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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 = fφi : i 2 Λg be an iterated function system on a com- pact metric space (X; d), where the index set Λ = f1; 2; : : : ; lg with l ≥ 2, or Λ = f1; 2;:::g. We denote by J the attractor of Φ, and by D the subset of points possessing multiple codings. For any x 2 JnD; there is a unique integer sequence f!n(x)gn≥1 ⊂ ΛN, called the digit sequence of x; such that \ fxg = φ!1(x) ◦ · · · ◦ φ!n(x)(X): n In this case we write x = [!1(x);!2(x);:::]: For x; y 2 JnD; we define the shortest distance function Mn(x; y) as Mn(x; y) = max k 2 N: !i+1(x) = !i+1(y); : : : ;!i+k(x) = !i+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 1: 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 dT n(x); x = 0 n!1 for µ-almost all x 2 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 : N ! R+ be a positive function such that (n) ! 0 as n ! 1, fzngn≥1 be a sequence of points in X: Define the \well approximable" set n Wfzng T; = x 2 X : d T (x); zn < (n) for infinitely many n 2 N : n It follows that the orbits fT (x)gn≥1 can be well approximated by the sequence fzngn≥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 Wfz g T; : n Li, Wang, Wu & Xu [16] computed the dimension of Wfzng T; 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 Wfzng T; has null µ-measure if n≥1 µ(Bn) < 1, and has full measure if X X X µ(Bn) = 1 with Rn;m ≤ C µ(Bn); n≥1 1≤n≤m≤N 1≤n≤N −n −m where Bn := B(zn; (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 2 X, we would like to investigate the n n quantitative properties of the distance between fT (x)gn≥1 and fT (y)gn≥1. We study the problem in the framework of iterated function systems. Let (X; d) be a compact metric space. Let Φ = fφi : i 2 Λg be a collection of injective mappings of X, where Λ = f1; 2; : : : ; lg (l ≥ 2) or Λ = f1; 2;:::g. We call Φ a uniformly contractive iterated function system (IFS for short) if there exists 0 < ρ < 1 such that for any i 2 Λ and x; y 2 X, d(φi(x); φi(y)) ≤ ρ · d(x; y): The limit set associated with an IFS fφi : i 2 Λg can be defined as the image ∗ n 1 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 ! = (!1;!2;:::;!n) 2 Λ , we set ∗ 1 φ! = φ!1 ◦ φ!2 ◦ · · · ◦ φ!n : For ! 2 Λ [ Λ and n ≥ 1; we denote by !jn the word (!1;!2;:::;!n) (when ! is a finite word, n ≤ j!j is required, where j!j denotes 1 the length of !). For ! 2 Λ ; fφ!jn(X)gn≥1 is a nested sequence of compact sets, whose intersection is a singleton; the point in the intersection will be denoted by π(!). The limit set (or the attractor) of the IFS is 1 [ \ \ [ J = π(Λ ) = φ!jn(X) = φ!(X): !2Λ1 n≥1 n≥1 ! : !2Λn The map π :Λ1 ! J is called the coding map, which is surjective but is not 1 necessarily injective. Any ! = (!1;!2;:::) 2 Λ such that x = π(!) is called a coding of x 2 J, and a point in J may have multiple codings. Let D = fx 2 J : x has multiple codingsg: Hence, for any x 2 JnD, there is a unique coding (!1(x);!2(x);:::) such that π(!) = x, whence we write x = [!1(x);!2(x);:::]: For x = [!1(x);!2(x);!3(x);:::] 2 JnD; we define T x = T ([!1(x);!2(x);!3(x);:::]) = [!2(x);!3(x);:::]: BEST APPROXIMATION OF ORBITS IN ITERATED FUNCTION SYSTEMS 4087 n For n ≥ 1 and (!1;:::;!n) 2 Λ ; we call Jn(!1;:::;!n) = φ!1 ◦ · · · ◦ φ!n (X) an n-th order cylinder. For x 2 JnD, let Jn(x) denote the unique n-th cylinder that contains x, i.e., Jn(x) = Jn(!1(x);:::;!n(x)): The σ-algebra over J generated n by all n-th order cylinders are denoted as C0 : For x = [!1(x);!2(x);:::]; y = [!1(y);!2(y);:::] 2 JnD; we define the shortest distance function Mn(x; y) as Mn(x; y) = max k 2 N: !i+1(x) = !i+1(y); : : : ;!i+k(x) = !i+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) 2 JnD × JnD: i i Remark 1. (1) Mn(x; y) = max k 2 N: T (x) 2 Jk(T (y)) for some 0 ≤ i ≤ n − k ; (2) We define a metric d on JnD: if x = [!1;!2;:::]; y = [ν1; ν2;:::] 2 JnD; the distance of x and y is defined as d(x; y) = exp − inffi ≥ 1: !i 6= νig : 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 2 J has a unique coding. For this system, we define the lower R´enyi entropy H∗ to be P 2 − log n µ(J (! ;:::;! )) (!1;:::;!n)2Λ n 1 n H∗ = lim inf ; n!1 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 -mixing if for any U 2 C0 ;V 2 C0 (m; n 2 N); −(n+k) µ U \ T V − µ(U)µ(V ) ≤ (k)µ(U)µ(V ); (1) where : N ! R+ vanishes at infinity. The system (J; µ, T ) is said to be -mixing with an exponential decay if moreover the function in (1) satisfies (n + 1) lim sup < 1: n!1 (n) We mention several examples which are -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 -mixing system with an exponential decay. (1) For µ ⊗ µ-almost all (x; y) 2 J × J; we have M (x; y) 1 lim sup n = : n!1 log n H∗ (2) For µ ⊗ µ-almost all (x; y) 2 J × J; we have M (x; y) 1 lim inf n = : n!1 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) 2 J × J; M (x; y) 1 lim n = : n!1 log n H 1 1 Here we use the convention that 0 = 1 and 1 = 0: Remark 2. Fix a point y0 = [!1(y0);!2(y0);:::] 2 JnD: For x = [!1(x);!2(x);:::]; we define a new quantity Mn(x; y0) similar to Mn(x; y) as follows: Mn(x; y0) = max k 2 N: !i+1(x) = !1(y0); : : : ;!i+k(x) = !k(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!1 n and define the upper local entropy Dµ(yo) similarly.
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