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Topological Entropy of Free Semigroup Actions for Noncompact Sets DISCRETE AND CONTINUOUS doi:10.3934/dcds.2019041 DYNAMICAL SYSTEMS Volume 39, Number 2, February 2019 pp. 995{1017 TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS FOR NONCOMPACT SETS Yujun Ju and Dongkui Ma ∗ School of Mathematics, South China University of Technology Guangzhou 510641, China Yupan Wang School of Computer Science and Engineering, South China University of Technology Guangzhou 510641, China (Communicated by Xiangdong Ye) Abstract. In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [10], by using the Carath´eodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew- product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with m generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [10], Ma et al. [26], and Misiurewicz [27]. 1. Introduction. Topological entropy was first introduced by Adler, Konheim and McAndrew [1]. Later, Bowen [7] and Dinaburg [17] defined topological entropy for a uniformly continuous map on metric space and proved that for a compact metric space, they coincide with that defined by Adler et al. Since the topological entropy appeared to be a very useful invariant in ergodic theory and dynamical systems, there were several attempts to find its suitable generalizations for other systems such as groups, pseudogroups, graphs, foliations, nonautonomous dynamical systems and so on [3,4,5,6, 10, 11, 12, 13, 19, 20, 21, 22, 23, 25, 34, 35]. Bowen [8] extended the concept of topological entropy for non-compact sets in a way which resembles the Hausdorff dimension. Pesin [28] gave a new characterization of topological entropy and topological pressure for non-compact sets by Carath´eodory structure, which we call Carath´eodory-Pesin structure or C-P structure for short. Topological entropy for non-compact sets can be used to investigate multifractal spectra and saturated 2010 Mathematics Subject Classification. Primary: 37B40, 37A35; Secondary: 37C45. Key words and phrases. C-P structure, free semigroup actions, topological entropy, skew- product, local entropy, Hausdorff dimension. Project supported by National Natural Science Foundation of China grant no.11771149, 11671149. ∗ Corresponding author: Dongkui Ma. 995 996 YUJUN JU, DONGKUI MA AND YUPAN WANG sets of dynamical systems, see, for example, [2, 14, 15, 18, 28, 29, 31, 32, 33]. Bi´s [3] and Bufetov [10] gave the definition of the topological entropy of free semigroup actions on a compact metric space, respectively. Related studies include [6, 11, 12, 13, 22, 23, 25, 30, 34, 35], etc. Similar to [25] , the main purpose of this paper is to introduce the topological entropy, lower and upper capacity topological entropies of a free semigroup action generated by m generators of continuous maps by using the C-P structure, which extends the notion of the topological entropy of the free semigroup action defined by Bufetov [10]. Some important properties of topological entropy and lower and upper capacity topological entropies are proved. We give some applications of these entropies. This paper is organized as follows. In section2, we give some preliminaries. In section3, by using the C-P structure we give the definitions of the topological entropy, lower and upper capacity topological entropies of a free semigroup action. Several of their properties are provided. In section4, two equivalent definitions of topological entropy are given. In section5, with respect to arbitrary subset of a compact metric space, we give a theorem which illustrates the relationship between the upper capacity topological entropy of a free semigroup action and the upper capacity topological entropy of a skew-product transformation. In section6, we estimate the topological entropy of a free semigroup action by local entropies. In section7, we prove that for any free semigroup action with m generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimen- sion of the subset multiplied by the maximum logarithm of the Lipschitz constants. These give generalizations of results of Bufetov [10], Ma et al. [26] and Misiurewicz [27]. 2. Preliminaries. 2.1. Carath´eodory-Pesin structure. Let X and S be arbitrary sets and F = fUs : s 2 Sg a collection of subsets in X. Following Pesin [28], we assume that there exist two functions η; : S! R+ satisfying the following conditions: (1) there exists s0 2 S such that Us0 = ;; if Us = ; then η(s) = (s) = 0; if Us 6= ; then η(s) > 0 and (s) > 0; (2) for any δ > 0 one can find " > 0 such that η(s) ≤ δ for any s 2 S with (s) ≤ "; (3) for any " > 0 there exists a finite or countable subcollection G ⊂ S which covers X (i.e., [s2GUs ⊃ X) and (G):=supf (s): s 2 Gg ≤ ". Let ξ : S! R+ be a function, we say that the set S, collection of subsets F, and the set functions ξ; η; satisfying Conditions (1), (2) and (3), introduce the Carath´eodory-Pesin structure or C-P structure τ on X and write τ = (S; F; ξ; η; ). Given a subset Z of X, α 2 R, and " > 0, we define ( ) X M(Z; α; ") := inf ξ(s)η(s)α ; G s2G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z with (G) ≤ ". By Condition (3) the function M(Z; α; ") is correctly defined. It is non-decreasing as " decreases. Therefore, the following limit exists: m(Z; α) = lim M(Z; α; "): "!0 TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 997 It was shown in [28] that there exists a critical value αC 2 [−∞; 1] such that m(Z; α) = 0; α > αC ; m(Z; α) = 1; α < αC : The number αC is called the Carath´eodory-Pesin dimension of the set Z. Now we assume that the following condition holds: 0 (3 ) there exists > 0 such that for any 0 < " ≤ there exists a finite or countable subcollection G ⊂ S covering X such that (s) = " for any s 2 G. Given α 2 R and " > 0, for any subset Z ⊂ X, define ( ) X R(Z; α; ") = inf ξ(s)η(s)α ; G s2G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z such that (s) = " for any s 2 G. Set r(Z; α) = lim inf R(Z; α; "); r(Z; α) = lim sup R(Z; α; "): "!0 "!0 It was shown in [28] that there exists αC ; αC 2 R such that r(Z; α) = 1; α < αC ; r(Z; α) = 0; α > αC ; r(Z; α) = 1; α < αC ; r(Z; α) = 0; α > αC : The numbers αC and αC are called the lower and upper Carath´eodory-Pesin capacities of the set Z respectively. For any " > 0 and subset Z ⊂ X, put ( ) X Λ(Z; ") := inf ξ(s) ; G s2G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z such that (s) = " for any s 2 G. Assume that the function η satisfies the following condition: (4) η(s1) = η(s2) for any s1; s2 2 S satisfying (s1) = (s2). It was shown in [28] that if the function η satisfies Condition (4) then for any subset Z ⊂ X, log Λ(Z; ") log Λ(Z; ") αC = lim inf ; αC = lim sup : "!0 log (1/η(")) "!0 log (1/η(")) Example 1. Let (X; d) be a metric space, Z ⊂ X. Setting F the collection of the balls fB(x; "): x 2 X; " > 0g; S = f(x; "): x 2 X; " > 0g; ξ(x; ") ≡ 1, and η(x; ") = (x; ") = diamB(x; "); we can get 8 9 < X α= M(Z; α; ") = inf (diamB(xi;"i)) ; G :(xi;"i)2G ; where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z with "i < ". Then αc equals to the Hausdorff dimension of Z. Moreover, the lower and upper Carath´eodory-Pesin capacities of Z also equal to the lower and upper box dimensions of the set Z respectively. 998 YUJUN JU, DONGKUI MA AND YUPAN WANG + 2.2. Words and sequences. Let Fm be the set of all finite words of symbols + 0; 1; : : : ; m − 1. For any w 2 Fm , jwj stands for the length of w, that is, the digits + of symbols in w. Obviously, Fm with respect to the law of composition is a free 0 00 + semigroup with m generators. We write w ≤ w if there exists a word w 2 Fm 00 0 + such that w = w w . For w = i1 : : : ik 2 Fm , denote w = ik : : : i1. Let Σm be the set of all two-side infinite sequences of symbols 0; 1; : : : ; m − 1, that is Σm = f! = (: : : ; i−1; i0; i1;:::): ij = 0; 1; : : : ; m − 1 for all integer jg: The metric on Σm is defined by 0 0 k 0 d (!; ! ) = 1=2 ; where k = inffjnj : in 6= ing: Obviously, Σm is compact with respect to this metric. The bernoulli shift σm : Σm ! Σm is a homeomorphism of Σm given by the formula (σm!)k = ik+1: + Suppose that ! 2 Σm; w 2 Fm ; a; b are integers, and a ≤ b.
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