Acta Mathematica Sinica, English Series Jul., 2010, Vol. 26, No. 7, pp. 1369–1382 Published online: June 15, 2010 DOI: 10.1007/s10114-010-7383-5 Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010

Hausdorff Dimension and Measure of a Class of Subsets of the General Sierpinski Carpets Yong Xin GUI Department of Mathematics, Xianning College, Xianning 437100, P. R. China and Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China E-mail : [email protected] Abstract In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite. Keywords

the general Sierpinski carpets, Hausdorff dimension, Hausdorff measure

MR(2000) Subject Classification 28A80, 28A78

1

Introduction

Let T be the expanding endomorphism of the 2-torus T2 = R2 /Z2 given by the matrix diag(n, m) where 2 ≤ m < n are integers. The simplest invariant sets for T have the form ⎫ ⎧  k ∞ ⎬ ⎨ 0 n−1 d : d ∈ D for all k ≥ 1 , K(T, D) = k k ⎭ ⎩ 0 m−1 k=1 where D ⊆ I × J is a set of digits with I = {0, 1, . . . , n − 1} and J = {0, 1, . . . , m − 1}. Alternatively, define a “representation” map KT : (I × J)N → T2 by  k ∞  0 n−1 N dk , x = (dk )∞ KT (x) = k=1 ∈ (I × J) . −1 0 m k=1 Then K(T, D) = KT (DN ). So each element of K(T, D) can be represented as an expansion in base diag(n−1 , m−1 ) with digits in D. The set K(T, D) is called the general Sierpinski carpet, which was first studied by McMullen [1] and Bedford [2] independently. They showed that the Hausdorff dimension is not equal to the box dimension of the general Sierpinski carpets in general. This fact is different from the property of self-similar sets. Therefore, it attracts lots of authors to investigate some further problems which are similar to that of the self-similar sets, for more details we refer to [3–8]. Recently some papers discuss the distribution of frequency of digits based on recent results concerning the multifractal analysis of dynamical systems or underlying invariant sets, for more details we refer to [9–14]. In fact, these results are obtained Received July 24, 2007, Accepted October 29, 2008 Supported by the Educational Office of Hubei Province #Q20082802 and the Science and Technology Commission of Shanghai Municipality #06ZR14029

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in the setting of self-similar or self-conformal case. Unfortunately, less analogous results have been revealed for the general self-affine sets. However, for a class of self-affine sets—the general Sierpinski carpets, Nielsen [15] studied a certain subset of K(T, D) by insisting that the allowed digits in the expansions occur with prescribed frequencies. The authors [16–17] extended it to subsets of K(T, D), which permit that frequencies of some allowed digits in the expansions may not exist. In fact, using the methods developed in [16], we can study the following more general subsets of K(T, D). Now we describe the setting in this paper. Let {Γi }li=1 be any partition of D, i.e., D = l i=1 Γi with disjoint union. In the following we fix two members of partition of {Γi }i=1 , say N Γs , Γt , 1 ≤ s = t ≤ l and to avoid triviality, we assume l ≥ 3. For any x = (xj )∞ j=1 ∈ D and d ∈ D, define l

Nk (x, d) = {1 ≤ j ≤ k : xj = d},

(1)

and Nk (x, Γi ) = {1 ≤ j ≤ k : xj ∈ Γi },

i = s, t,

(2)

where and throughout this paper we use A to denote the cardinality of a finite set A. Whenever there exist the limits Nk (x, Γi ) f (x, Γi ) = lim , i = s, t, (3) k→∞ k it is called the frequency of the group Γi in the coding x. When we write the symbol f (x, Γi ) we are already assuming the existence of the limit in (3). For any β > 0 we consider now the set

N Λ(s, t, β) = x = (xi )∞ i=1 ∈ D : f (x, Γs ) = βf (x, Γt ) .

(4)

Then Λ(s, t, β) is a subset of DN such that the frequency of group Γs in the coding x is proportional to that of Γt . And so KT (Λ(s, t, β)) is the subset of the K(T, D) whose elements have their codings with a ratio relationship about two specifically described groups. We will study KT (Λ(s, t, β)) which is the image of Λ(s, t, β) under the “representation” map KT . For any Borel subset E of R2 , let dimH E and dimB E, respectively, denote its Hausdorff and Box dimensions, and H γ (E) denote its γ-dimensional Hausdorff measure. Let α = logn m 2 and θ = α−1 α . Let σ denote the projection of R onto its second coordinate and B = σ(D). For  each point b ∈ B put nb = {d ∈ D : σ(d) = b} and qb = {pd : σ(d) = b}. Let      Σ = p = (pd )d∈D : pd ∈ [0, 1] and (5) pd = 1 and pd = β pd . d∈D

For each p ∈ Σ, let f (p) = −α

 d∈D

pd logm pd − (1 − α)

d∈Γs



pd logm qσ(d) ,

d∈Γt

(6)

d∈D

where f (p) is in fact the Hausdorff dimension of the measure μ p on KT (Ω(s, t, β)) determined by p = (pd )d∈D ∈ Σ (see Proposition 3.1), and 0 logm 0 is interpreted as 0. In this paper, we obtain the following results.

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Theorem 1.1 Let p∗ = (pd )∗d∈D be the unique point which satisfies the system of equations (12). Then dimH KT (Λ(s, t, β)) = max f (p) p∈Σ    β 1 ∗θ ∗θ logm logm = α qσ(d) + qσ(d) 1+β 1+β d∈Γs d∈Γt   β logm β − logm − p∗d 1+β d∈Γt

and dimB KT (Λ(s, t, β)) = dimB K(T, D) = (1 − α) logm B + α logm D,  where qb∗ = d∈D,σ(d)=b p∗d for b ∈ B.

(7)

D is said to have uniform horizontal fibres if nb = nb for all b, b ∈ B. As to the relative Hausdorff measure, we have the following theorem. Theorem 1.2 Let γ = dimH KT (Λ(s, t, β)). γ s (I) If β = Γ Γt and D has uniform horizontal fibres then 0 < H (KT (Λ(s, t, β))) < ∞; Γs (II) If β = Γt or D does not have uniform horizontal fibres then H γ (KT (Λ(s, t, β))) = ∞. The rest of this paper is organized as follows. In Section 2, we introduce approximate squares and a density theorem. Proofs of Theorems 1.1 and 1.2 are arranged in Section 3 and Section 4 respectively. 2

Preliminaries

We first introduce the so-called approximate squares which are used to calculate the various N dimensions of the general Sierpinski carpet and its subsets. For each x = (xj )∞ j=1 ∈ (I × J) and each positive integer k, let N Qk (x) = KT (y) : y = (yj )∞ j=1 ∈ (I × J) , yj = xj for 1 ≤ j ≤ [αk] and σ(yj ) = σ(xj ) for [αk] + 1 ≤ j ≤ k} , where, as usual, [x] with x ∈ R denotes the greatest integer function. The sets Qk (x) are called the approximate squares in [0, 1]2 , whose sizes have length n−[αk] and m−k . Note that the ratio of the sizes of Qk (x) is at most n, and their diameters diamQk (x) satisfy √ −k √ 2m ≤ diamQk (x) ≤ 2nm−k . So in the definition of Hausdorff measure, we can restrict attention to covers by such approximate squares since any set of diameter less than m−k can be covered by a bounded number of approximate squares Qk (x). The following lemma appeared in [15], which is just a reformulation of the Rogers–Taylor density theorem as stated by Peres in Section 2 of [4]. Lemma 2.1 [15, Lemma 4] Suppose that δ is a positive number, that μ is a finite Borel measure in [0, 1]2 , and that E is a subset of (I × J)N such that KT (E) is a Borel subset of [0, 1]2 , and μ(KT (E)) > 0, put A(x) = lim sup(kδ + logm μ(Qk (x))) k→∞

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for each point x ∈ E, 1) If A(x) = −∞ for all x ∈ E, then H δ (KT (E)) = +∞, 2) If A(x) = +∞ for all x ∈ E, then H δ (KT (E)) = 0, 3) If there are numbers a and b such that a ≤ A(x) ≤ b for all x ∈ E, then 0 < δ H (KT (E)) < +∞. The Borel measures on [0, 1]2 to which the above lemma will be applied are constructed as  follows. Let p = (pd )d∈D be a probability vector on D, i.e., d∈D pd = 1 with each pd ∈ [0, 1]. Then p determines a unique infinite product Borel probability measure, denoted by μp , on DN . For any finite sequence (x1 , x2 , . . . , xk ) ∈ Dk μp (C(x1 , x2 , . . . , xk )) =

k 

pxj ,

j=1 N N where C(x1 , x2 , . . . , xk ) := {d = (dj )∞ j=1 ∈ D : dj = xj for 1 ≤ j ≤ k} is a cylinder set of D with base (x1 , x2 , . . . , xk ). Let μ p be the Borel probability measure on KT (DN ) which is the image measure of μp under KT . From the fact that the approximate square Qk (x) is a finite union of cylinder sets, it follows that for any x ∈ DN (cf. formula (4) in [15], also formula (4.4) in [7]) [αk] k   pxj · qσ(xj ) . (8) μ p (Qk (x)) = j=1

j=[αk]+1

p -measure for some properly The following lemma shows that KT (Λ(s, t, β)) is of full μ selected p. Lemma 2.2 Let Λ(s, t, β) and Σ be defined as (4) and (5). If the probability vector p = p (KT (Λ(s, t, β))) = 1. (pd )d∈D satisfies p ∈ Σ, then μ Proof For any probability vector p = (pd )d∈D ∈ Σ, let  Nk (x, d) = pd Λp = x ∈ DN : lim k→∞ k

 for all d ∈ D ,

and for each point d ∈ D let

  Nk (x, d) N = pd . Λp (d) = x ∈ D : lim k→∞ k   Then Λ(s, t, β) ⊃ p∈Σ Λp and Λp = d∈D Λp (d). So it suffices to show that μp (Λp (d)) = 1 for each d ∈ D. Fix a d ∈ D and define a sequence of radon variables {Xj }∞ j=1 on the probability N space (D , F , μp ) (F is the Borel σ-algebra) by letting  1, xj = d, Xj (x) = 0, xj = d. Then X1 , X2 , . . . are independent and identically distributed random variables with μp (X1 = 1) = pd and μp (X1 = 0) = 1 − pd . By Kolmogrov strong law of large numbers, we have that for μp -a.e. x ∈ DN k Nk (x, d) 1 = lim Xj (x) = E(X1 ) = pd , k→∞ k→∞ k k j=1

lim

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implying μp (Λp (d)) = 1.

It will be convenient to refer to the Hausdorff dimension of a measure μ. This is defined as the minimum of the dimensions of sets of full μ-measure. Lemma 2.3 (The modification of Billingsley lemma, cf. [4]) p are defined as above. If μp and μ lim inf k→∞

For any p = (pd )d∈D ∈ Σ,

log μ p (Qk (x)) =ζ log m−k

for μp -almost every x ∈ Λ(s, t, β), then dimH μ p = ζ. 3

Proof of Theorem 1.1

Since KT (Λ(s, t, β)) is dense in K(T, D) it follows that the box dimensions of these two sets are equal. And Mcmullen [1] has shown that the right side of (7) is equal to dimB K(T, D) and this proves (7).    β 1 ∗θ ∗θ logm logm qσ(d) + qσ(d) 1) dimH KT (Λ(s, t, β)) ≥ α 1+β 1+β d∈Γs d∈Γt   β logm β − logm − p∗d . 1+β d∈Γt

We use two propositions to estimate the lower bound of dimH KT (Λ(s, t, β)) for convenience.   Proposition 3.1 Let Σ = {p = (pd )d∈D : 0 ≤ pd ≤ 1, d∈D pd = 1 and d∈Γs pd =  β d∈Γt pd }. For every p ∈ Σ, we have   dimH ( μp ) = −α pd logm pd − (1 − α) qb logm qb , (9) where qb =

 σ(d)=b

d∈D

b∈B

pd .

p be the Borel probability measure on Λ(s, t, β) Proof For any p = (pd )d∈D ∈ Σ, let μp and μ and KT (Λ(s, t, β)) respectively as above. For any point x = (xj )∞ j=1 ∈ Λ(s, t, β) and any integer k ∈ N, taking logarithm in (8), we have p (Qk (x)) = logm μ

[αk] 

logm pxj +

j=1

k 

logm qσ(xj ) .

(10)

j=[αk]+1

From the constructure of μp and μ p , μp is the distribution of i.i.d. random vector X1 , X2 , . . . , p = μp ◦ T −1 . each of which has distribution μp and μ Obviously, [αk] → α, k → ∞. k By Kolmogrov strong law of large numbers, we have [αk]  1  logm pxj → pd logm pd [αk] j=1 d∈D

a.e.-μp ,

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Then

[αk]  1 logm pxj → α pd logm pd k j=1

a.e.-μp .

d∈D

Note that

k 

logm qσ(xj ) =

k 

logm qσ(xj ) −

j=1

j=[αk]+1

[αk] 

logm qσ(xj ) .

j=1

Again by Kolmogrov strong law of large numbers, we have k  1 logm qσ(xj ) −→ qb logm qb k j=1

a.e.-μp .

b∈B

Then

[αk]  1 logm qσ(xj ) −→ α qb logm qb k j=1

a.e.-μp ,

b∈B

so 1 k Therefore

k 

logm qσ(xj ) −→ (1 − α)



qb logm qb

a.e.-μp .

b∈B

j=[αk]+1

  log μ p (Qk (x)) = −α pd logm pd − (1 − α) qb logm qb −k k→∞ log m lim

d∈D

a.e.-μp .

b∈B

By Billinsley Lemma 2.3, this proves Proposition 3.1.



Our next target is to maximize the expression (9) under the constraint p ∈ Σ.  μp ) = −α d∈D pd logm pd − (1 − Proposition 3.2 For every p ∈ Σ, let f (p) = dimH (  α) b∈B qb logm qb . Denote D = D\(Γs ∪ Γt ). There exists a unique probability vector p∗ = (p∗d )d∈D ∈ Σ such that μp ). (11) f (p∗ ) = max dimH ( p∈Σ

∗

Furthermore, p must be an interior point of Σ, and precisely p∗ is the unique solution of the system of equations: ⎧  θ   qσ(d) ⎪ ⎪ ⎪  pd = 1 − (β + 1) for d ∈ D , pd , ⎪ θ ⎪ ⎪ d∈D  qσ(d) ⎪ d∈Γt ⎪ ⎪ θ ⎪  βqσ(d) ⎪ ⎪ ⎪ pd , for d ∈ Γs , ⎪ pd =  θ ⎪ ⎪ qσ(d) ⎪ d∈Γ s d∈Γ ⎪ t ⎪ θ ⎪  ⎪ qσ(d) ⎨ pd , for d ∈ Γt , pd =  θ (12) d∈Γt qσ(d) d∈Γt ⎪     ⎪ ⎪     ⎪ ⎪ θ θ ⎪ log + β log β + log pd − log qσ(d) pd − log qσ(d) ⎪ ⎪ ⎪ ⎪ d∈Γt  d∈Γt d∈Γs ⎪  d∈Γt ⎪    ⎪ ⎪ θ ⎪ ⎪ pd − log qσ(d) = 0, −(1 + β) log 1 − (β + 1) ⎪ ⎪ ⎪  ⎪ d∈Γ d∈D t ⎪ ⎩ 0 < pd < 1, for all d ∈ D,  where, as before, qb = d∈D,σ(d)=b pd for b ∈ B.

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Proof Note that f (p) is a strictly concave function of a probability vector p. In fact the first summand of f (p) is strictly concave and the second is concave. However, Σ is convex. By a well-known property of convex programming, there exists a unique probability vector p∗ in Σ such that f (p) attains its maximum at p = p∗ . Next we show that p∗ ∈ int(Σ), i.e., p∗d =  0 for all d ∈ D. Let   pd logm pd and Z2 (p) = (α − 1) qb logm qb . Z1 (p) = −α d∈D

b∈B

Then f (p) = Z1 (p) + Z2 (p). Suppose p∗ = (p∗d )d∈D ∈ Σ \ int(Σ). Let D1 = {d ∈ D : p∗d = 0}  = ( pd )d∈D ∈ int(Σ). Let and D2 = D \ D1 . Then both D1 and D2 are nonempty. Take p ∗ ∗ p + (1 − t)p = (t pd + (1 − t)pd )d∈D , t ∈ [0, 1]. Then pt ∈ int(Σ) for t ∈ (0, 1] and pt = t p0 = p∗ . Note that   d d  Z1 (pt ) = Z1 (pt ) = −α (t pd + (1 − t)p∗d ) logm (t pd + (1 − t)p∗d ) dt dt d∈D  ∗ ( pd − pd ) logm (t pd + (1 − t)p∗d ) = −α d∈D

= −α

 

pd logm (t pd + (1 −

t)p∗d )

d∈D1

+



( pd −

p∗d ) logm (t pd

+ (1 −

t)p∗d )

 .

d∈D2

Thus we have limt→0+ Z1 (pt ) = +∞. The same argument shows that limt→0+ Z2 (pt ) = +∞ if qb∗ = 0 for some b ∈ B, or equals a finite real number. Therefore, limt→0+ f  (pt ) = +∞. Note that limt→0+ f (pt ) = f (p∗ ). Thus, f (pt ) > f (p∗ ) = maxp∈Σ f (p) when t is small enough, leading to a contradiction. Therefore p∗ must be an interior point of Σ. Since f (p) is strictly concave on Σ, and p∗ must be an interior points of Σ, then the point which satisfies Kuhn–Turker condition on Σ is just the unique maximum point. Let      pd log pd − (1 − α) qb log qb + λ1 pd − β pd L(p, λ, u, v) = −α d∈D b∈B d∈Γ d∈Γ s t     pd − 1 + ud p d + vd (1 − pd ). +λ2 d∈D

d∈D

d∈D

In our setting Kuhn–Turker condition on Σ can be written as: ⎧ −α(log pd + 1) − (1 − α)(log qσ(d) + 1) + λ2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −α(log pds + 1) − (1 − α)(log qσ(ds ) + 1) + λ1 + λ2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −α(log pdt + 1) − (1 − α)(log qσ(dt ) + 1) − βλ1 + λ2 = 0, ⎪ ⎨   pd − β pd = 0, ⎪ ⎪ d∈Γ d∈Γ s t ⎪  ⎪ ⎪ ⎪ pd − 1 = 0, ⎪ ⎪ ⎪ ⎪ d∈D ⎪ ⎩ 0 < pd < 1,

for d ∈ D , for d ∈ Γs , for d ∈ Γt ,

for all d ∈ D,

i.e., an admissible solution on Σ which satisfies Lagrange multipliers method is just the only maximum point.

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By the method of successive elimination, it is equivalent to the following system of equations: ⎧  θ   qσ(d) ⎪ ⎪ ⎪  p 1 − (β + 1) for d ∈ D , = pd , d ⎪ θ ⎪ q  ⎪ d∈D σ(d) ⎪ d∈Γt ⎪ ⎪ θ ⎪  q ⎪ ⎪ ⎪ pd =  σ(d) β pd , for d ∈ Γs , ⎪ θ ⎪ ⎪ ⎪ d∈Γs qσ(d) d∈Γt ⎪ ⎪ θ ⎪  ⎪ qσ(d) ⎨ pd , for d ∈ Γt , pd =  θ d∈Γt qσ(d) d∈Γt ⎪    ⎪  ⎪     ⎪ ⎪ θ θ ⎪ log + β log β + log p − log q p − log q ⎪ d d σ(d) σ(d) ⎪ ⎪ ⎪ d∈Γt  d∈Γt d∈Γt d∈Γs ⎪    ⎪   ⎪ ⎪ θ ⎪ ⎪ = 0, pd − log qσ(d) −(1 + β) log 1 − (β + 1) ⎪ ⎪ ⎪  ⎪ d∈Γ d∈D t ⎪ ⎩ 0 < pd < 1, for all d ∈ D.  Let p∗ = (p∗d )d∈D be the unique solution of the above system of equations, then from (12) a straightforward calculation will show that      β 1 β ∗θ ∗θ logm logm logm β − logm qσ(d) + qσ(d) − p∗d . f (p∗ ) = α 1+β 1+β 1+β d∈Γs

d∈Γt

d∈Γt

∗

For the rest of this paper we fix this choice of p . From the definition of the dimension of a measure and the above propositions, we have already shown that μp∗ ) dimH KT (Λ(s, t, β)) ≥ dimH (    1 β ∗θ ∗θ logm logm qσ(d) + qσ(d) = α 1+β 1+β d∈Γs d∈Γt   β logm β − logm − p∗d . 1+β d∈Γt

2) Next we will use Lemma 2.1 to show that    β 1 ∗θ ∗θ logm logm qσ(d) + qσ(d) dimH KT (Λ(s, t, β)) ≤ α 1+β 1+β d∈Γs d∈Γt   β logm β − logm − p∗d . 1+β d∈Γt

For any x = x1 x2 · · · ∈ Λ(s, t, β) and any positive integer k, denote  ∗ Nk (x, d) logm qσ(d) , S1 (x, k) = d∈Γt

S2 (x, k) =



∗ Nk (x, d) logm qσ(d) ,

d∈Γs

S3 (x, k) =



∗ Nk (x, d) logm qσ(d) .

d∈D 

Recalling the definition of Nk (x, Γi ) and Nk (x, d) as (2) and (1), we have  Nk (x, d), Nk (x, Γs ) = d∈Γs

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Nk (x, Γt ) =



Nk (x, d).

d∈Γt

It implies that Nk (x, D ) = k −





Nk (x, d) −

d∈Γs

Nk (x, d),

d∈Γt

where we recall that D = D\(Γs ∪ Γt ). Denote      β 1 β ∗θ ∗θ logm logm logm β − logm γ=α qσ(d) + qσ(d) − p∗d . 1+β 1+β 1+β d∈Γs

d∈Γt

d∈Γt

∗

We will apply Lemma 2.1 to obtaining the upper bound by taking p = p . From the system of equations (12), we have for any d ∈ Γs , ∗ = logm p∗d − logm qσ(d)

 −

   1 ∗ ∗θ logm qσ(d) − logm qσ(d) + logm β + logm p∗d , α d∈Γs

for any d ∈ Γt , logm p∗d

−

∗ logm qσ(d)

 =

d∈Γs

   1 ∗ ∗θ − logm qσ(d) − logm qσ(d) + logm p∗d , α d∈Γt

for any d ∈ D , ∗ = logm p∗d − logm qσ(d)

 −

d∈Γt

    1 ∗ ∗θ logm qσ(d) . − logm qσ(d) + logm 1 − (1 + β) p∗d α  d∈D

d∈Γt

Using this fact, rewrite (8) in the form logm μ p∗ (Qk (x)) =

[αk] 

logm p∗xj +

j=1

=



k 

∗ logm qσ(x j)

j=[αk]+1

N[αk] (x, d) logm p∗d +

d∈Γs

+



N[αk] (x, d) logm p∗d

d∈Γt



N[αk] (x, d) logm p∗d

d∈D 

+



∗ (Nk (x, d) − N[αk] (x, d)) logm qσ(d)

d∈Γs

+



∗ (Nk (x, d) − N[αk] (x, d)) logm qσ(d)

d∈Γt

+



∗ (Nk (x, d) − N[αk] (x, d)) logm qσ(d)

d∈D 

=



∗ N[αk] (x, d)(logm p∗d − logm qσ(d) )

d∈Γs

+



∗ N[αk] (x, d)(logm p∗d − logm qσ(d) )

d∈Γt

+



∗ N[αk] (x, d)(logm p∗d − logm qσ(d) )+

d∈D 

+



d∈Γt

∗ Nk (x, d) logm qσ(d)

+

 d∈D 



∗ Nk (x, d) logm qσ(d)

d∈Γs ∗ Nk (x, d) logm qσ(d)

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=

  1 ∗ ∗θ logm qσ(d) − logm qσ(d) + logm β + logm p∗d α d∈Γs d∈Γs d∈Γs      1 ∗ ∗θ + N[αk] (x, d) − logm qσ(d) − logm qσ(d) + logm p∗d α d∈Γt d∈Γt d∈Γt       1 ∗ ∗θ + N[αk] (x, d) − logm qσ(d) − logm qσ(d) + logm 1 − (1 + β) p∗d α d∈D  d∈D  d∈Γt    ∗ ∗ ∗ Nk (x, d) logm qσ(d) + Nk (x, d) logm qσ(d) + Nk (x, d) logm qσ(d) + 

N[αk] (x, d) −

d∈Γs

=







+



N[αk] (x, d) − logm





N[αk] (x, d) − logm

d∈D 



∗θ qσ(d) + logm

d∈Γt





∗θ qσ(d) + logm β + logm

d∈Γs



d∈Γt

+



N[αk] (x, d) − logm

d∈Γs

d∈D 

d∈Γt





p∗d



d∈Γs

p∗d



d∈Γt

∗θ qσ(d)

   + logm 1 − (1 + β) p∗d

d∈D 

  1 ∗ ∗ + Nk (x, d) logm qσ(d) − N[αk] (x, d) logm qσ(d) α d∈Γs    1 ∗ ∗ + Nk (x, d) logm qσ(d) − N[αk] (x, d) logm qσ(d) α d∈Γt    1 ∗ ∗ + Nk (x, d) logm qσ(d) − N[αk] (x, d) logm qσ(d) α 

d∈Γt

d∈D

Using the symbol of Si (x, k), i = 1, 2, 3 and Nk (x, Γs ), Nk (x, Γt ), we have     ∗θ logm μ p∗ (Qk (x)) = N[αk] (x, Γs ) − logm qσ(d) + logm β + logm p∗d 

d∈Γs

+ N[αk] (x, Γt ) − logm



∗θ qσ(d) + logm

d∈Γt





d∈Γs

p∗d

d∈Γt

+ ([αk] − N[αk] (x, Γs ) − N[αk] (x, Γt )) − logm





∗θ qσ(d)

d∈D 

  3     1 + Si (x, k) − Si (x, [αk]) . + logm 1 − (1 + β) p∗d α i=1 d∈Γt

Therefore 1 logm μ p∗ (Qk (x)) k k→∞     ∗θ qσ(d) + logm β + logm p∗d = αf (x, Γs ) − logm

lim sup



d∈Γs

− αf (x, Γt ) − logm



d∈Γt

∗θ qσ(d) + logm



d∈Γs

p∗d



d∈Γt

      ∗θ + α 1 − f (x, Γs ) − f (x, Γt ) − logm qσ(d) + logm 1 − (1 + β) p∗d d∈D 

d∈Γt

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 Si (x, [αk]) − + lim sup k αk k→∞ i=1      ∗θ = α − logm qσ(d) + logm 1 − (1 + β) p∗d 3   Si (x, k)

d∈D 

+ lim sup k→∞



3   i=1

Si (x, k) Si (x, [αk]) − k αk

+ β logm β + log 





p∗d − logm

d∈Γt

d∈Γt





− αf (x, Γt )

− (1 + β) logm 1 − (β + 1)

p∗d



logm

d∈Γt



p∗d

− logm



 θ∗ qσ(d)

d∈Γt

θ∗ qσ(d)

d∈Γs





 − logm



θ∗ qσ(d)

 .

d∈D 

d∈Γt

Note that in the second equality of the above expression the last summand is equal to zero and the first summand      ∗θ α − logm qσ(d) + logm 1 − (1 + β) p∗d 

d∈D 

d∈Γt

    β 1 β ∗θ ∗θ = −α logm logm logm β − logm qσ(d) + qσ(d) − p∗d 1+β 1+β 1+β = −γ,

d∈Γs

d∈Γt

d∈Γt

since p∗ is the maximum point which should satisfy formula (12). Then we have  3   1 Si (x, k) Si (x, [αk]) − . lim sup logm μ p∗ (Qk (x)) = −γ + lim sup k αk k→∞ k k→∞ i=1 Next, we will show that for every point x ∈ Λ(s, t, β),  3   Si (x, k) Si (x, [αk]) lim sup ≥ 0. − k αk k→∞ i=1

(13)

(14)

This can essentially be derived from Lemma 4.1 in [5] and the details are shown in [16]. For reader’s convenience, we give its proof in detail. For every point x ∈ Λ(s, t, β) and any k ∈ N, from the definition of Si (x, k), i = 1, 2, 3, it is obvious that for every i = 1, 2, 3, sup |Si (x, k + 1) − Si (x, k)| < ∞.

(15)

k

For a fixed x = (xj )∞ j=1 ∈ Λ(s, t, β), let Ti (k)= Si (x, k), i = 1, 2, 3. We extend each Ti , i = 1, 2, 3 to [1, +∞) by piecewise linear interpolation. Then each Ti , i = 1, 2, 3 is a Lipschitz function by (15). Now define gi : [0, ∞) → R, i = 1, 2, 3 by gi (z) = e−z Ti (ez ). We claim that each gi (z), i = 1, 2, 3 is bounded and uniformly continuous on [0, ∞). In fact, |gi (z)| ≤ |gi (0)|e−z + |gi (z) − gi (0)e−z | ≤ |Ti (1)| + e−z |Ti (ez ) − Ti (1)| ≤ |Ti (1)| + LipTi ,

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and for any δ > 0, |gi (z + δ) − gi (z)| = |e−(z+δ) Ti (ez+δ ) − e−z Ti (ez )| ≤ e−(z+δ) |Ti (ez+δ ) − Ti (ez )| + |gi (z)|(1 − e−δ ) ≤ (1 − e−δ )LipTi + (1 − e−δ )(|T (1)| + LipTi ). Now for any v > − log α,  v   

3  

− log α i=1

   3  = i=1

   3  = i=1

   3 =  i=1

≤

   gi (z) − gi (z + log α) dz  

v − log α

gi (z)dz −

− log α

gi (z)dz −

i=1

  gi (z + log α)dz 

v+log α

0



− log α

  gi (z)dz 

v+log α

gi (z)dz +

0

3   − log α  

 

− log α



v

v

0

v

    gi (z)dz  + 

  gi (z)dz 

v+log α v

  gi (z)dz 

< +∞, since each gi is bounded on [0, +∞). Therefore, lim sup z→+∞

3 

(gi (z) − gi (z + log α)) ≥ 0.

i=1

By letting z = log t, this gives lim sup t→+∞

3   Ti (t) i=1

t

Ti (αt) − αt

 ≥ 0.

Note that     Ti (t) Ti (αt) Ti (t) − Ti ([t]) Ti (αt) − Ti ([αt]) Ti ([t]) [t] − = − + −1 t αt t αt [t] t     Si (x, [t]) Si (x, [α[t]]) Si (x, [α[t]]) Si (x, [αt]) − + − , + α[t] αt [t] α[t]

(16)

where, as before, [t] with t ∈ R denotes the greatest integer function. However, the first three terms on the right side of (16) tend to zero as t → +∞ by the fact that both functions |Ti (t) − Ti ([t])| and gi (z) are bounded, and gi (z) is uniformly continuous for all i = 1, 2, 3. Hence (14) holds. Therefore, for every x = (xj )∞ j=1 ∈ Λ(s, t, β) we have lim sup k→∞

1 logm μ p∗ (Qk (x)) ≥ −γ, k

which leads to

  1 lim sup(kδ + logm μ p∗ (Qk (x))) = lim sup k δ + logm μ p∗ (Qk (x)) = +∞, k k→∞ k→∞

for any δ > γ. Now Lemma 2.1 2) and Lemma 2.2 imply that dimH KT (Λ(s, t, β)) ≤ γ.



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4

1381

Proof of Theorem 1.2

(I) Let p∗ be the unique point such that f (p) attain its maximum on Σ, it should satisfy the s system of equations (12). We will use Lemma 2.1 3) to prove by taking p = p∗ . If β = Γ Γt and D has uniform horizontal fibres, we first claim that p∗ = (p∗d )d∈D is uniformly distributed   1 1 1 . This is done by simply checking that the probability vector , D , . . . , D on D, i.e., p∗ = D   1 1 1 ∈ Σ satisfies the system of equations (12) because of its uniqueness. In D , D , . . . , D   1 1 1 s ∈ Σ from the condition of β = Γ , D , . . . , D fact, it is easily concluded that p∗ = D Γt .  Γt ∗ ∗ ∗ And d∈Γt pd = D if p = (pd )d∈D is uniformly distributed on D. Thus we can verify   1 1 1 ∗ satisfies the system of equations (12). straightforwardly that p = D , D , . . . , D At this moment, we have γ = f (p∗ ) = (1 − α) logm B + α logm D. Therefore, p∗ (Qk (x)) = k((1 − α) logm B + α logm D) kγ + logm μ 1 1 + (k − [αk]) logm + [αk] logm D B D = (αk − [αk]) logm B for every x = (xj )∞ j=1 ∈ Λ(s, t, β) and all k ∈ N. Then (I) is justified by Lemma 2.1 3) and Lemma 2.2. (II) For any probability vector p = (pd )d∈D , let   Nk (x, d) N = p ∈ D : lim for all d ∈ D , Λp = x = (xj )∞ d j=1 k→∞ k where Nk (x, d) is defined as (1). The vector p = (pd )d∈D is said to be uniformly distributed on 1 for all d ∈ D. Nielsen (cf. Theorem 1 and 3 in [15]) proved that D if pd = D   Lemma 4.1 (a) dimH KT (Λp ) = −α d∈D pd logm pd − (1 − α) d∈D pd logm qσ(d) . (b) If p is not uniformly distributed on D or if D does not have uniform horizontal fibres then H δ (KT (Λp )) = ∞, where δ = dimH KT (Λp ). Let p∗ be the unique point such that f (p) attains its maximum on Σ. In Theorem 1.1 we have shown that dimH KT (Λ(s, t, β)) = f (p∗ ) = dimH KT (Λp∗ ). So KT (Λ(s, t, β)) has the same Hausdorff dimension γ as KT (Λp∗ ). A basic observation is that KT (Λp∗ ) is a subset of KT (Λ(s, t, β)), it is important to simplify our proof. There are two case: D does not have uniformly horizontal fibres or D has uniformly ∗ s horizontal fibres but β = Γ Γt . In the second case we claim that the probability vector p = (p∗d )d∈D would not be uniformly distributed on D. This is done by simply checking that the   1 1 1 does not satisfy the system of equations (12) because of its , D , . . . , D probability vector D uniqueness. In fact uniformly distributed probability vector does not satisfy the third equation

Gui Y. X.

1382

in the system of equations (12), since for any d ∈ Γs the left hand side of the third equation in 1 if p∗ = (p∗d )d∈D is uniformly distributed on D, however, the system of equations (12) is p∗d = D the right hand side of the third equation in the system of equations (12) reads 

θ∗ βqσ(d)



θ∗ d∈Γs qσ(d) d∈Γt

This leads to a contradiction since β =

p∗d = β

Γt 1 . Γs D

Γs Γt . γ

So both the case one and the case two we can deduce directly from Lemma 4.1 (b) that H (KT (Λp∗ )) = ∞. Therefore (II) can be deduced by the fact that KT (Λ(s, t, β)) ⊃ KT (Λp∗ ), and dimH KT (Λ(s, t, β)) = dimH KT (Λp∗ ) = γ. Acknowledgements The author would like to thank the anonymous referees for their significant suggestions and comments which lead to the improvement of the manuscript. References [1] McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J., 96, 1–9 (1984) [2] Bedford, T.: Crinkly Curves, Markov Partitions and Box Dimension in Self-similar Sets, Ph.D Thesis, University of Warwick, 1984 [3] Peres, Y.: The packing measure of self-affine carpets. Math. Proc. Camb. Phil. Soc., 115, 437–450 (1994) [4] Peres, Y.: The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Camb. Phil. Soc., 116, 513–526 (1994) [5] Kenyon, R., Peres, Y.: Measures of full dimension on affine-invariant sets. Ergodic Theory Dyn. Systems. 16, 307–323 (1996) [6] Kenyon, R., Peres, Y.: Hausdorff dimensions of sofic affine-invariant sets. Israel J. Math., 122, 540–574 (1996) [7] Gatzouras, D., Lalley, S.: Hausdorff and box dimensions of certain self-affine fractals. Indiana University Math. J., 41, 533–568 (1992) [8] Bara´ nski, K.: Hausdorff dimension of the limit sets of some planar geometric constructions. Advances in Mathematics, 210, 215–245 (2007) [9] Barreira, L., Saussol, B., Schmeling, J.: Distribution of frequencies of digits via multifractal analysis. Journal of Number Theory, 97, 410–438 (2002) [10] Fan, A. H., Feng, D. J., Jun, W.: Recurence, dimension and entropy. J. London Math. Soc., 64, 229–244 (2001) [11] Fan, A. H., Feng, D. J.: On the distribution of long-term time average on the symbolic space. J. Stat. Phys., 99, 813–856 (2000) [12] Olsen, L.: Applications of multifractal divergence points to sets of d-tuples of numbers defined by their N -adic expansion. Bulletin des Sciences Mathematiques, 128, 265–289 (2004) [13] Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl., 82, 1591–1649 (2003) [14] Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical birkhoff averages, III. Aequationes Mathematicae, 71, 29–53 (2006) [15] Nielsen, O.: The Hausdorff and packing dimensions of some sets related to Sierpinski carpets. Canad. J. Math., 51, 1073–1088 (1999) [16] Gui, Y. X., Li, W. X.: The Hausdorff dimension of sets related to the general Sierpinski carpets. Acta Mathematica Sinica, English Series, 26(4), 731–742 (2010) [17] Gui, Y. X., Li, W. X.: Hausdorff dimension of subsets with proportional fibre frequencies of the general sierpinski carpet. Nonlinearity, 20, 2353–2364 (2007)