Best Approximation of Orbits in Iterated Function Systems

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 compact 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, Lürothsystem, 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 Poincarérecurrence 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 dT 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 Poincaré 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 Rényi 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 Rényi entropy H similarly. When H∗ and H coincide, we say that the Rényi 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: continued fractions system, Lürothsystem, 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 dT i(x),T i(y) 1 lim sup 1≤i≤n ≥ n→∞ log n H∗ and − log min dT i(x),T i(y) 1 lim inf 1≤i≤n ≥ . n→∞ log n H∗ The existence of Rényi entropy has been proved only for a few special measures: Bernoulli measures, Markov measures and, more generally, Gibbs measures with Höldercontinuous 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, Lürothsystem, 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 & Urbański[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:

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 ⊗ µU ∩ T −(n+k)V × U 0 ∩ T −(n+k)V 0 =µU ∩ T −(n+k)V · µU 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 µJ 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

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

4 0 2 log K −log αc Writing αc = sup φc(x) , x ∈ [0, 1] and p0 = 1 , we obtain that log ρ when p ≥ p0, P (dimH JM − ) ≥ 0.

Let {kn}n≥1 be an inﬁnite subsequence of N, and {Bn}n≥1 be a sequence of ﬁnite blocks over {1, 2,...,M}, say B = (an, . . . , an ). For x ∈ J , we write n 1 bn M,p,c its coding as

(ω1, . . . , ωk1 , ωk1+1, . . . , ωk2 , ωk2+1,...). We insert the blocks {Bn}n≥1 after the positions {kn}n≥1 in the coding of x, and obtain a new sequence (ω , . . . , ω , a1, . . . , a1 , ω , . . . , ω , a2, . . . , a2 , ω ,...). 1 k1 1 b1 k1+1 k2 1 b2 k2+1 There exists a unique point y := τ(x) in [0, 1] which possesses the above sequence as its coding. In this way, we deﬁne a mapping τ : JM,p,c → [0, 1], and we denote the range by JM,p,c {kn}, {Bn} , i.e., JM,p,c {kn}, {Bn} = τ(JM,p,c).

Lemma 4.2. Let Φ = {φi : [0, 1] → [0, 1], i ∈ Λ} be a conformal IFS such that 0 0 φi(x) > 0 or φi(x) < 0 for all i ∈ Λ. If b + b + ... + b lim 1 2 n = 0, (9) n→∞ kn 4096 SAISAI SHI, BO TAN AND QINGLONG ZHOU where bn = |Bn| is the length of Bn, then dimH JM,p,c = dimH JM,p,c {kn}, {Bn} . Proof. Obviously, the mapping τ establishes a one-to-one correspondence between JM,p,c and JM,p,c {kn}, {Bn} . We write $ for the inverse of τ. Write 0 γ = min sup φi(x) , x ∈ [0, 1] . 1≤i≤M

By OSC, the intervals φ1((0, 1)), φ2((0, 1)), . . . , φM ((0, 1)) are pairwise disjoint. Reindexing them if necessary, we may assume these intervals are arranged from left to right. Step 1. We show dimH JM,p,c {kn}, {Bn} ≥ dimH JM,p,c by establishing the (1 − )-H¨olderproperty of the mapping $ for any ∈ (0, 1), since by Lemma 2.3, the H¨olderproperty implies that dimH JM,p,c {kn}, {Bn} ≥ (1 − ) dimH JM,p,c.

By (9), we can choose l0 large enough such that for any l > l0, 4 1kl K b1+...+bl ≥ . (10) ρ γ Take x1, x2 ∈ JM,p,c {kn}, {Bn} with x1 < x2. There is a greatest integer n such that x1 and x2 belong to a common cylinder of order n. And an integer l exists such that kl + b1 + ... + bl ≤ n < kl+1 + b1 + ... + bl+1.

There is no loss of generality in assuming that l ≥ l0. We consider two cases. Case 1: φ0 (x) > 0 for all i ∈ Λ. i From the construction of JM,p,c {kn}, {Bn} , we know that x1 and x2 are sepa- rated by the cylinder

Jn+p(ω1(x1), . . . , ωn+p−1(x1),M) or Jn+p(ω1(x2), . . . , ωn+p−1(x2), 1), and thus

|x1 − x2| ≥ min |Jn+p(ω1(x1), . . . , ωn+p−1(x1),M)|, |Jn+p(ω1(x2), . . . , ωn+p−1(x2), 1)| −2p p ≥K γ |Jn(ω1(x1), ω2(x1), . . . , ωn(x1))|, where the second inequality follows from (2) and (3). On the other hand, we know that $(x1), $(x2) ∈ Jn−(b1+...+bl) c1, c2, . . . , cn−(b1+...+bl) , l where the block c1, c2, . . . , cn−(b1+...+bl) is obtained by deleting the blocks {Bn}n=1 from (ω1(x1), . . . , ωn(x1)). By Proposition1 and (10), we deduce that |$(x1) − $(x2)| ≤ Jn−(b1+...+bl) c1, c2, . . . , cn−(b1+...+bl) 4 K b1+...+bl ≤ |J (ω (x ), ω (x ), . . . , ω (x ))| γ n 1 1 2 1 n 1 1n ≤ |J (ω (x ), ω (x ), . . . , ω (x ))| ρ n 1 1 2 1 n 1 1− |x1 − x2| .

0 Case 2: φi(x) < 0 for all i ∈ Λ. BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4097

In this case, we note that the gap of x1 and x2 is larger than the length of the cylinder

Jn+p(ω1(xi), . . . , ωn+p−1(xi), 1) or Jn+p(ω1(xi), . . . , ωn+p−1(xi),M), i = 1, 2. And the similar arguments as in Case 1 apply. Step 2. We establish the (1 − )-H¨olderproperty of τ to obtain dimH JM,p,c {kn}, {Bn} ≤ dimH JM,p,c. For y1, y2 ∈ JM,p,c, let x1, x2 be the corresponding points in JM,p,c {kn}, {Bn} respectively. Let n be the smallest integer such that ωn+1(y1) 6= ωn+1(y2), and let L be an integer such that kL ≤ n < kL+1. By (9), we may assume that

1kL (K3ρ)b1+b2+...+bL ≤ . ρ By the choice of n, we have −2p p |y1 − y2| ≥ K γ |Jn(ω1(y1), ω2(y1), . . . , ωn(y1))|.

By the deﬁnition of τ, x1 and x2 have a common preﬁx up to the position n + b1 + b2 + ... + bL. By Proposition1, we deduce that

|x1 − x2| ≤ |Jn+b1+b2+...+bL (ω1(x1), ω2(x1), . . . , ωn+L(x1))|

3 b1+b2+...+bL ≤ (K ρ) |Jn(ω1(y1), ω2(y1), . . . , ωn(y1))| 1n ≤ |J (ω (y ), ω (y ), . . . , ω (y ))| ρ n 1 1 2 1 n 1 1− |y1 − y2| .

We are now in a position to prove Theorem 1.2. The upper bound estimation is obvious by Lemma 2.4, and the lower bound estimation relies on Lemmas 4.1& 4.2 and constructing cantor-like subsets with Hausdorﬀ dimensions approaching to 2 dimH J, and the proof will be divided into several cases according to the values of α and β. We provide a detailed proof for the case 0 < α < β < +∞, and only sketches of proofs for the remaining cases. case 1: 0 < α < β < ∞. Since limn→∞ ϕ(n) = ∞ and limn→∞ ϕ(n + 1) − ϕ(n) = 0, we may assume that for all n ≥ 1, βϕ(n) ≥ 1 and ϕ(n + 1) − ϕ(n) < 1.

We choose two integer sequences {nk}k≥0 and {mk}k≥0 satisfying that β k β k n > 1, − 1 ≤ ϕ(n ) < , m = bβϕ(n )c. 0 α k α k k It is clear that ϕ(n ) β lim k+1 = . (11) k→∞ ϕ(nk) α Note that

ϕ(nk + mk) = ϕ(nk) + ϕ(nk + 1) − ϕ(nk) + ... + ϕ(nk + mk) − ϕ(nk + mk − 1) 4098 SAISAI SHI, BO TAN AND QINGLONG ZHOU

≤ ϕ(nk) + mk max {ϕ(n + 1) − ϕ(n)}. nk≤n mk for all k ≥ 0.

Now we construct Cantor-type subsets of JM . To this end, for k ≥ 0, we write

nk+1 − nk = mk · ιk + θk, where jnk+1 − nk k ιk = , 0 ≤ θk < mk. mk We then define a position set P . By just fixing digits on all positions in P , we obtain the Cantor-type subsets. The position set is defined as [ P = {nk, nk + 1, nk + 2, . . . , nk + mk, nk + 2mk, nk + 3mk, . . . , nk + ιkmk}. k≥0

The digits on the positions in P are given by ank = 1, ank+1 = ... = ank+mk−1 =

M, ank+mk = ank+2mk = ... = ank+ιkmk = 1, and bn = M for n ∈ P. The Cantor-type subsets are deﬁned as \ [ EM ({nk}, {mk}) = Jn(ε1, . . . , εn),

n≥1 (ε1,...,εn)∈Dn

\ [ 0 0 FM ({nk}, {mk}) = Jn(ε1, . . . , εn), 0 0 0 n≥1 (ε1,...,εn)∈Dn where n Dn = (ε1, . . . , εn) ∈ {1, 2,...,M} : εk = ak if k ∈ P , 0 n Dn = (ε1, . . . , εn) ∈ {1, 2,...,M} : εk = bk if k ∈ P . We make two claims. ϕ Claim 1: EM ({nk}, {mk}) × FM ({nk}, {mk}) ⊂ Eα,β. Fix (x, y) ∈ EM ({nk}, {mk}) × FM ({nk}, {mk}), and nk ≤ n < nk+1.

(1) if nk ≤ n < nk + mk, then

mk−1 − 1 ≤ Mn(x, y) ≤ mk − 1;

(2) if nk + mk ≤ n < nk+1, then

Mn(x, y) = mk − 1. From (11), (12) we deduce that M (x, y) n m − 1 m − 1 o lim inf n ≥ min lim k−1 , lim k = α n→∞ ϕ(n) k→∞ ϕ(nk + mk) k→∞ ϕ(nk+1) and M (x, y) n m − 1 m − 1 o lim sup n ≤ max lim k , lim k = β. n→∞ ϕ(n) k→∞ ϕ(nk) k→∞ ϕ(nk + mk) BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4099

On the other hand, since Mnk+1 (x, y) = Mnk+mk (x, y) = mk − 1, it follows from (11), (12) again that M (x, y) m − 1 lim nk+1 = lim k = α k→∞ ϕ(nk+1) k→∞ ϕ(nk+1) and M (x, y) m − 1 lim nk+mk = lim k = β. k→∞ ϕ(nk + mk) k→∞ ϕ(nk + mk) Therefore, M (x, y) M (x, y) lim inf n = α and lim sup n = β. n→∞ ϕ(n) n→∞ ϕ(n)

Claim 2: The density of P ⊂ N is zero. (1) If nk ≤ n < nk + mk, then k−1 X ]{i ≤ n, i ∈ P } = (mj + ιj) + n − nk + 1. j=0

Writing N1 = ∪k≥0{nk, nk + 1, . . . , nk + mk}, we have Pk−1 ]{i ≤ n, i ∈ P } (mj + ιj) m lim sup ≤ lim sup j=0 + k n∈N1,n→∞ n k→∞ nk nk Pk−1 (mj + ιj) m ≤ lim sup j=0 + lim sup k k→∞ nk k→∞ nk m + ι ≤ lim sup k−1 k−1 = 0, k→∞ nk − nk−1 where the third inequality follows from the Stolz-Ces`arotheorem, and the last one holds by (13). (2) If nk + lmk ≤ n < nk + (l + 1)mk for some 0 < l < ιk, then k−1 X ]{i ≤ n, i ∈ P } = (mj + ιj) + mk + l. j=0

(3) If nk + ιkmk ≤ n < nk+1, then k X ]{i ≤ n, i ∈ P } = (mj + ιj). j=0 Claim 2 follows from these estimations.

For j ≥ 0, 1 ≤ t ≤ ιj, we put j−1 X kι0+...+ιj−1+t = nj + (t − 1)mj − (mi + ιi) − 1, i=0  (1 M...M 1), t = 1;  | {z } Bι0+...+ιj−1+t = mj −1  (1), 1 < t ≤ ιj. 0 0 and B = (M...M) for any n ≥ 1 (so |Bn| and |B | have the same length). n | {z } n |Bn| 4100 SAISAI SHI, BO TAN AND QINGLONG ZHOU

For p > 1 and c ∈ {1, 2,...,M}, we can check that 0 JM,p,c({kn}, {Bn}) ⊂ EM ({nk}, {mk}),JM,p,c({kn}, {Bn}) ⊂ FM ({nk}, {mk}), and thus by Claim 1 we get that 0 ϕ JM,p,c({kn}, {Bn}) × JM,p,c({kn}, {Bn}) ⊂ Eα,β. By Lemmas 2.4, 4.2 and Claim 2, ϕ 0 dimH Eα,β ≥ dimH (JM,p,c({kn}, {Bn}) × JM,p,c({kn}, {Bn})) ≥ 2 dimH JM,p,c. ϕ Letting p → ∞ we obtain by Lemma 4.1 that dimH Eα,β ≥ 2 dimH JM . On account of Lemma 2.2 we complete the proof for this case. Similar arguments apply to the remaining cases. In the following, we only give the constructions for the proper sequences {nk}k≥0 and {mk}k≥0. Let 1 − 2 η(n) = sup{ϕ(k + 1) − ϕ(k)} . k≥n Since ϕ(n) is increasing and ϕ(n+1)−ϕ(n) → 0 as n → ∞, it follows that η(n) → ∞ as n → ∞. case 2: 0 < α = β < ∞.

nk+1 = nk + bη(nk)ϕ(nk)c, mk = bαϕ(nk)c for k ≥ 0. case 3: α = 0, 0 < β < ∞. 2k 2k nk = min n ≥ 1: 2 − 1 ≤ ϕ(n) < 2 , mk = bβϕ(nk)c for k ≥ 0. case 4: α = 0, β = ∞. p nk+1 = min{n ≥ 1: nkϕ(nk)−1 ≤ ϕ(n) < nkϕ(nk)}, mk = b nkϕ(nk)c for k ≥ 0. case 5: 0 < α < β = ∞.

nk+1 = min{n ≥ 1: η(nk)ϕ(nk) − 1 ≤ ϕ(n) < η(nk)ϕ(n)},

mk = bαη(nk)ϕ(nk)c for k ≥ 0. case 6: α = β = ∞. p nk+1 = nk + bη(nk)ϕ(nk)c, mk = b η(nk)ϕ(nk)c for k ≥ 0. case 7: α = β = 0. 2 p nk+1 = (nk) , mk = b ϕ(nk)c for k ≥ 0. 5. Applications. 5.1. Best approximation in continued fractions system. Let ([0, 1),T ) be the continued fractions system, where the Gauss transformation T is deﬁned as 1 − 1 , x ∈ (0, 1), T (x) = x x 0, x = 0. Then, every real number x ∈ [0, 1) can be expanded as the continued fraction expansion 1 x = = [a (x), a (x),...]. 1 1 2 a1(x) + a2(x)+. .. BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4101

1 n−1 The digits a1(x) = b x c and an(x) = a1(T (x)) (n ≥ 2) are called the partial n quotients of x. For n ≥ 1 and (a1, . . . , an) ∈ N , we call In(a1, . . . , an) = {x ∈ [0, 1): a1(x) = a1, . . . , an(x) = an} an n-th cylinder. For more information on the continued fraction expansion, the readers are referred to [5, 10, 15, 28]. The corresponding IFS consists of 1 φ (x) = (i ∈ ). i i + x N 0 It is worth pointing out that the system is not uniformly contractive, since φ1(0) =

−1; this can be overcome by considering the system {φi1i2 : i1, i2 ∈ N} which is uniformly contractive. It is well known (see for example [29]) that Gauss transformation T is measure- preserving and ergodic with respect to the Gauss measure G, where G is defined as 1 1 dG = dx. log 2 x + 1 Lemma 5.1. The Rényientropy H of the continued fractions system exists. Proof. Denoting X 2 bn = log G(In(a1, . . . , an)) , n (a1,...,an)∈N We check that bn+m ≤ bn + bm + C −bn for some constant C. So the Rényi entropy H = limn→∞ n exists. Theorem 5.2. Let ([0, 1),T ) be the continued fractions system with the Gauss measure G. Then n M (x, y) 1 o G ⊗ G (x, y) ∈ [0, 1) × [0, 1): lim n = = 1 n→∞ log n H and for 0 ≤ α ≤ β ≤ ∞, ϕ dimH Eα,β = 2. 5.2. Best approximation in Lürothsystem. Let ([0, 1),T ) be the Lürothex- pansion system, where T is the Lürothtransformation given by n(n + 1)x − n, x ∈ 1 , 1 , T (x) = n+1 n 0, x = 0. Then every x ∈ [0, 1) has a Lürothexpansion 1 X 1 x = + , d1(x) d1(x)(d1(x) − 1) . . . dn−1(x)(dn−1(x) − 1)dn(x) n≥2 1 n−1 where the digits d1(x) = x + 1 and dn(x) = d1(T (x)) for n ≥ 2. The corresponding IFS is given by n 1 1 o φ (x) = x + : i ≥ 2 . i i(i − 1) i Jager & de Vroedt [14] proved that the transformation T is invariant and ergodic with respect to the Lebesgue measure L. The Rényi entropy ∞ X 1 2 H = − log . n(n − 1) n=2 4102 SAISAI SHI, BO TAN AND QINGLONG ZHOU

Theorem 5.3. Let ([0, 1),T ) be the L¨uroth expansion system. We have

n M (x, y) 1 o L ⊗ L (x, y) ∈ [0, 1) × [0, 1): lim n = = 1, n→∞ log n H and ϕ dimH Eα,β = 2, 0 ≤ α ≤ β ≤ ∞. 5.3. Best approximation in N-ary system. Let N ≥ 2 be an integer and T be the N-adic transformation deﬁned by T (x) = Nx − bNxc for x ∈ [0, 1]. It is well known that every x admits a unique non-terminating N-ary expansion ε (x) ε (x) x = 1 + 2 + ... = [ε (x), ε (x),...], N N 2 1 2 where εk(x) ∈ {0,...,N − 1}. The IFS is n x + i o φ (x) = : i ∈ {0,...,N − 1} , i N and the R´enyi entropy H = log N.

Theorem 5.4. ([17]) Let ([0, 1),T ) be the N-ary expansion system. We have

n M (x, y) 1 o L ⊗ L (x, y) ∈ [0, 1) × [0, 1): lim n = = 1 n→∞ log n log N and ϕ dimH Eα,β = 2, 0 ≤ α ≤ β ≤ ∞. 5.4. Best approximation in triadic Cantor system. The triadic Cantor set C is an attractor of the finite IFS n x x + 2o φ (x) = , φ (x) = . 1 3 2 3 Define the corresponding map T : C → C as T x = 3x (mod 1). Let µ be the Cantor measure which is the restriction to C of s-dimensional Hausdorff measure, with log 2 s = dimH C = log 3 . Then the Rényi entropy H = log 2. Theorem 5.5. Let (C,T ) be the triadic Cantor system and µ be the Cantor measure. We have n M (x, y) 1 o µ ⊗ µ (x, y) ∈ [0, 1] × [0, 1]: lim n = = 1 n→∞ log n log 2 and 2 log 2 dim Eϕ = , 0 ≤ α ≤ β ≤ ∞. H α,β log 3

Acknowledgments. We would like to thanks for reviewers’ comments and suggestions. Those comments and suggestions are all valuable and very helpful for revising and improving our paper. BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4103

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[29] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer- Verlag, New York-Berlin, 1982. [30] Q.-L. Zhou, Dimensions of recurrent sets in β-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762–1776. Received October 2020; revised December 2020. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]