EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPI ...

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 Sierpi´ nski gasket. Furthermore a law of iterated logarithm for the discrepancy of trajectories of Brownian motion is proved. The main tools for this result are regularity properties of the heat kernel on the Sierpi´ nski gasket. Some of the results can be generalized to arbitrary nested fractals in the sense of T. Lindstrøm.

1. Introduction As a starting point we consider the Sierpi´ nski gasket, a well known planar fractal set introduced by W. Sierpi´ nski [Si]. Let A0 be a closed equilateral triangle of unit √ 3 1 sides e1 , e2 , e3 with vertices P1 ( 2 , 2 ), P2 (0, 0), P3 (1, 0). Let A1 be the set obtained by deleting the open equilateral triangle whose vertices are the midpoints of the edges of A0 . Thus A1 consists of 3 equilateral triangles with side 21 . Repeating this procedure we obtain successively A2 , A3 , . . . . An consists of 3n equilateral triangles of side 2−n , which are called elementary triangles of level n. Furthermore we denote the set of all vertices of An by Vn and the boundary of An by En . Thus Fn = (Vn , En ) is defining a finite graph. T∞ Definition 1. The set G = n=0 An is called the (bounded) Sierpi´ nski gasket.

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

∞ (i) X ε ℓ=1

(1)

(2)

(3)

and εℓ + εℓ + εℓ p to the side ei .

ℓ 2ℓ

,

(i)

εℓ = 0

or 1

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

√ 3 2

is just the distance of

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

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PETER J. GRABNER AND ROBERT F. TICHY

Figure 1 By standard techniques as discribed in [Fa] it is easy to see that the Sierpi´ nski log 3 gasket has Hausdorff dimension α = log 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 the Euclidean space were extended to the Sierpi´ nski gasket, the Sierpi´ nski carpet, where the whole potential theory is developed in a series of papers (cf. [BB] for further references) or more generally to so called nested fractals. We want to mention here the fundamental paper of M.T. Barlow and E.A. Perkins [BP], where a systematic theory of Brownian motion onS the Sierpi´ nski gasket is developed. Let F˜ be the infinite graph defined by F˜ = k 2k Fk .

´ EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPINSKI GASKET

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Definition 2. The topological closure of ∞ [

2−k F˜

k=0

˜ is called the infinite Sierpi´ nski gasket G. The Brownian motion is introduced as a suitable limit process of discrete random walks on the vertices of F˜ : Let Yk be the random walk on F˜ with transitition probabilities 41 from each vertex to its neighbour. In order to give a proper definition of the limiting process one has to rescale the time according to the eigenvalues of the transition matrix. Thus we consider the processes X (n) (t) = 2−n Y[5n t]

for t ≥ 0, n ∈ N.

In [BP, Theorem 2.8] it is shown, that the processes X (n) converge weakly to a ˜ process X, where X is a continuous, non-constant, G-valued, strong Markov process starting at O. This process X = Xt can be considered as the Brownian motion ˜ The Laplacian is obtained as the infinitesimal generator of the semi-group on G. describing this process. Among other very interesting results the authors obtain regularity properties of the heat kernel. Lindstrøm [Li] studies Brownian motion on compact nested fractals. In the case of the finite (=compact) Sierpi´ nski gasket G a trajectory of Brownian motion ˜ ˜ modulo the equivalence is obtained from a trajectory on G just by factorizing G ˜ Notice that ρ relation ρ, which identifies all the translates of G whose union is G. identifies the points P1 , P2 and P3 . Another way to obtain the Brownian motion on G is to consider the limit process of random walks on the vertices of An , where the transition probabilities in each point is 41 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 Laplacian is again defined as the infinitesimal generator of this process. The main results of Lindstrøm are concerned with the asymptotic behaviour of the eigenvalues of the Laplacian. A completely different approach is due to a Japanese school. In [Ki] Kigami considers finite difference operators, so called harmonic differences on Fn . The Laplacian then is defined as a certain limit of these operators. The key idea is to use the step by step construction of the graphs to investigate the “evolution” of the eigenvalues (cf. [Sh]). Kigami starts with a detailed investigation of harmonic functions on the Sierpi´ nski gasket and its N -dimensional generalizations. Fukushima and Shima [FS] use this approach in order to develop a precise spectral analysis. A survey on this developments can be found in the monographs [DK] and [EI]. In classical papers on uniformly distributed sequences and functions the distribution behaviour of the trajectories of the Brownian motion on the p-dimensional torus 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] this problem 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

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general technique due to W. Philipp [Ph] and bounds for the eigenvalues of the Laplace-Beltrami-operator on the manifold. In section 2 we develop the basic properties of uniformly distributed sequences on the Sierpi´ nski gasket. We define a natural metric and introduce various concepts of discrepancy and obtain inequalities comparing these discrepancies. We discuss special sequences including irregularities of distribution. Furthermore we prove explicit formulæ for the Hausdorff measure of certain triangles contained in the Sierpi´ nski gasket. For related subsets of G this measure was computed in [Gr] as an application of summation formulæ for special q-multiplicative arithmetic functions. This is a consequence of the digital description of the gasket, which we have presented above. In section 3 we conclude by proving a law of iterated logarithm for the discrepancy of the trajectories of the Brownian motion on the compact gasket G using a method of Bl¨ umlinger [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) = lim 2−k dk (ak , bk ), k→∞

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 curve in 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 (i) (i) a = (εℓ ), b = (δℓ ), i = 1, 2, 3, ℓ = 1, 2, . . . . Let L be the first index such that the (i) (i) (i) triples (εL ) and (δL ) are distinct and define the indices i and j by εL = 0 and (j) δL = 0. Then the distance of a and b is given by ∞ X

ℓ=L

(j) (i) 2−ℓ εℓ + δℓ − 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 level 1. Blowing up this triangle by a factor 2 yields d(a, b) = 21 d(˜ a, ˜b), where a ˜ and ˜b are the homothetic images of a and b. This procedure can be continued as far as these iterated homothetic images of a and b lie in two different elementary triangles of level 1. This happens after L − 1 iterations. In this case we have a, Pj ) + 12 d(˜b, Pi ). Thus we only have to compute the distances of d(a, b) = 21 d(˜ 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) ≥ 12 , we obtain d(a, b) = kj (a) + ki (b) − 1. Inserting the digital representations of Remark 1 we obtain the desired result.


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Remark 2. Let p ∈ G be a point different from Pi , i = 1, 2, 3 and let ε > 0 be sufficiently 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 common vertex. Obviously, B(Pi , ε) consists of one triangle. Thus the metric d induces the topology 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)

Z N 1 X f (xn ) = f (x)dµ(x) lim N→∞ N G n=1

holds for all continuous functions f on G. By [KN, Theorem 1.2, p. 175] (xn ) is uniformly distributed if and only if (2.1) is satisfied for all functions f = χM , where M is a Borel set with negligable boundary. In order to describe the distribution behaviour 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)

N 1 X D χA (xn ) − µ(A) , DN (xn ) = DN (xn ) = sup A∈D N n=1

where χA is the characteristic function of the set A. Of special interest are discrepancy 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) with p ∈ 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 which are intersections of G with triangles the sides of which are parallel to the sides of G A0 and whose vertices are elements of G and define DN as in (2.2). Furthermore S we consider the star discrepancy DN , which is defined via the discrepancy system S consisting of triangles of G that have one side in the boundary of A0 (see Figure 2). E Finally we introduce the elementary discrepancy DN . In this case the supremum in (2.2) is extended over all elementary triangles. In the following we establish some easy relations between these four types of discrepancy. Let y ∈ G be arbitrary and let ∆1 (y), ∆2 (y) and ∆3 (y) be three triangles as defined in Figure 2. Note that one side of ∆i is a part of the side ei of

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

P1

e3

∆3(y)

∆2(y)

e2

y ∆1(y) e1

P2

P3

Figure 2 We introduce three discrepancy functions of the sequence (xn ): (i) DN (xn , y)

(2.3)

N 1 X = χ∆ (y) (xn ) − µ(∆i (y)), N n=1 i

for i = 1, 2, 3.

Furthermore we define the corresponding discrepancies of a given sequence (xn ) (i) (i) (2.4) DN (xn ) = sup DN (xn , y) . y∈G

The measures µ(∆i (p)) can be computed explicitely using the digital representation of 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 in Remark 1 ∞ (i) X εℓ . ki = 2ℓ ℓ=1

Then the Hausdorff measure of ∆1 (p) is given by ∞ X ℓ=2

3

−ℓ

ℓ−1 X

n=1

(3) ε(2) n εn

(3) (3) (2) (2) (3) (2) . 1 + εn+1 · · · 1 + εℓ−1 εℓ + 1 + εn+1 · · · 1 + εℓ−1 εℓ

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


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

(2) ε1 , . . .

(2) (3) , εN , ε1 , . . .

(3) , εN

=#

(

(2) δ1 , . . .

(2) (3) , δN , δ1 , . . .

(3) , δN

|

) (i) (i) (i) (i) (2) (3) δ1 , . . . , δN ≤ ε1 , . . . , εN for i = 2, 3 and δn + δn > 0 for n ≤ N . An easy observation shows the following recurrence relation (2.15) (2) (2) (3) (2) (3) (2) (3) (3) ΦN ε1 , . . . , εN , ε1 , . . . , εN = ΦN−1 ε2 , . . . , εN , ε2 , . . . , εN + (2) (3) (2) (2) (3) (3) ε1 ε1 ΦN−1 ε2 , . . . , εN , 1, . . . , 1 + ΦN−1 1, . . . , 1, ε2 , . . . , εN . By inserting special values we get ΦN (1, . . . , 1, ε1 , . . . , εN ) = (1 + ε1 )ΦN−1 (1, . . . , 1, ε2 , . . . , εN ) + ε1 ΦN−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 X

k=1

(1 + ε1 ) · · · (1 + εk−1 )εk 3N−k .

Inserting this into (2.15) yields the following explicit formula ΦN

(2) ε1 , . . .

N−n X

(2) (3) , εN , ε1 , . . .

(2)

(3)

1 + εn+1

k=1

N−n X k=1

1 + εn+1

(3) , εN

=

N X

n=1

(3) ε(2) n εn

(2) (2) 1 + εn+1 · · · 1 + εN +

(2) (2) (3) (3) N−n−k · · · 1 + εn+k−1 εn+k 3 + 1 + εn+1 · · · 1 + εN +

! (3) (3) · · · 1 + εn+k−1 εn+k 3N−n−k .

We use this formula and the fact that (2) (2) (3) (3) µ(∆1 (p)) = lim 3−N ΦN ε1 , . . . , εN , ε1 , . . . , εN N→∞

to obtain the desired result. We note that any triangle with sides parallel to the sides of A0 can be represented as set-theoretic sum or difference of at most six triangles of types ∆1 , ∆2 , ∆3 . Thus we have (2.5)

S S G (xn ) ≤ 6DN (xn ). DN (xn ) ≤ DN

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In order to compare the elementary and the gasket discrepancy we have to estimate how many elementary triangles are necessary to approximate a given gasket triangle T contained in G. For this purpose we define Pn as the union of all elementary 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 elementary triangles of level n which intersect the boundary of T . The number of elementary triangles of level n, which intersect the boundary of T is at most 3 · 2n . Thus T can be approximated by a union of 6 · 2n elementary triangles of level ≤ n with an error of at most 6 · ( 23 )n . From this we derive the inequality G E DN ≤ 6 · 2m DN + 3−m

for any m. By inserting m = [log3 (2.6)

1 E DN

] we get

E G E DN ≤ DN ≤ 24 DN

α−1 α

,

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 be exhausted 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 triangle of sidelength 2−k−1 and we can iterate the procedure. Thus we need K balls to approximate an elementary triangle with accuracy µ(∆)3−K . This yields E B DN ≤ KDN + 3−K

for any integer K ≥ 0. We set K = [log3 (2.7)

E DN

≤

B DN

1 B DN

] to obtain

1 log3 B + 3 . DN

In order to obtain an inequality in the opposite direction we have to exhaust a given ball by elementary triangles. The procedure to do this is quite the same as in the proof of (2.6) and yields (2.8)

B E DN ≤ 72 DN

α−1 α

.

Remark 3. For any triangle ∆ ∈ G and arbitrary k ∈ N there exist two triangles ∆′ and ∆′′ with vertices in Vk such that ∆′ ⊆ ∆ ⊆ ∆′′ and µ(∆′′ \ ∆′ ) = O(( 32 )k ). Since Vk contains only finitely many points and µ(∆) is a continuous function of the vertices of ∆, compactness and uniform continuity immediately yield G lim DN (xn ) = 0

N→∞

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


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2.3. Lp -Discrepancy. We introduce the Lp -discrepancy of (xn ) for arbitrary p ≥ 1: (2.9)

(p)

LN =

Z

1

0

(p)

p ! p1 Z X N 1 χB(y,r) (xn ) − µ(B(y, r) dµ(y) dr . G N n=1

B Obviously LN ≤ DN (xn ). In order to prove an opposite inequality we sketch a procedure used in [Ti] to derive a general inequality of this type on compact metric spaces endowed with a Borel probability measure. This is a more general version of an inequality between the usual discrepancy on [0, 1)s and the corresponding Lp -discrepancy proved in [NTT].

Theorem 1. Let (X, d) be a compact metric space and λ and ζ be two Borel probability 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))| ≤ L2 d(x1 , x2 )β λ(B(x, r) ≥ L0 r s .

Then the discrepancy function D(y, r) = ζ(B(y, r)) − λ(B(y, r)) satisfies the following inequality Z Z ϑ s+1 1 β kDk∞ ϕ (|D(y, r)|) dr dλ(y) ≥ ckDk∞ ϕ 6 X 0 for any increasing function ϕ on [0, 1], where c is a positive constant only depending on X, L0 , L1 , L2 , β and s. ϑ denotes the diameter of X. (p)

Corollary 1. Let LN denote the Lp -discrepancy defined in (2.9). Then the following inequality holds α+1 (p) B (α−1)p +1 LN ≥ c DN (p)

for a suitable positive constant c depending only on p. Thus lim LN (xn ) = 0 is N→∞

equivalent with the uniform distribution of the sequence (xn ). Sketch Proof of Theorem 1. Let D = kDk∞ . 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) and show that 2 1 m(a) := sup |D(x0 , r)| ≥ D − L1 aβ 3 3 r∈[a,ϑ(x0 )−a] for arbitrary 0 < a < 21 ϑ(x0 ). We choose a = min

! β1 − β1 1 1 L1 + L2 1 β D ϑ(x0 ) ϑ(x ) , 4L1 + 6 , 0 2 2β 2

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

1 2 D − L1 a β − ε 3 3

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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 ≥ a β ≥ ζ(B(x0 , r0 )) − λ(B(x0 , r0 )) − L1 (r − r0 )β − L2 4 a β D(x0 , r0 ) − (L1 + L2 ) , 4

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

|D(y, r)| ≥

(2.10)

1 D. 6

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)| ≥ 16 D. Combining (2.10) and the last condition on the measure λ yields Z Z X

L0

ϑ 0

ϕ (|D(y, r)|) dr dλ(y) ≥

a s+1 4

ϕ

Z

Z

ϕ

I B(x0 , a 4) a

1 D 6

dr dλ(y) ≥

1 D , 6

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

Proof of Corollary 1. In order to prove the Corollary we notice that N 1 X χE (xn ) ζ(E) = N n=1

is a Borel measure, and λ = µ satisfies the conditions of Theorem 1 with β = α − 1 and s = α and some suitable constants L0 , L1 , L2 . 2.4. Special Sequences and Irregularities of Distribution. Obviously we have (2.11)

1 ≤ DN (xn ) ≤ 1 N

for the four discrepancy systems under consideration in section 2.2. We note here that in the case of the elementary discrepancy it is possible to find sequences (xn ) E such that N DN (xn ) is bounded. Such sequences can be compared with the wellknown net-sequences in the unit cube. We remark here that these net-sequences have recently been used for various applications in quasi Monte Carlo methods (cf. [Ni]). For the other two notions of discrepancy there is the phenomenon of irregularities of distribution. In the following we want to describe a gasket-analogon of the well-known van der Corput sequence γ = (γn ). For this purpose we note that the digital expansion described in Remark 1 can also be given as follows: let δℓ ∈ {0, 1, 2} be the index i

´ EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPINSKI GASKET 11 (i)

mod 3 such that εℓ = 0. Then every point in the gasket can be encoded (not necessarily uniquely) as an infinite triadic string (cf. [Cu]). For defining the sequence γ we expand every integer n in triadic expansion n=

L X

δℓ+1 (n)3ℓ

ℓ=0

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 residue class mod 3k . Thus the elementary discrepancy is O( N1 ). By (2.5), (2.6) and (2.8) we immediately derive 1 S G B (2.12) DN (γ), DN (γ), DN (γ) = O . α−1 N α By a standard technique due to W. Philipp [Ph] the average rate of growth of the discrepancy of an arbitrary sequence xn ∈ G can be determined. Proposition 3. The following law of iterated logarithm holds for D = G, S, B √ D 1 DN (xn ) N = (2.13) lim sup √ 2 2 log log N N→∞ for almost all (with respect to the infinite product measure generated by µ) sequences on 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 32 . The example of van der Corput sequence shows that there is a gap between 1 . A simple application of the lower bound N1 and the upper bound O α−1 N α W. Schmidt’s theorem on irregularities of distribution [Sch] yields Proposition 4. Let xn be a sequence in G. Then S DN (xn ) ≥ c

log N 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 distribution function F (x) = µ(∆2 (p)), where p is given by k1 (p) = x and k3 (p) = 1, see Proposition 2 and [Gr]. Clearly this function is continuous and strictly increasing. Applying Schmidt’s lower bound to the sequence F −1 (k1 (xn )) yields S ∗ DN (xn ) ≥ DN F −1 (k1 (xn )) ≥

1 log N , 66 log 4 N

∗ where DN denotes the usual star-discrepancy in the unit interval.

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Remark 6. Clearly, this is a very weak bound, since we have used only very special gasket triangles to derive this inequality. It remains as an interesting open problem to improve this lower bound. Since we have no natural group structure it seems to be very hard to apply Beck’s Fourier transform approach to the gasket. Concluding this section we present a probabilistic approach for constructing a set ΓN of N points in G with small discrepancy. Theorem 2. For every positive integer N > 1 there exists a point set ΓN consisting of N points such that 1 1 B DN (ΓN ) ≤ cN 2α −1 (log N ) 2 , 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 integer such that 3k−1 < N ≤ 3k and take Q1 , . . . , QN1 as elementary triangles of level k, QN1 +1 , . . . , QN1 +N2 as the union of two elementary triangles of level k and QN1 +N2 +1 , . . . , QN1 +N2 +N3 as the union of three elementary triangles of level k, where N1 +N2 +N3 = N and N1 +2N2 +3N3 = 3k . We choose N3 = 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 on Qn (with respect to µ), n = 1, . . . , N . We observe that o n m Bℓ = B(x, r) | x ∈ Vℓ , r = ℓ , m = 0, . . . , 2ℓ 2 has the property that for any ball B(x, r) there exist two balls B ′ , B ′′ ∈ Bℓ such that B ′ ⊆ B ⊆ B ′′ and µ(B ′′ \ B ′ ) ≪ ( 32 )ℓ . Furthermore #Bℓ ≤ 2 · 6ℓ . Now we compute the expected value of the random variables Xn (S) = χS (Zn ) for a ball n) S ∈ Bℓ for some ℓ, which will be chosen later. Clearly EXn (S) = µ(S∩Q µ(Qn ) . Thus we obtain that Xn (S) ≡ EXn (S) if S ∩ Qn = ∅ or S ∩ Qn = Qn . As in the proof of (2.6) we have #{n | ∅ = 6 S ∩ Qn 6= Qn } ≪ N log3 2 ,

(2.14)

where the implied constant is absolute. By [BC, Lemma 8.2] we derive ! N X Prob (Xn (S) − EXn (S)) ≥ γ ≤ 2 exp −Cγ 2 N − log3 2 . n=1

√ 1 2 log 3−log 2 Setting γ = C ′ N 2 log3 2−1 log N and ℓ = [ 2(log 3−log 2) log 3 log N ], a suitable choice ′ of C yields N 1 X p 1 χS (Zn ) − µ(S) ≥ C ′ N 2 log3 2−1 log N , Prob N n=1

for some S ∈ Bℓ

!

≤

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

1 . 2

´ EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPINSKI GASKET 13

Remark 7. The main ingredient of the proof is the approximation of the discrepancy 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 the gasket: Z 1 T D (2.15) DT (x(t)) = DT (x(t)) = sup χA (x(t))dt − µ(A) . A∈D T 0

If we interpret x(t) as the motion of a particle on the gasket, the discrepancy can be considered as the deviation of the mean with respect to time and the spatial mean. The motion is called equidistributed if limT →∞ DT (x(t)) = 0. The theory of uniform distribution for continuous functions was developed in a series of papers by Hlawka [Hl], Kuipers et al. (cf. [KN]). We consider all discrepancy systems D introduced above. An application of a general result in [DT] yields Proposition 5. Let D be one discrepancy systems considered in section 2.2 and x(t) a continuous function R+ 0 → G with finite arclength s(T ) and lim s(T ) = ∞.

Then there exists a constant c(D) > 0 such that α α−1 1 D DT (x(t)) ≥ c(D) s(T )

T →∞

for T ≥ T0 .

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

N−1 X

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

n=0

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]. We only note, that in the technical condition (2.2) in [DT] it is not necessary to take balls 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 continuous 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 the trajectories of Brownian motion. For Brownian motion on manifolds similar results can be found in [BDT] and [Bl]. In the introduction we have defined Brownian motion on G as a limit process of a discrete random walk. Barlow and Perkins [BP] have shown for the corresponding ˜ that this is a symmetric Markov process with process on the infinite gasket G ˜ × G. ˜ Furthermore jointly continuous transition densities p˜(t, x, y) on [0, ∞) × G ˜ the following estimate holds t 7→ p˜(t, x, y) is C ∞ , and for all t > 0, x, y ∈ G log 3 log 5 log 2 c1 t log 5 exp −c2 d(x, y) log 5−log 2 t− log 5−log 2 ≤ p˜(t, x, y) ≤ (3.1) log 5 log 2 log 3 c3 t log 5 exp −c4 d(x, y) log 5−log 2 t− log 5−log 2

14


with suitable positive constants c1 , c2 , c3 , c4 . ˜ As described in the introduction the Brownian motion on G by factorizing G S ˜ as modulo a suitable equivalence relation ρ. Thus we can decompose G k Tk G, where the Tk are translations, which originate from the definition of ρ. Since p(t, x, y) =

X

p˜(t, x, Tk y),

k

we have to combine the estimates (3.1) in order to give upper and lower bounds. ˜ whose points p have a It follows from [Gr] that the number of copies of G in G distance ℓ ≤ d(p, 0) ≤ ℓ + 1 is 2s(ℓ)+1 , where s(ℓ) denotes the binary sum-of-digits function. Therefore we get the bounds γt = 2c1 t

log 3 log 5

∞ X

2

s(ℓ)

ℓ=0

(3.2) p(t, x, y) ≤

γt′

= 2c3 t

log 5 2 − log log log 5−log 2 5−log 2 exp −c2 (ℓ + 1) t ≤

log 3 log 5

∞ X

2

s(ℓ)

ℓ=0

exp −c4 ℓ

log 5 log 5−log 2

t

2 − log log 5−log 2

.

We want to prove that the functions γt and γt′ are bounded from above and below by positive constants for t ≥ 1. In order to prove this and to estimate γt from below we apply partial summation to the first sum in (3.2). This yields log 3

γt = 2c1 t log 5

∞ X ℓ−1 X ℓ=1 k=0

2s(k) ×

log 5 log 2 log 2 log 5 . × exp −c2 ℓ log 5−log 2 t− log 5−log 2 − exp −c2 (ℓ + 1) log 5−log 2 t− log 5−log 2

From [Ha], [FGKPT] and [Gr] we know that N−1 X 1 log2 3 N ≤ 2s(n) ≤ N log2 3 , 2 n=0

which implies (3.3) γt ≥ c1 t

log 3 log 5

∞ X ℓ=1

ℓlog2 3 ×

log 5 log 5 log 2 log 2 × exp −c2 ℓ log 5−log 2 t− log 5−log 2 − exp −c2 (ℓ + 1) log 5−log 2 t− log 5−log 2 ≥ log 3

c1 t log 5

∞ X ℓ=1

log 5 log 2 ℓlog2 3−1 exp −c2 ℓ log 5−log 2 t− log 5−log 2 .

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

∞ X

n=1

log 5 nlog2 3−1 exp −c2 n log 5−log 2 u ,


which yields ∗

f (s) =

Z

∞

f (u)u

s−1

du = ζ

0

log 5 s − log2 3 + 1 Γ(s). log 5 − log 2

By the well-known correspondence between the singularities of the transform and log 5(log 5−log 2) the asymptotic behaviour of the function we obtain f (u) ∼ c5 u log 2 log 5 with 2 − log log 5−log 2 ) and applying some explicit constant c5 . Inserting this into (3.3) (u = t ′ the same procedure to γt we derive 0 < c6 ≤ p(t, x, y) ≤ c7

(3.4)

for t ≥ 1.

Now we introduce a gasket analogon of the classical Wiener measure. Let Cw be the set of all continuous curves in G starting in a given point w ∈ G. Then for fixed 0 < t1 < t2 < . . . < tn and a Borel set E ⊆ Gn the corresponding Wiener measure is given by µw ({x ∈ Cw | (x(t1 ), x(t2 ), . . . , x(tn )) ∈ E}) = Z (3.5) p(tn − tn−1 , xn , xn−1 ) · · · p(t2 − t1 , x2 , x1 )p(t1 , x1 , w) dµ(x1 ) · · · dµ(xn ). E

Proposition 6. Let P (t, x, S) = Brownian motion on G. Then

R

S

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

(3.6)

|P (t, x, S) − µ(S)| ≤ Ae−at

(3.7)

kP (t, x, dy) − dµ(y)k ≤ 2Ae−at ,

with positive constants A, a, independent of x and the measurable set S. (k.k denotes the uniform norm with respect to x.) Proof. From [Do, p. 197] it follows that there exists a measure ν t such that P (nt, x, S) − ν t (S) ≤ (1 − γt )n−1 (3.8)

for t > 0. Applying the Chapman-Kolmogorov equation we obtain Z (3.9) P (nt, y, S)P (ns, x, dy) = P (n(s + t), x, S). G

For n → ∞, P (nt, x, S) converges to ν t (S) uniformly in x ∈ G by (3.8). Thus the integral in (3.9) converges to ν t (S), whereas the right hand side converges to ν s+t (S). Hence ν = ν t is independent of t. Next we identify ν as the Hausdorff measure µ. As p(t, x, y) is bounded by an absolute constant ν is absolutely continuous with respect to µ. Let f be the density 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 that f is non-constant. Let now C be the essential supremum of f and set Aε = {x | f (x) < C − ε} .

16


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 ∈ AC ε .

Next we choose ε so small that µ(Aε ) > 0. Thus we obtain Z (q(t, x, y0 ) − q(t, y0 , x))dν(x) ≤ 0 AC ε

and

Z

Aε

(q(t, x, y0 ) − q(t, y0 , x))dν(x) > 0.

Since µ(AC ε ) → 0 for ε → 0 we have Z (q(t, x, y0 ) − q(t, y0 , x))dν(x) > 0. G

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 an immediate consequence of the Hahn decomposition theorem. Let I be a time interval and FI the σ-algebra generated by events in I. A process is called ϕ-mixing if |P (E2 | E1 ) − P (E2 )| ≤ ϕ(t),

(3.10) and ψ-mixing if

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

(3.11)

for events E1 , E2 , with E1 being F[0,s] -measurable, E2 being F[s+t,∞] -measurable and 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 |p(s + t, x, y) − 1| ≤ 2γs′ Ae−at .

(3.12)

Using the Markov property, a simple computation yields for t ≥ 1 |P (E2 | E1 ) − P (E2 )| ≤ P (E2 )Ke−at , 4γ ′

A

where K = inf x,y∈G1/2p(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. It follows from a general result of W. Philipp [Ph] and isRa gasket analogon of Theorem 4 in [Bl]. As usual we will use the notation E(f ) = G f (x)dµ(x).


Theorem 4. Let θ be a positive integer and let Aθ be a family of real - valued uniformly bounded measurable functions on G such that 1 ≤ #Aθ ≤ eθk1 with some constant k1 . Let A be the set of all functions on G, uniformly bounded by 1 and having the following approximation property: For all f ∈ A there exists a sequence of functions hθ , hθ ∈ A such that for all positive integers L L L X X hθ ≤ f ≤ hθ θ=1

k

θ=1

L X (hθ − hθ k1 ≤ e−k2 L , θ=1

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 R T 0 f (x(t))dt − T E(f ) √ < σ(f ) + ǫ 2T log log T

for all f ∈ A and all T > T0 . Furthermore, for µw -almost all curves x(t) in Cw R T 0 f (x(t))dt − T E(f ) √ lim sup = σ(f ) 2T log log T T →∞

holds uniformly for all f ∈ A.

Rn Sketch Proof. We use the notation Xn = n−1 f (x(t))dt. Since the functions in A are uniformly bounded we mayR consider integer values for T only. Furthermore, w. l. o. g. we may assume that G f dµ = 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] and Proposition 7, if we can verify the following conditions (3.13), (3.14), (3.15): (3.13)

E(

N X

Xn (f ))2 = N σ 2 (f ) + O(N )

n=1

σ 2 (f ) = O(1)

(3.14)

(3.15)

E(

N X

Xn (hθ ))2 = O(N e−k1 θ ),

n=1

where the O-constants are absolute ones. (3.13) and (3.14) follow from Proposition 6 and (3.12) after some standard calculations and estimates, see [Bl]. (3.15) is a direct consequence of the approximation property stated in the theorem.

18


Remark 11. As in [Bl, Theorem 5] it can be shown that 0 < σ(f ) < ∞ for all non-zero f ∈ L∞ . Corollary 2. The following law of iterated logarithm holds for the discrepancy systems D = G, S, B √ D DN (x(t)) N =σ lim sup √ 2 log log N N→∞ for µw -almost all functions x(t) ∈ Cw , where w is a given point in G and σ is some positive constant. Proof. As in section 2.2 the exponential approximation property can be derived for all discrepancy systems. Acknowledgements. We are indebted to Martin Bl¨ umlinger and Wolfgang Woess for valuable discussions and for providing some recent references. Note added in proof: The first author has studied further properties of the Brownian on the Sierpi´ nski gasket and the random walk on the Sierpi´ nski graph in two subsequent papers, one of them jointly with W. Woess (to appear in Mathematika and Stochastic Processes Appl.). References [BB]

M.T. Barlow and R.F. Bass, Coupling and Harnack Inequalities for Sierpinski Carpets, Bull. AMS 29 (1993), 208–212. [BP] M.T. Barlow and E.A. Perkins, Brownian Motion on the Sierpinski Gasket, Probab. Th. Rel. Fields 79 (1988), 543–623. [BC] J. Beck and W. Chen, Irregularities of Distribution, Cambridge University Press, Cambridge, 1987. [Be] B. Berndt, Ramanujan’s Notebooks, Part II, Springer Verlag, 1989. [Bl] M. Bl¨ umlinger, Sample Path Properties of Diffusion Processes on Compact Manifolds, Number-Theoretic Analysis (E. Hlawka and R.F. Tichy, eds.), vol. 1452, Springer Lecture Notes in Mathematics, 1989. [BDT] M. Bl¨ umlinger, M. Drmota and R.F. Tichy, A Uniform Law of the Iterated Logarithm for Brownian Motion on Compact Riemannian Manifolds, Math. Z. 201 (1989), 495– 507. [Cu] A.A. Cuoco, Visualizing the p-adic Integers, Amer. Math. Monthly 98 (1991), 355–364. [DK] R.L. Dobrushin and S. Kusuoka, Statistical Mechanics and Fractals, vol. 1567, Springer Lecture Notes, 1993. [DT] M. Drmota and R.F. Tichy, C-Uniform Distribution on Compact Metric Spaces, J. Math. Analysis Appl. 129 (1988), 284–292. [EI] K.D. Elworthy and N. Ikeda, Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, vol. 283, Pitman Research Notes in Mathematics, 1993. [Fa] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985. [FGKPT] P. Flajolet, P.J. Grabner, P. Kirschenhofer, H. Prodinger and R.F. Tichy, Mellin transforms and asymptotics: digital sums, Theor. Comput. Sci. 123 (1994), 291–314. [Fl] W. Fleischer, Das Wienersche Maß einer gewissen Menge von Vektorfunktionen, Mh. Math. 75 (1971), 193–197. [FS] M. Fukushima and T. Shima, On a Spectral Analysis for the Sierpinski Gasket, Potential Analysis 1 (1992), 1–35. [Gr] P.J. Grabner, Completely q-Multiplicative Functions: the Mellin Transform Approach, Acta Arith. 65 (1993), 85–96. [Ha] H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62 (1977), 19–22. ¨ [Hl] E. Hlawka, Uber C-Gleichverteilung, Ann. Math. Pura Appl. 49 (1960), 311–326.

´ EQUIDISTRIBUTION AND BROWNIAN MOTION ON THE SIERPINSKI GASKET 19 [Ki] [KN] [Li] [Ni] [NTT] [Ph] [Sch] [Sh] [Si] [St] [Ti]

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¨r Mathematik Institut fu ¨t Graz Technische Universita Steyrergasse 30 8010 Graz, Austria E-mail address: [email protected]

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