ǫ > 0 and for µw-almost all

curves x(t) in Cw there exists a T0 > 0 such that

∣

∣

∣

∣

∣

∫ T

0f(x(t))dt− TE(f)√

2T log logT

∣

∣

∣

∣

∣

< σ(f) + ǫ

for all f ∈ A and all T > T0. Furthermore, for µw-almost all curves x(t) in Cw

lim supT→∞

∣

∣

∣

∣

∣

∫ T

0f(x(t))dt− TE(f)√

2T log logT

∣

∣

∣

∣

∣

= σ(f)

holds uniformly for all f ∈ A.

Sketch Proof. We use the notation Xn =∫ n

n−1f(x(t))dt. Since the functions in A

are uniformly bounded we may consider integer values for T only. Furthermore, w.l. o. g. we may assume that

∫

Gfdµ = 0 , E(Xn(f)) = 0 for all positive integers

n. The theorem is an immediate consequence of theorems 1.3.1, 1.3.2 in [Ph] andProposition 7, if we can verify the following conditions (3.13), (3.14), (3.15):

(3.13) E(N∑

n=1

Xn(f))2 = Nσ2(f) +O(N)

(3.14) σ2(f) = O(1)

(3.15) E(N∑

n=1

Xn(hθ))2 = O(Ne−k1θ),

where the O-constants are absolute ones.(3.13) and (3.14) follow from Proposition 6 and (3.12) after some standard calcu-

lations and estimates, see [Bl]. (3.15) is a direct consequence of the approximationproperty stated in the theorem. �


Remark 11. As in [Bl, Theorem 5] it can be shown that 0 < σ(f) < ∞ for allnon-zero f ∈ L∞.

Corollary 2. The following law of iterated logarithm holds for the discrepancysystems D = G,S,B

lim supN→∞

DDN (x(t))

√N√

2 log logN= σ

for µw-almost all functions x(t) ∈ Cw, where w is a given point in G and σ is somepositive constant.

Proof. As in section 2.2 the exponential approximation property can be derived forall discrepancy systems. �

Acknowledgements. We are indebted to Martin Blumlinger and WolfgangWoess for valuable discussions and for providing some recent references.

Note added in proof: The first author has studied further properties of theBrownian on the Sierpinski gasket and the random walk on the Sierpinski graph intwo subsequent papers, one of them jointly with W. Woess (to appear in Mathe-matika and Stochastic Processes Appl.).

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Institut fur Mathematik

Technische Universitat Graz

Steyrergasse 30

8010 Graz, Austria

E-mail address: [email protected] [email protected]

Equidistribution and Brownian motion on the Sierpi?ski gasket

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