DISCRETE AND CONTINUOUS doi:10.3934/dcds.2021029 DYNAMICAL SYSTEMS Volume 41, Number 9, September 2021 pp. 4085–4104
BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS
Saisai Shi and Bo Tan∗ School of Mathematics and Statistics Huazhong University of Science and Technology, Wuhan 430074, China Qinglong Zhou School of Science Wuhan University of Technology, Wuhan 430074, China
(Communicated by Sylvain Crovisier)
Abstract. Let Φ = {φi : i ∈ Λ} be an iterated function system on a com- pact metric space (X, d), where the index set Λ = {1, 2, . . . , l} with l ≥ 2, or Λ = {1, 2,...}. We denote by J the attractor of Φ, and by D the subset of points possessing multiple codings. For any x ∈ J\D, there is a unique integer sequence {ωn(x)}n≥1 ⊂ ΛN, called the digit sequence of x, such that \ {x} = φω1(x) ◦ · · · ◦ φωn(x)(X). n
In this case we write x = [ω1(x), ω2(x),...]. For x, y ∈ J\D, we define the shortest distance function Mn(x, y) as Mn(x, y) = max k ∈ N: ωi+1(x) = ωi+1(y), . . . ,ωi+k(x) = ωi+k(y) for some 0 ≤ i ≤ n − k , which counts the run length of the longest same block among the first n digits of (x, y). In this paper, we are concerned with the asymptotic behaviour of Mn(x, y) as n tends to ∞. We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance function. As applications, we study the exceptional sets in several concrete systems such as continued fractions system, L¨urothsystem, N-ary system, and triadic Cantor system.
1. Introduction. Ergodic theory is the study of the long-term behaviour of a measure-preserving system. One of the classical results is the Poincar´erecurrence theorem, which states that almost all points in a prescribed set of positive measure are infinitely recurrent. In a metric space (X, d), the theorem implies that lim inf d T n(x), x = 0 n→∞ for µ-almost all x ∈ X. The theorem provides a qualitative rather than quantitative description of the long-term behaviour, and the quantitative behaviour of Poincar´e recurrence has been studied by Boshernitzan [1], Ornstein & Weiss [20] and Barreira & Saussol [3]. Other notions, such as the recurrence time, hitting time, etc., have
2020 Mathematics Subject Classification. Primary: 11K55; Secondary: 37F35, 28A80. Key words and phrases. Best approximation, Shortest distance function, Iterated function systems. ∗ Corresponding author: Bo Tan.
4085 4086 SAISAI SHI, BO TAN AND QINGLONG ZHOU been introduced to study the quantitative nature of recurrence [9, 21, 22, 23, 26, 30]. Among others, let us mention the shrinking target problem introduced by Hill & Velani [11]. Let ψ : N → R+ be a positive function such that ψ(n) → 0 as n → ∞, {zn}n≥1 be a sequence of points in X. Define the “well approximable” set n W{zn} T, ψ = x ∈ X : d T (x), zn < ψ(n) for infinitely many n ∈ N . n It follows that the orbits {T (x)}n≥1 can be well approximated by the sequence {zn}n≥1. For an expanding rational map of a Riemann sphere acting on its Julia set, Hill & Velani [11, 12] estimated the Hausdorff dimension of the set W{z } T, ψ . n Li, Wang, Wu & Xu [16] computed the dimension of W{zn} T, ψ in the dynamical system of continued fractions. Bugeaud & Wang [4] studied the case of β-dynamical systems. The metric result has been obtained by Chernov & Kleinbock [6]. They P proved that the set W{zn} T, ψ has null µ-measure if n≥1 µ(Bn) < ∞, and has full measure if X X X µ(Bn) = ∞ with Rn,m ≤ C µ(Bn), n≥1 1≤n≤m≤N 1≤n≤N −n −m where Bn := B(zn, ψ(n)), and Rn,m := µ T Bn ∩ T Bm) − µ(Bn)µ(Bm) is the decay of the correlations. For more information about the shrinking target problems, the readers are referred to [8, 11, 12, 24] and the references therein. n Taking zn = T (y)(n ≥ 1) for a fixed y ∈ X, we would like to investigate the n n quantitative properties of the distance between {T (x)}n≥1 and {T (y)}n≥1. We study the problem in the framework of iterated function systems. Let (X, d) be a compact metric space. Let Φ = {φi : i ∈ Λ} be a collection of injective mappings of X, where Λ = {1, 2, . . . , l} (l ≥ 2) or Λ = {1, 2,...}. We call Φ a uniformly contractive iterated function system (IFS for short) if there exists 0 < ρ < 1 such that for any i ∈ Λ and x, y ∈ X,
d(φi(x), φi(y)) ≤ ρ · d(x, y).
The limit set associated with an IFS {φi : i ∈ Λ} can be defined as the image ∗ n ∞ of the coding space. Let Λ = ∪n≥0Λ , the space of finite words, and let Λ be n the collection of all infinite words over Λ. For ω = (ω1, ω2, . . . , ωn) ∈ Λ , we set ∗ ∞ φω = φω1 ◦ φω2 ◦ · · · ◦ φωn . For ω ∈ Λ ∪ Λ and n ≥ 1, we denote by ω|n the word (ω1, ω2, . . . , ωn) (when ω is a finite word, n ≤ |ω| is required, where |ω| denotes ∞ the length of ω). For ω ∈ Λ , {φω|n(X)}n≥1 is a nested sequence of compact sets, whose intersection is a singleton; the point in the intersection will be denoted by π(ω). The limit set (or the attractor) of the IFS is
∞ [ \ \ [ J = π(Λ ) = φω|n(X) = φω(X). ω∈Λ∞ n≥1 n≥1 ω : ω∈Λn The map π :Λ∞ → J is called the coding map, which is surjective but is not ∞ necessarily injective. Any ω = (ω1, ω2,...) ∈ Λ such that x = π(ω) is called a coding of x ∈ J, and a point in J may have multiple codings. Let D = {x ∈ J : x has multiple codings}. Hence, for any x ∈ J\D, there is a unique coding (ω1(x), ω2(x),...) such that π(ω) = x, whence we write
x = [ω1(x), ω2(x),...].
For x = [ω1(x), ω2(x), ω3(x),...] ∈ J\D, we define
T x = T ([ω1(x), ω2(x), ω3(x),...]) = [ω2(x), ω3(x),...]. BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4087
n For n ≥ 1 and (ω1, . . . , ωn) ∈ Λ , we call
Jn(ω1, . . . , ωn) = φω1 ◦ · · · ◦ φωn (X) an n-th order cylinder. For x ∈ J\D, let Jn(x) denote the unique n-th cylinder that contains x, i.e., Jn(x) = Jn(ω1(x), . . . , ωn(x)). The σ-algebra over J generated n by all n-th order cylinders are denoted as C0 . For x = [ω1(x), ω2(x),...], y = [ω1(y), ω2(y),...] ∈ J\D, we define the shortest distance function Mn(x, y) as Mn(x, y) = max k ∈ N: ωi+1(x) = ωi+1(y), . . . ,ωi+k(x) = ωi+k(y) for some 0 ≤ i ≤ n − k , which counts the longest run length of the longest same block among the first n digits of (x, y) ∈ J\D × J\D.
i i Remark 1. (1) Mn(x, y) = max k ∈ N: T (x) ∈ Jk(T (y)) for some 0 ≤ i ≤ n − k ; (2) We define a metric d on J\D: if x = [ω1, ω2,...], y = [ν1, ν2,...] ∈ J\D, the distance of x and y is defined as d(x, y) = exp − inf{i ≥ 1: ωi 6= νi} . i i Then Mn(x, y) ≤ − log min1≤i≤n d T (x),T (y) . Let µ be a T -invariant complete Borel probability measure µ on J. We then have a measure-preserving system (J, µ, T ), and µ-almost every point x ∈ J has a unique coding. For this system, we define the lower R´enyi entropy H∗ to be P 2 − log n µ(J (ω , . . . , ω )) (ω1,...,ωn)∈Λ n 1 n H∗ = lim inf , n→∞ n ∗ ∗ and define the upper R´enyi entropy H similarly. When H∗ and H coincide, we say that the R´enyi entropy exists and denote the common value as H. n m We say a system (J, µ, T ) is ψ-mixing if for any U ∈ C0 ,V ∈ C0 (m, n ∈ N), −(n+k) µ U ∩ T V − µ(U)µ(V ) ≤ ψ(k)µ(U)µ(V ), (1) where ψ : N → R+ vanishes at infinity. The system (J, µ, T ) is said to be ψ-mixing with an exponential decay if moreover the function ψ in (1) satisfies ψ(n + 1) lim sup < 1. n→∞ ψ(n) We mention several examples which are ψ-mixing with an exponential decay: con- tinued fractions system, L¨urothsystem, N-ary system, triadic Cantor system (see Section 5). With the notation above, we state our main results. Theorem 1.1. Let (J, µ, T ) be a ψ-mixing system with an exponential decay. (1) For µ ⊗ µ-almost all (x, y) ∈ J × J, we have M (x, y) 1 lim sup n = . n→∞ log n H∗ (2) For µ ⊗ µ-almost all (x, y) ∈ J × J, we have M (x, y) 1 lim inf n = . n→∞ log n H∗ 4088 SAISAI SHI, BO TAN AND QINGLONG ZHOU
If the R´enyi entropy exists, then for µ ⊗ µ-almost all (x, y) ∈ J × J, M (x, y) 1 lim n = . n→∞ log n H 1 1 Here we use the convention that 0 = ∞ and ∞ = 0.
Remark 2. Fix a point y0 = [ω1(y0), ω2(y0),...] ∈ J\D. For x = [ω1(x), ω2(x),...], we define a new quantity Mn(x, y0) similar to Mn(x, y) as follows: Mn(x, y0) = max k ∈ N: ωi+1(x) = ω1(y0), . . . ,ωi+k(x) = ωk(y0) for some 0 ≤ i ≤ n − k .
We call Mn(x, y0) the maximal hitting depth of x to y0, which reflects the degree how the trajectories of x can approach y0. We define the lower local entropy of y0 − log µ(Jn(y0)) to be Dµ(yo) = lim infn→∞ n and define the upper local entropy Dµ(yo) similarly. By using the same method as in proving Theorem 1.1, we have that for µ-almost all x ∈ J, M (x, y ) 1 M (x, y ) 1 lim sup n 0 = , lim inf n 0 = . n→∞ log n Dµ(yo) n→∞ log n Dµ(yo)
By Theorem 1.1 and Remark1(2), we have that for µ⊗µ-almost all (x, y) ∈ J ×J, − log min d T i(x),T i(y) 1 lim sup 1≤i≤n ≥ n→∞ log n H∗ and − log min d T i(x),T i(y) 1 lim inf 1≤i≤n ≥ . n→∞ log n H∗ The existence of R´enyi entropy has been proved only for a few special measures: Bernoulli measures, Markov measures and, more generally, Gibbs measures with H¨oldercontinuous potentials ϕ (see [2, 13, 27]). It is natural to study the Hausdorff dimensions of exceptional sets of points violating the metric properties in Theorem 1.1. Let Φ be a conformal IFS on X = [0, 1] (see Section 2 for the formal definition). Due to the OSC in [0, 1], there are at most countably many points x ∈ J having multiple codings. A countable set is negligible in the sense of the Hausdorff dimension, so as far as Hausdorff dimensions are concerned, we may assume that every point in J has a unique coding. Let ϕ: N → (0, ∞) be a function. For 0 ≤ α ≤ β ≤ ∞, we define the set
ϕ n Mn(x, y) Mn(x, y) o Eα,β = (x, y) ∈ J × J : lim inf = α, lim sup = β . n→∞ ϕ(n) n→∞ ϕ(n)
Theorem 1.2. Let Φ = {φi : [0, 1] → [0, 1], i ∈ Λ} be a conformal IFS that satisfies 0 0 φi(x) > 0 or φi(x) < 0 for all i ∈ Λ. Let ϕ: N → (0, ∞) be an increasing function satisfying limn→∞ ϕ(n) = ∞ and limn→∞ ϕ(n + 1) − ϕ(n) = 0. Then for 0 ≤ α ≤ β ≤ ∞ we have ϕ 2 dimH J ≤ dimH Eα,β ≤ dimH J + dimP J.
Throughout this paper, dimH and dimP denote the Hausdorff dimension and the packing dimension respectively. Corollary 1. Under the hypotheses of Theorem 1.2, if moreover Λ is finite, then ϕ dimH Eα,β = 2 dimH J. BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4089
In fact, if Λ is finite, then dimH J = dimP J (see [18]). We remark that if Λ is infinite, dimH J = dimP J does not hold in general. Our paper is organised as follows. In Section 2 we introduce the conformal IFSs and cite some lemmas that will be used later. We then prove Theorems 1.1 and 1.2 in Sections 3 and 4 respectively. In Section 5 we study the asymptotic behaviour of Mn(x, y) and compute the Hausdorff dimensions of the exceptional sets in the continued fractions system, L¨urothsystem, N-ary system, and triadic Cantor system by applying Theorems 1.1 and 1.2.
2. Preliminaries. This section is devoted to recalling the definition of conformal IFSs and introducing some properties for conformal IFSs. For more information on conformal IFSs, the readers are referred to [18, 19, 25]. We also cite some results on estimating the Hausdorff dimension of a fractal set (see [7]).
Definition 2.1. We call IFS Φ = {φi : [0, 1] → [0, 1], i ∈ Λ} a conformal IFS if the following conditions are satisfied.
(1) (OSC) For i ∈ Λ, φi((0, 1)) ⊂ (0, 1) and for i, j ∈ Λ, i 6= j, φi((0, 1)) ∩ φj((0, 1)) = ∅. (2) There exists an open connected set V ⊂ R such that [0, 1] ⊂ V and all maps 1 φi extend to C -diffeomorphisms of V into V. (3) (BDP) There exists K ≥ 1 such that for any x, y ∈ V and ω ∈ Λ∗ 0 0 |φω(x)| ≤ K|φω(y)|. The topological pressure function P (t) for a conformal IFS Φ is defined as
1 X 0 t P (t) = lim log sup |φω(x)| . n→∞ n ω∈Λn x∈[0,1] Mauldin & Urba´nski[18] proved that P (t) is non-increasing on [0, ∞). They also established a continuity property of a conformal system with respect to its finite subsystems and showed the relationship between the topological pressure function P (t) and the Hausdorff dimension of attractor J. Let F be a finite subset of Λ, and let JF = x ∈ J : there exists a coding ω ∈ F N such that x = π(ω) . Lemma 2.2 ([18]).
dimH J = sup{dimH JF : F is a finite subset of Λ} = inf{t: P (t) ≤ 0}. BDP property (3) in Definition 2.1 provides an estimation on the diameter of a cylinder. n Proposition 1 ([25]). For (ω1, . . . , ωn) ∈ Λ with n ≥ 1, ξ ∈ [0, 1], we have:
(1) The diameter of Jn(ω1, . . . , ωn) satisfies K−1 φ0 (ξ) ≤ |J (ω , . . . , ω )| ≤ K φ0 (ξ) . (2) (ω1,...,ωn) n 1 n (ω1,...,ωn) (2) For every 1 ≤ k ≤ n, |J (ω , . . . , ω )| K−1 ≤ n 1 n ≤ K. (3) |Jk(ω1, . . . , ωk)| · |Jn−k(ωk+1, . . . , ωn)| (3) n |Jn(ω1, . . . , ωn)| ≤ ρ . (4) The following result can be obtained directly from the ψ-mixing property (1). 4090 SAISAI SHI, BO TAN AND QINGLONG ZHOU
0 n 0 m Proposition 2. For m, n, k ∈ N, U, U ∈ C0 ,V,V ∈ C0 , if µ is ψ-mixing, then µ⊗µ (U ×U 0)∩(T ×T )−(n+k)(V ×V 0) ≤ (1+ψ(k))2µ⊗µ(U ×U 0)·µ⊗µ(V ×V 0). Proof. It is readily checked that µ ⊗ µ (U × U 0) ∩ (T × T )−(n+k)(V × V 0) =µ ⊗ µ U ∩ T −(n+k)V × U 0 ∩ T −(n+k)V 0 =µ U ∩ T −(n+k)V · µ U 0 ∩ T −(n+k)V 0 ≤(1 + ψ(k))µ(U)µ(V ) · (1 + ψ(k))µ(U 0)µ(V 0) ≤(1 + ψ(k))2µ ⊗ µ(U × U 0) · µ ⊗ µ(V × V 0).
We cite two properties of Hausdorff dimension, namely H¨olderproperty and Product formulae.
Lemma 2.3 ([7]). Let E ⊂ R. If f : E → R is η-H¨older,i.e., there exists a constant C > 0 such that for all x, y ∈ E, |f(x) − f(y)| ≤ C|x − y|η, then 1 dim f(E) ≤ dim E. H η H
Lemma 2.4 ([7]). If E ⊂ Rd,F ⊂ Rn, then
dimH E + dimH F ≤ dimH (E × F ) ≤ dimH E + dimP F. We end this section by listing some notation of words. n m For n, m ≥ 1, let ω = (ω1, . . . , ωn) ∈ B ⊆ Λ , τ = (τ1, . . . , τm) ∈ C ⊆ Λ ,
• ωτ = (ω1, . . . , ωn, τ1, . . . , τm); • BC = {ωτ : ω ∈ B, τ ∈ C}; • Bm = B...B . | {z } m times
3. Proof of Theorem 1.1. We divide the proof of Theorem 1.1 into a sequence of lemmas. Throughout bxc denotes the integer part of the real number x, and X Y (or Y X) means that X ≤ CY for some absolute constant C > 0. Lemma 3.1. Let (J, µ, T ) be a measure-preserving system. Then for almost all (x, y) ∈ J × J, we have M (x, y) 1 lim sup n ≤ . n→∞ log n H∗ 1 Proof. With the convention 0 = ∞, we only need to consider the case H∗ > 0. Fix s1 < s2 < H∗. By the definition of the H∗, we have that X s1 + s2 µ(J (ω , . . . , ω ))2 < exp − n (5) n 1 n 2 n (ω1,...,ωn)∈Λ log n for n large enough. Let un = b c. For k ≥ 1, we have that s1 µ ⊗ µ (x, y) ∈ J × J : Mn(x, y) > un BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4091
∞ X ≤ µ ⊗ µ (x, y) ∈ J × J : Mn(x, y) = k
k=un+1 ∞ n−k X X ≤ µ ⊗ µ (x, y) ∈ J × J : ωj+1(x) = ωj+1(y), . . . , ωj+k(x) = ωj+k(y) .
k=un+1 j=0 Since T is measure-preserving, we deduce that µ ⊗ µ (x, y) ∈ J × J : Mn(x, y) > un ∞ X ≤n µ ⊗ µ (x, y) ∈ J × J : ω1(x) = ω1(y), . . . , ωk(x) = ωk(y)
k=un+1 ∞ X X 2 =n µ(Jk(ω1, . . . , ωk)) k k=un+1 (ω1,...,ωk)∈Λ n s + s o n · exp − 1 2 u (by (5)) 2 n − s2−s1 n 2s1 .
k Taking nk = 2 for k ≥ 1, we readily check that ∞ X µ ⊗ µ (x, y) ∈ J × J : Mnk (x, y) > unk < ∞. (6) k=1 Combining (6) with the Borel-Cantelli lemma, we have that for almost all (x, y) ∈ J × J, M (x, y) 1 lim sup nk ≤ . k→∞ log nk s1 The monotonicity of Mn(x, y) yields that
Mn(x, y) M2k+1 (x, y) M2k+1 (x, y) 1 lim sup ≤ lim sup k = lim sup k+1 ≤ . n→∞ log n k→∞ log 2 k→∞ log 2 s1
Therefore, by the arbitrariness of s1, we have that M (x, y) 1 lim sup n ≤ . n→∞ log n H∗
Lemma 3.2. Let (J, µ, T ) be a ψ-mixing system with an exponential decay. Then for almost all (x, y) ∈ J × J, we have M (x, y) 1 lim sup n ≥ . n→∞ log n H∗ ∗ Proof. We only need to consider the case H < ∞. By the definition of H∗, we can choose an integer sequence {uk}k≥1 such that P 2 − log uk µ(Juk (ω1, . . . , ωuk )) (ω1,...,ωuk )∈Λ lim = H∗. k→∞ uk
For any s > H∗, we have X s + H∗ µ(J (ω , . . . , ω ))2 > exp − u (7) uk 1 uk 2 k uk (ω1,...,ωuk )∈Λ 4092 SAISAI SHI, BO TAN AND QINGLONG ZHOU for k large enough. suk nk Taking nk = e and lk = 2 for k ≥ 1, we claim that uk µ ⊗ µ (x, y) ∈ J × J : Mnk (x, y) < uk for infinitely many k ∈ N = 0. If the claim is proved, for almost all (x, y) ∈ J × J, we have that M (x, y) 1 lim sup n ≥ , n→∞ log n s which completes the proof of the lemma by the arbitrariness of s. We proceed to show the claim. For n > m ≥ 1, set M[m,n](x, y) = max k : ωi+1(x) = ωi+1(y), . . . ,ωi+k(x) = ωi+k(y) for some m − 1 ≤ i ≤ n − k .
Writing {(x, y) ∈ J ×J : Mn(x, y) < k} by {Mn(x, y) < k} for brevity, we have that Mn (x, y) < uk ⊂ M 2 2 (x, y) < uk, 0 ≤ i < lk k [iuk+1,iuk+uk] −u2 ⊂ Mu (x, y) < uk ∩ (T × T ) k M 2 2 (x, y) < uk, 0 ≤ i < lk − 1 . k [iuk+1,iµk+uk] By Proposition2, we deduce that µ ⊗ µ {Mnk (x, y) < uk} ≤µ ⊗ µ {Muk (x, y) < uk} −u2 ∩ (T × T ) k M 2 2 (x, y) < uk, 0 ≤ i < lk − 1 [iuk+1,iµk+uk] l k 2 2lk ≤µ ⊗ µ Muk (x, y) < uk (1 + ψ(uk − uk)) .
2 2nk The exponential decay implies that ψ(uk − uk) 2 → 0 as k → 0. So, uk µ ⊗ µ {Mnk (x, y) < uk} lk X 2 2 2lk ≤ 1 − µ(Juk (ω1, . . . , ωuk )) (1 + ψ(uk − uk)) uk (ω1,...,ωuk )∈Λ n s+H∗ o n o nk − 2s 2 2nk exp − 2 · nk · exp ψ(uk − uk) 2 (by (7)) uk uk s−H∗ 2s nk exp − 2 . uk As a result, we have ∞ X µ ⊗ µ (x, y) ∈ J × J : Mnk (x, y) < uk < ∞, k=1 as desired.
Mn(x,y) We study the behaviour of lim infn→∞ log n in the following two lemmas. Lemma 3.3. Let (J, µ, T ) be a ψ-mixing system with an exponential decay. Then for almost all (x, y) ∈ J × J, we have M (x, y) 1 lim inf n ≥ . n→∞ log n H∗ BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4093
Proof. We may assume H∗ < ∞. For s > H∗, we have that X s + H∗ µ(J (ω , . . . , ω ))2 > exp − n n 1 n 2 n (ω1,...,ωn)∈Λ
log n n for n large enough. Let un = and ln = 2 . As in the proof of Lemma 3.2, s un we deduce that l n 2 2ln µ ⊗ µ {Mn(x, y) < un} ≤ µ ⊗ µ Mun (x, y) < un (1 + ψ(un − un)) and s−H∗ n 2s µ ⊗ µ {Mn(x, y) < un} exp − 2 . un Thus, we have ∞ X µ ⊗ µ (x, y) ∈ J × J : Mn(x, y) < un < ∞. n=1 As a result, we have that for almost all (x, y) ∈ J × J, M (x, y) 1 lim inf n ≥ . n→∞ log n s
Lemma 3.4. Let (J, µ, T ) be a measure-preserving system. Then for almost all (x, y) ∈ J × J, we have M (x, y) 1 lim inf n ≤ . n→∞ log n H∗ ∗ Proof. Likewise, we assume that H > 0. Let {uk}k≥1 be a subsequence of N such that P 2 − log (ω ,...,ω )∈Λuk µ Juk ω1, . . . , ωuk lim 1 uk = H∗. k→∞ uk ∗ For s1 < s2 < H , we have that X 2 s1 + s2 µ J ω , . . . , ω < exp − u uk 1 uk 2 k uk (ω1,...,ωuk )∈Λ
s1u for k large enough. Letting nk = e k , we have that µ ⊗ µ Mnk (x, y) ≥ uk ∞ X ≤ µ ⊗ µ Mnk (x, y) = i
i=uk
∞ nk−i−1 X [ ≤ µ ⊗ µ (x, y): ωj+1(x) = ωj+1(y), . . . , ωj+i(x) = ωj+i(y), ωj+i+1(x)
i=uk j=0 6= ωj+i+1(y) ∪ (x, y): ωnk−i(x) 6= ωnk−i(y), ωnk−i+1(x) = ωnk−i+1(y), . . . , ωnk (x) = ωnk (y) ∩ (J × J) 4094 SAISAI SHI, BO TAN AND QINGLONG ZHOU
∞ X ≤nk µ ⊗ µ (x, y): ω1(x) = ω1(y), . . . , ωi(x) = ωi(y), ωi+1(x) 6= ωi+1(y)
i=uk + µ ⊗ µ (x, y): ω1(x) 6= ω1(y), ω2(x) = ω2(y), . . . , ωi+1(x) = ωi+1(y) ≤2nk · µ ⊗ µ (x, y) ∈ J × J : ω1(x) = ω1(y), . . . , ωuk (x) = ωuk (y) − s2−s1 2s1 ≤2nk . Then we can pick a subsequence uki i≥1 from {uk}k≥1 such that ∞ X µ ⊗ µ (x, y) ∈ J × J : M (x, y) ≥ u < ∞, nki nki i=1 and thus for almost all (x, y) ∈ J × J, M (x, y) 1 lim inf n < . n→∞ log n s1
4. Proof of Theorem 1.2. We begin the proof of Theorem 1.2 with two key lemmas. Let J be the attractor of the 1-dimensional conformal IFS
{φi : [0, 1] → [0, 1], i ∈ Λ}. Let 1 < M ≤ ]Λ be a positive integer and
JM = {x ∈ J : 1 ≤ ωn(x) ≤ M, n ≥ 1}, where ] means the cardinality of a set. Then JM is the attractor of the finite IFS
{φi : [0, 1] → [0, 1], 1 ≤ i ≤ M}. For 1 ≤ c ≤ M, p > 1, set
JM,p,c = {x ∈ JM : ωkp+1(x) = c, k ≥ 0}.
The Hausdorff dimension of JM can be approximated by that of JM,p,c.
Lemma 4.1. Let Φ = {φi : [0, 1] → [0, 1], i ∈ Λ} be a conformal IFS. For > 0, 1 ≤ c ≤ M, there exists a positive integer p0 such that
dimH JM,p,c ≥ dimH JM − for all p ≥ p0. Proof. We write p IM,p,c = (ω1, . . . , ωp) ∈ Λ : ω1 = c, 1 ≤ ωj ≤ M for 1 < j ≤ p . By Lemma 2.2, we only need to show that
1 X 0 dimH JM − P (dimH JM − ) = lim log sup φ (x) ≥ 0. n→∞ (ω1,...,ωnp) n n x∈[0,1] (ω1,...,ωnp)∈IM,p,c n We take a block (ω1, . . . , ωnp) in IM,p,c, that is, (ωkp+1, . . . , ω(k+1)p) ∈ IM,p,c for any 0 ≤ k < n. From (ω1, . . . , ωnp) we delete all the (kp+1)-st terms for 0 ≤ k < n, and denote by (ω1, . . . , ωnp) the caused block (which is of length n(p−1)). Applying inequality (3) repeatedly, we deduce that 1 3n−2 |J (ω , . . . , ω )| ≥ |J (c)|n|J (ω , . . . , ω )|. (8) np 1 np K 1 np−n 1 np BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4095
By (2) and (4) we have that
P (dimH JM − )
1 X 0 dimH JM − = lim log sup φ (x) n→∞ (ω1,...,ωnp) n n x∈[0,1] (ω1,...,ωnp)∈IM,p,c
1 X − dimH JM − = lim log |Jnp(ω1, . . . , ωnp)| 2 |Jnp(ω1, . . . , ωnp)| 2 n→∞ n n (ω1,...,ωnp)∈IM,p,c np 1 1 2 X dimH JM − ≥ lim log |Jnp(ω1, . . . , ωnp)| 2 . n→∞ n n ρ (ω1,...,ωnp)∈IM,p,c This estimation yields that
P (dimH JM − ) ≥ np dim J − 1 X 1 2 2−3n n H M 2 lim log K |J1(c)| |Jnp−n(ω1, . . . , ωnp)| n→∞ n n ρ (ω1,...,ωnp)∈IM,p,c 1 1 ≥ p log + dim J − log K−3|J (c)| 2 ρ H M 2 1
1 X dimH JM − + lim log |Jnp−n(ν1, . . . , νn(p−1))| 2 (by (8)) n→∞ n 1≤ν1,...,νnp−n≤M 1 1 K3 ≥ p log − log , 2 ρ |J1(c)| −3 where the last inequality follows from that log K |J1(c)| < 0 and
1 X dimH JM − lim log |Jnp−n(ν1, . . . , νn(p−1))| 2 ≥ 0. n→∞ n 1≤ν1,...,νnp−n≤M