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 Carathéodory-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 Carathéodory structure, which we call Carathéodory-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 dimension 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 = {Us : s ∈ S} a collection of subsets in X. Following Pesin [28], we assume that there exist two functions η, ψ : S → R+ satisfying the following conditions:

(1) there exists s0 ∈ S such that Us0 = ∅; if Us = ∅ then η(s) = ψ(s) = 0; if Us 6= ∅ then η(s) > 0 and ψ(s) > 0; (2) for any δ > 0 one can find ε > 0 such that η(s) ≤ δ for any s ∈ S with ψ(s) ≤ ε; (3) for any ε > 0 there exists a finite or countable subcollection G ⊂ S which covers X (i.e., ∪s∈GUs ⊃ X) and ψ(G):=sup{ψ(s): s ∈ G} ≤ ε. Let ξ : S → R+ be a function, we say that the set S, collection of subsets F, and the set functions ξ, η, ψ satisfying Conditions (1), (2) and (3), introduce the Carathéodory-Pesin structure or C-P structure τ on X and write τ = (S, F, ξ, η, ψ). Given a subset Z of X, α ∈ R, and ε > 0, we define ( ) X M(Z, α, ε) := inf ξ(s)η(s)α , G s∈G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z with ψ(G) ≤ ε. By Condition (3) the function M(Z, α, ε) is correctly defined. It is non-decreasing as ε decreases. Therefore, the following limit exists: m(Z, α) = lim M(Z, α, ε). ε→0 TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 997

It was shown in [28] that there exists a critical value αC ∈ [−∞, ∞] such that

m(Z, α) = 0, α > αC ,

m(Z, α) = ∞, α < αC .

The number αC is called the Carathéodory-Pesin dimension of the set Z. Now we assume that the following condition holds: 0 (3 ) there exists > 0 such that for any 0 < ε ≤ there exists a finite or countable subcollection G ⊂ S covering X such that ψ(s) = ε for any s ∈ G. Given α ∈ R and ε > 0, for any subset Z ⊂ X, define ( ) X R(Z, α, ε) = inf ξ(s)η(s)α , G s∈G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z such that ψ(s) = ε for any s ∈ G. Set r(Z, α) = lim inf R(Z, α, ε), r(Z, α) = lim sup R(Z, α, ε). ε→0 ε→0

It was shown in [28] that there exists αC , αC ∈ R such that

r(Z, α) = ∞, α < αC , r(Z, α) = 0, α > αC ;

r(Z, α) = ∞, α < αC , r(Z, α) = 0, α > αC .

The numbers αC and αC are called the lower and upper Carathéodory-Pesin capacities of the set Z respectively. For any ε > 0 and subset Z ⊂ X, put ( ) X Λ(Z, ε) := inf ξ(s) , G s∈G where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z such that ψ(s) = ε for any s ∈ G. Assume that the function η satisfies the following condition: (4) η(s1) = η(s2) for any s1, s2 ∈ S satisfying ψ(s1) = ψ(s2). It was shown in [28] that if the function η satisfies Condition (4) then for any subset Z ⊂ X, log Λ(Z, ε) log Λ(Z, ε) αC = lim inf , αC = lim sup . ε→0 log (1/η(ε)) ε→0 log (1/η(ε))

Example 1. Let (X, d) be a metric space, Z ⊂ X. Setting F the collection of the balls {B(x, ε): x ∈ X, ε > 0}, S = {(x, ε): x ∈ X, ε > 0}, ξ(x, ε) ≡ 1, and η(x, ε) = ψ(x, ε) = diamB(x, ε), we can get    X α M(Z, α, ε) = inf (diamB(xi, εi)) , G (xi,εi)∈G  where the infimum is taken over all finite or countable subcollections G ⊂ S covering Z with εi < ε. Then αc equals to the Hausdorff dimension of Z. Moreover, the lower and upper Carathéodory-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 ∈ Fm , |w| 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 ∈ Fm 00 0 + such that w = w w . For w = i1 . . . ik ∈ 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 = {ω = (. . . , i−1, i0, i1,...): ij = 0, 1, . . . , m − 1 for all integer j}.

The metric on Σm is deﬁned by 0 0 k 0 d (ω, ω ) = 1/2 , where k = inf{|n| : in 6= in}.

Obviously, Σm is compact with respect to this metric. The bernoulli shift σm : Σm → Σm is a homeomorphism of Σm given by the formula

(σmω)k = ik+1. + Suppose that ω ∈ Σm, w ∈ Fm , a, b are integers, and a ≤ b. We write ω|[a,b] = w if w = iaia+1 . . . ib−1ib. + Let Σm be the set of all one-side inﬁnite sequences of symbols 0, 1, . . . , m − 1: + Σm = {ω = (i0, i1,...): ij = 0, 1, . . . , m − 1 for all integer j}. 2.3. Topological entropy for a free semigroup action in [10]. In this section, we recall the topological entropy for a free semigroup action. Our presentation follows Bufetov [10]. Let X be a compact metric space with metric d. Suppose a free semigroup with m generators acts on X. Denote the maps corresponding to the generators by f0, f1, . . . , fm−1. Assume that these maps are continuous. + Let w = i1i2 . . . ik ∈ Fm , where ij = 0, 1, . . . , m − 1 for all j = 1, . . . , k. Let −1 −1 −1 −1 f = f f . . . f , f = f f . . . f . Obviously, f 0 = f f 0 . w i1 i2 ik w ik ik−1 i1 ww w w + For each w ∈ Fm , a new metric dw on X (named Bowen metric) is given by

dw(x1, x2) = max d(fw0 (x1), fw0 (x2)). w0≤w

0 00 Clearly, if w ≤ w , then dw0 (x1, x2) ≤ dw00 (x1, x2) for all x1, x2 ∈ X. Let ε > 0, a subset E of X is said to be a (w, ε, f0, . . . , fm−1)-spanning subset if for any x ∈ X, there exists y ∈ E with dw(x, y) < ε. The minimal cardinality of a (w, ε, f0, . . . , fm−1)-spanning subset of X is denoted by B(w, ε, f0, . . . , fm−1). A subset F of X is said to be a (w, ε, f0, . . . , fm−1)-separated subset if for any x, y ∈ F, x 6= y implies dw(x, y) ≥ ε. The maximal cardinality of a (w, ε, f0,..., fm−1)-separated subset of X is denoted by N(w, ε, f0, . . . , fm−1). Let 1 X B(n, ε, f , . . . , f ) = B(w, ε, f , . . . , f ), 0 m−1 mn 0 m−1 |w|=n 1 X N(n, ε, f , . . . , f ) = N(w, ε, f , . . . , f ). 0 m−1 mn 0 m−1 |w|=n In [10], Bufetov deﬁned the topological entropy of a free semigroup action by the formula 1 h(f0, . . . , fm−1) = lim lim sup log B(n, ε, f0, . . . , fm−1) ε→0 n→∞ n TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 999

1 = lim lim sup log N(n, ε, f0, . . . , fm−1). ε→0 n→∞ n

We also denote h(f0, . . . , fm−1) by h(G) with G := {f0, . . . , fm−1}. Remark 1. If m = 1, this deﬁnition coincides with the Bowen entropy for the classical dynamical system (X, f)[7, 36]. 2.4. Local entropy. Let (X, d) be a compact metric space, f : X → X a continuous map and µ a Borel probability measure on X. For any n ≥ 1, ε > 0 and x ∈ X, deﬁne the Bowen ball centered at x by i i Bn(x, ε) = {y ∈ X : d(f (x), f (y)) < ε, for 0 ≤ i ≤ n − 1}. Brin and Katok [9] introduced the notion of local entropy for a single map f in the following way. L 1 hµ (x) = lim lim inf − log µ(Bn(x, ε)) ε→0 n→∞ n is called the lower local entropy of µ at point x ∈ X while the quantity

U 1 hµ (x) = lim lim sup − log µ(Bn(x, ε)) ε→0 n→∞ n is called the upper local entropy of µ at point x ∈ X.

2.5. Hausdorff dimension. Let (X, d) be a metric space. Given a subset Z ⊂ X, s ≥ 0 and δ > 0, we define ( ) s X s Hδ (Z) = inf (diam(U)) : B is a cover of Z and diam(U) < δ for all U ∈ B . U∈B s As δ decreases, Hδ (Z) increases, therefore there exists the limit s s H (Z) = lim Hδ (Z) δ→0 called the s-dimensional Hausdorff measure of Z. The number HD(Z) = inf{s : Hs(Z) = 0} = sup{s : Hs(Z) = ∞} is called the Hausdorff dimension of Z.

3. Topological entropy, lower and upper capacity topological entropies of a free semigroup and their properties. In this section, we introduce the deﬁnitions of topological entropy, lower and upper capacity topological entropies of a free semigroup by using C-P structure and provide some properties of them.

3.1. Topological entropy and lower and upper capacity topological entropies. Let X be a compact metric space with metric d, f0, f1, . . . , fm−1 continuous transformations from X to itself. Suppose that a free semigroup with m generators G = {f0, f1, . . . , fm−1} acts on X. Considering a ﬁnite open cover U of X, let n+1 Sn+1(U) := {U = (U0,U1,...,Un): U ∈ U }, n+1 n+1 Q where U = U and n ≥ 0. For any string U ∈ Sn+1(U), deﬁne the length i=1 of U to be m(U) := n + 1. We put S = S(U) = ∪n≥1Sn(U). For any ω = 1000 YUJUN JU, DONGKUI MA AND YUPAN WANG

+ (i1, i2, . . . , in,...) ∈ Σm, n ≥ 1, and a given string U = (U0,U1,...,Un) ∈ Sn+1(U), we associate the set

Xω(U) = {x ∈ X : x ∈ U0, fij ◦ ... ◦ fi1 (x) ∈ Uj, j = 1, 2, . . . , n}. + If w = i1i2 . . . in = ω|[0,n−1] ∈ Fm , we also denote Xω(U) by Xw(U) for the sake of convenience. Define + F = {Xω(U): U ∈ S(U) and ω ∈ Σm}. and three functions ξ, η, ψ : S → R+ as follows ξ(U) ≡ 1, η(U) = exp(−m(U)), ψ(U) = m(U)−1. It is easy to verify that the set S, collection of subsets F, and the functions ξ, η, and ψ satisfy the Conditions (1), (2) and (3) in section 2.1 and hence they determine a C-P structure τ = (S, F, ξ, η, ψ) on X. + Given w ∈ Fm , |w| = N,Z ⊂ X and α ≥ 0, we define ( ) X Mw(Z, α, U,N) := inf exp(−αm(U)) , Gw U∈Gw where the infimum is taken over all finite or countable collections of strings Gw ⊂ S(U) such that m(U) ≥ N + 1 for all U ∈ Gw and Gw covers Z (i.e., for any + S U ∈ Gw, there exists ωU ∈ Σm such that ωU|[0,N−1] = w and XωU (U) ⊃ U∈Gw + Z, or equivalently, for any U ∈ Gw, there is wU ∈ Fm such that w ≤ wU and S XwU (U) ⊃ Z). U∈Gw Let 1 X M(Z, α, U,N) = M (Z, α, U,N). mN w |w|=N We can easily verify that the function M(Z, α, U,N) is non-decreasing as N increases. Therefore there exists the limit m(Z, α, U) = lim M(Z, α, U,N). N→∞ + 0 Furthermore, given w ∈ Fm and |w| = N, by the Condition (3 ) in section 2.1, we can define ( ) X Rw(Z, α, U,N) = inf exp(−α(N + 1)) Gw U∈Gw

= Λw(Z, U,N) exp(−α(N + 1)), where Λw(Z, U,N) = inf{card(Gw)}, the inﬁmum is taken over all ﬁnite or countable Gw collections of strings Gw ⊂ S(U) such that m(U) = N + 1 for all U ∈ Gw and + Gw covers Z (i.e., for any U ∈ Gw, there is wU ∈ Fm such that wU = w and S XwU (U) ⊃ Z). U∈Gw Let 1 X R(Z, α, U,N) = R (Z, α, U,N), mN w |w|=N and 1 X Λ(Z, U,N) = Λ (Z, U,N). mN w |w|=N TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1001

It is easy to see that R(Z, α, U,N) = Λ(Z, U,N) exp(−α(N + 1)). We set r(Z, α, U) = lim inf R(Z, α, U,N), N→∞ r(Z, α, U) = lim sup R(Z, α, U,N). N→∞ The C-P structure τ generates the Carath´eodory-Pesin dimension of Z and the lower and upper Carath´eodory-Pesin capacities of Z with respect to G. We denote them by hZ (U,G), ChZ (U,G), and ChZ (U,G) respectively. We have

hZ (U,G) = inf{α : m(Z, α, U) = 0} = sup{α : m(Z, α, U) = ∞},

ChZ (U,G) = inf{α : r(Z, α, U) = 0} = sup{α : r(Z, α, U) = ∞}, and ChZ (U,G) = inf{α : r(Z, α, U) = 0} = sup{α : r(Z, α, U) = ∞}. Theorem 3.1. For any set Z ⊂ X, the following limits exist:

hZ (G) = lim hZ (U,G), |U|→0

ChZ (G) = lim ChZ (U,G), |U|→0

ChZ (G) = lim ChZ (U,G). |U|→0 Proof. We use the analogous method as that of [28] . Let V be a ﬁnite open cover of X with diameter smaller than the Lebesgue number of U. Then each element V ∈ V is contained in some element U(V ) ∈ U. To any V = (V0,V1,...,Vk) ∈ Sk+1(V) ⊂ S(V), we associate the string U(V) = {U(V0),U(V1),...,U(Vk)} ∈ + Sk+1(U) ⊂ S(U). Let w = i1i2 . . . iN ∈ Fm . If Gw ⊂ S(V) covers Z, where, for + each V ∈ Gw, m(V) ≥ N + 1 and there is wV ∈ Fm such that w ≤ wV. We denote the word that corresponds to U(V) by wU(V) such that wU(V) = wV, then U(Gw) = {U(V): V ∈ Gw} ⊂ S(U) also covers Z. Then for every α > 0 and N > 0. One can easily see that

Mw(Z, α, U,N) ≤ Mw(Z, α, V,N). It follows that M(Z, α, U,N) ≤ M(Z, α, V,N). Moreover, m(Z, α, U) ≤ m(Z, α, V). This implies that hZ (U,G) ≤ hZ (V,G). Since X is compact it has ﬁnite open covers of arbitrarily small diameter. Therefore,

hZ (U,G) ≤ lim inf hZ (V,G). |V|→0 Let |U| → 0, we have

lim sup hZ (U,G) ≤ lim inf hZ (V,G). |U|→0 |V|→0 This implies the existence of the ﬁrst limit. The existence of the two other limits can be proved in similar ways. 1002 YUJUN JU, DONGKUI MA AND YUPAN WANG

The quantities hZ (G), ChZ (G), and ChZ (G) are called the topological entropy and lower and upper capacity topological entropies of G on the set Z respectively.

Remark 2. (1) It is easy to see that hZ (G) ≤ ChZ (G) ≤ ChZ (G). (2) Indeed, let f : X → X be a continuous transformation and G = {f}. Then hZ (G) = hZ (f), ChZ (G) = ChZ (f), ChZ (G) = ChZ (f), for any set Z ⊂ X, where hZ (f), ChZ (f) and ChZ (f) are the topological entropy and lower and upper capacity topological entropies deﬁned by Pesin [28]. If Z = X, then h(G) = h(f) =

ChX (f) = ChX (f), i.e., the classical topological entropy defined by Adler et al [1]. 3.2. Properties of topological entropy and lower and upper capacity topological entropies. Using the basic properties of the Carathéodory-Pesin dimension [28] and definitions, we get the following basic properties of topological entropy and lower and upper capacity topological entropies of a free semigroup action.

Proposition 1. (1) h∅(G) ≤ 0.

(2) hZ1 (G) ≤ hZ2 (G) if Z1 ⊂ Z2.

(3) hZ (G) = supi≥1 hZi (G) where Z = ∪i≥1Zi and Zi ⊂ X, i = 1, 2,....

Proposition 2. (1) Ch∅(G) ≤ 0, Ch∅(G) ≤ 0. (2) Ch (G) ≤ Ch (G) and Ch (G) ≤ Ch (G) if Z ⊂ Z . Z1 Z2 Z1 Z2 1 2 (3) Ch (G) ≥ sup Ch (G) and Ch (G) ≥ sup Ch (G), where Z = Z i≥1 Zi Z i≥1 Zi ∪i≥1Zi and Zi ⊂ X, i = 1, 2,.... (4) If g : X → X is a homeomorphism which commutes with G (i.e., fi ◦ g = g ◦ fi, ∀fi ∈ G = {f0, f1, . . . , fm−1}) then

hZ (G) = hg(Z)(G), ChZ (G) = Chg(Z)(G), ChZ (G) = Chg(Z)(G). Obviously, the function η and ψ satisfy Condition (4) in section 2.1. Therefore, similar to the Theorem 2.2 in [28], we obtain the following lemma. Lemma 3.2. For any open cover U of X and any set Z ⊂ X, there exist the limits log Λ(Z, U,N) ChZ (U,G) = lim inf , N→∞ N log Λ(Z, U,N) ChZ (U,G) = lim sup . N→∞ N Proof. We will prove the ﬁrst equality; the second one can be proved in a similar fashion. Put log Λ(Z, U,N) α = ChZ (U,G), β = lim inf . N→∞ N Given γ > 0, choose a sequence Ni → ∞ such that

0 = r(Z, α + γ, U) = lim R(Z, α + γ, U,Ni). i→∞

It follows that R(Z, α + γ, U,Ni) ≤ 1 for all suﬃciently large i. Therefore, for such numbers i, Λ(Z, U,Ni) exp(−(α + γ)(Ni + 1)) ≤ 1. Moreover, log Λ(Z, U,N ) α + γ ≥ i . Ni + 1 Therefore, log Λ(Z, U,N) α + γ ≥ lim inf . N→∞ N TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1003

Hence, α ≥ β − γ. (1) 0 Let us now choose a sequence Ni such that 0 log Λ(Z, U,Ni ) β = lim 0 . i→∞ Ni We have that 0 lim R(Z, α − γ, U,Ni ) ≥ r(Z, α − γ, U) = ∞. i→∞ 0 This implies that R(Z, α − γ, U,Ni ) ≥ 1 for all suﬃciently large i. Therefore, for such i, 0 0 Λ(Z, U,Ni ) exp(−(α − γ)(Ni + 1)) ≥ 1, and hence, 0 log Λ(Z, U,Ni ) α − γ ≤ 0 . Ni + 1 Taking the limit as i → ∞ we obtain that log Λ(Z, U,N) α − γ ≤ lim inf = β, N→∞ N and consequently, α ≤ β + γ. (2) Since γ can be chosen arbitrarily small the inequalities (1) and (2) imply that α = β.

Remark 3. By the Theorem 3.1 and Lemma 3.2, we can obtain log Λ(Z, U,N) ChZ (G) = lim lim inf , |U|→0 N→∞ N log Λ(Z, U,N) ChZ (G) = lim lim sup . |U|→0 N→∞ N For a free semigroup with m generators acting on X, denote the maps corresponding to the generators by G = {f0, f1, . . . , fm−1}, a set Z ⊂ X is called G-invariant −1 if fi (Z) = Z for all fi ∈ G. For invariant sets, similar to the lower and upper capacity topological entropies of a single map [28], we have the following theorems. Theorem 3.3. For any G-invariant set Z ⊂ X, we have

ChZ (G) = ChZ (G); moreover, for any open cover U of X, we have

ChZ (U,G) = ChZ (U,G).

(1) (2) + (1) Proof. Let Z ⊂ X be a G-invariant set. For any w , w ∈ Fm where |w | = p (2) and |w | = q, we choose two collections of strings Gw(1) ⊂ Sp+1(U) and Gw(2) ⊂ Sq+1(U) which cover Z. Supposing that U = (U0,U1 ...,Up) ∈ Gw(1) and V = (V0,V1 ...,Vq) ∈ Gw(2) , we deﬁne

UV = (U0,U1,...,Up,V0,V1,...,Vq). 1004 YUJUN JU, DONGKUI MA AND YUPAN WANG

For a ﬁxed i ∈ {0, 1, . . . , m − 1}, we consider

Gw := {UV : U ∈ Gw(1) , V ∈ Gw(2) } ⊂ Sp+q+2(U), where w = w(1)iw(2). Then −1 Xw(UV) = Xw(1) (U) ∩ (fi ◦ fw(1) ) (Xw(2) (V)), and m(UV) = m(U) + m(V). Since Z is a G-invariant set, the collection of strings Gw also covers Z. By the deﬁnition of Λw(Z, U, p + q + 1). We have

Λw(Z, U, p + q + 1) ≤ card(Gw) ≤ card(Gw(1) ) × card(Gw(2) ), which implies m−1 1 X Λ (Z, U, p + q + 1) ≤ Λ (1) (Z, U, p) × Λ (2) (Z, U, q). m w w w i=0 It follows that Λ(Z, U, p + q + 1) ≤ Λ(Z, U, p) × Λ(Z, U, q). ap−1 Let ap = log Λ(Z, U, p). Note that Λ(Z, U, p) ≥ 1. Therefore, inf > −∞. The p>1 p desired result is now a direct consequence of the following Lemma 3.4 (The proof follows the Theorem 4.9 in [36]).

ap−1 Lemma 3.4. Let {ap}, p = 1, 2,... be a sequence of numbers satisfying inf > p>1 p ap −∞ and ap+q+1 ≤ ap + aq for all p, q > 1. Then the limit lim exists and p→∞ p coincides with inf ap−1 . p>1 p Next, we discuss the relationship between the topological entropy and upper capacity topological entropy of a free semigroup action generated by G on Z when Z is a compact G-invariant set. Given a compact G-invariant set Z ⊂ X and an open cover U of X, we choose any α > hZ (U,G), then m(Z, α, U) = lim M(Z, α, U,N) = 0. N→∞ Since M(Z, α, U,N) is non-decreasing as N increases and non-negative, it follows + that M(Z, α, U,N) = 0 for any N. Therefore, for any w ∈ Fm and |w| = N, we have Mw(Z, α, U,N) = 0. For Mw(Z, α, U, 2) = 0, there exists Aw ⊂ S(U) + such that Aw covers Z (i.e., for any U ∈ Aw, there exists wU ∈ Fm such that S |wU| = m(U) − 1, w ≤ wU and XwU (U) ⊃ Z) and U∈Gw X Q(Z, α, Aw) = exp(−α · m(U)) < p < 1, (3)

U∈Aw where p is a constant. Since Z is compact we can choose Aw to be ﬁnite and K ≥ 3 to be a constant and K [ Aw ⊂ Sm(U). (4) m=3 (1) (2) + (1) (2) For any w , w ∈ Fm , |w | = |w | = 2 and j ∈ {0, 1, . . . , m − 1}, we can construct

Aw(1)jw(2) = {UV : U ∈ Aw(1) and V ∈ Aw(2) }, where Aw(1) , Aw(2) satisfy (3), (4). Then −1 X(UV) = XwU (U) ∩ (fj ◦ fwU ) (XwV (V)), TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1005 where the word corresponds to UV is wUjwV and m(UV) = m(U) + m(V) ≥ 6. Since Z is G-invariant, then Aw(1)jw(2) covers Z. It is easy to see that 2 Q(Z, α, Aw(1)jw(2) ) ≤ Q(Z, α, Aw(1) ) · Q(Z, α, Aw(2) ) < p .

By mathematical induction, for each n ∈ N and j1, . . . , jn−1 ∈ {0, 1, . . . , m − 1}, we (1) (2) (n−1) (n) can deﬁne Aw j1w j2...w jn−1w which covers Z and satisﬁes n (1) (2) (n−1) (n) Q(Z, α, Aw j1w j2...w jn−1w ) < p .

(1) (2) (1) (1) (2) (1) (2) Let Γw j1w ... = Aw ∪Aw j1w ∪.... Since Z is G-invariant, then Γw j1w ... covers Z and ∞ X n (1) (2) Q(Z, α, Γw j1w ...) ≤ p < ∞. n=1 + Therefore, for any ω ∈ Σm, there exists Γω covering Z and Q(Z, α, Γω) < ∞. Put + F = {Γω : ω ∈ Σm}. + Condition 3.5. For any N > 0 and any w = i1i2 . . . iN ∈ Fm , there exists Γω ∈ F such that for any U ∈ Γω, w ≤ wU and N + 1 ≤ m(U) ≤ N + K, where wU is the word corresponds to U and K is a constant as that in (4). Theorem 3.6. Under the condition 3.5, for any compact G-invariant set Z ⊂ X, we have

hZ (G) = ChZ (G) = ChZ (G); moreover, for any open cover U of X, we have

hZ (U,G) = ChZ (U,G) = ChZ (U,G).

+ Proof. Under the condition 3.5. For any N > 0 and any w = i1i2 . . . iN ∈ Fm , there is Γω ∈ F covering Z such that for any U ∈ Γω, the word corresponds to U is wU and w ≤ wU. Then for any x ∈ Z, there exists a string U =

(U0,U1,...,UN ,...,UN+P ) ∈ Γω such that x ∈ XwU (U), where 0 ≤ P < K. Let ∗ ∗ ∗ U = {U0,U1,...,UN }. Then XwU (U) ⊂ Xw(U ). If Γw denotes the collection of all substrings U∗ constructed above then

−α(N+1) −α(N+1) ∗ αK e Λw(Z, U,N) ≤ e · card(Γw) ≤ max{1, e }Q(Z, α, Γω) ∞ X ≤ max{1, eαK }· pn < ∞, n=1 Therefore, 1 X e−α(N+1)Λ(Z, U,N) = e−α(N+1)Λ (Z, U,N) < ∞. mN w |w|=N

By Lemma 3.2 we obtain that α > ChZ (U,G), and hence the desired result follows Remark2(1).

4. Two equivalent deﬁnitions of topological entropy in the present paper. Now, we describe two other approaches to redeﬁne the topological entropy and lower and upper capacity topological entropies of G = {f0, . . . , fm−1} on any subset of X. 1006 YUJUN JU, DONGKUI MA AND YUPAN WANG

+ 4.1. Definition using Bowen balls. Fix a number δ > 0. Given w ∈ Fm and a point x ∈ X, define the (w, δ)-Bowen ball at x by 0 Bw(x, δ) = {y ∈ X : d(fw0 (x), fw0 (y)) ≤ δ, for w ≤ w}. Put S = X × N. We define the collection of subsets + F = {Bw(x, δ): x ∈ X, w ∈ Fm , |w| = n and n ∈ N}. and three functions ξ, η, ψ : S → R as follows ξ(x, |w|) ≡ 1, η(x, |w|) = exp(−(n + 1)), ψ(x, |w|) = (n + 1)−1, |w| = n. We can easily verify that the collection of subsets F and the functions ξ, η, ψ satisfy Conditions (1), (2), (3) and (3’) in section 2.1. Therefore, they determine a + C-P structure τ = (S, F, ξ, η, ψ) on X. Given w ∈ Fm , |w| = N,Z ⊂ X and α ≥ 0, we define    X 0  M w(Z, α, δ, N) := inf exp(−α · (|w | + 1)) , Gw   Bw0 (x,δ)∈Gw where the infimum is taken over all finite or countable subcollections Gw ⊂ F 0 S covering Z (i.e., for any Bw0 (x, δ) ∈ Gw, w ≤ w and Bw0 (x, δ) ⊃ Z). Bw0 (x,δ)∈Gw Let 1 X M(Z, α, δ, N) = M (Z, α, δ, N). mN w |w|=N We can easily verify that the function M(Z, α, δ, N) is non-decreasing as N increases. Therefore, there exists the limit m(Z, α, δ) = lim M(Z, α, δ, N). N→∞

0 Furthermore, by the Condition (3 ) in section 2.1, we can deﬁne    X  Rw(Z, α, δ, N) = inf exp(−α · (N + 1)) Gw Bw(x,δ)∈Gw 

= Λw(Z, δ, N) exp(−α · (N + 1)), where Λw(Z, δ, N) = inf{card(Gw)}, the inﬁmum is taken over all ﬁnite or countable Gw subcollections Gw ⊂ F covering Z and the words correspond to every ball in Gw are all equal. Let 1 X R(Z, α, δ, N) = R (Z, α, δ, N). mN w |w|=N 1 X Λ(Z, δ, N) = Λ (Z, δ, N). mN w |w|=N We set r(Z, α, δ) = lim inf R(Z, α, δ, N), N→∞ r(Z, α, δ) = lim sup R(Z, α, δ, N). N→∞ TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1007

The C-P structure τ generates the Carath´eodory-Pesin dimension of Z and the lower and upper Carath´eodory-Pesin capacities of Z with respect to G. We denote them by hZ (δ, G), ChZ (δ, G), and ChZ (δ, G) respectively. We have that

hZ (δ, G) = inf{α : m(Z, α, δ) = 0} = sup{α : m(Z, α, δ) = ∞}.

ChZ (δ, G) = inf{α : r(Z, α, δ) = 0} = sup{α : r(Z, α, δ) = ∞}.

ChZ (δ, G) = inf{α : r(Z, α, δ) = 0} = sup{α : r(Z, α, δ) = ∞}. Theorem 4.1. For any set Z ⊂ X, the following limits exist:

hZ (G) = lim hZ (δ, G), δ→0

ChZ (G) = lim ChZ (δ, G), δ→0

ChZ (G) = lim ChZ (δ, G). δ→0 Proof. Let U be a ﬁnite open cover of X, and δ(U) is the Lebesgue number of U. It is easy to see that for every x ∈ X, if x ∈ Xω(U) for some U ∈ Sk+1(U) and some + ω ∈ Σm then 1 B (x, δ(U)) ⊂ X (U) ⊂ B (x, 2|U|). ω|[0,k−1] 2 ω ω|[0,k−1]

If ω|[0,N−1] = w, then we have 1 M (Z, α, 2|U|,N) ≤ M (Z, α, U,N) ≤ M (Z, α, δ(U),N). w w w 2 It follows that 1 M(Z, α, 2|U|,N) ≤ M(Z, α, U,N) ≤ M(Z, α, δ(U),N). 2 This implies that 1 m(Z, α, 2|U|) ≤ m(Z, α, U) ≤ m(Z, α, δ(U)). 2 Hence 1 h (2|U|,G) ≤ h (U,G) ≤ h ( δ(U),G). Z Z Z 2 Let |U| → 0, then δ(U) → 0 and hence the theorem is proved. The existence of the two other limits can be proved in a similar fashion.

4.2. Definition using Bowen’s approach. Let (X, d) be a compact metric space and f : X → X be a continuous map. Bowen [8] used a way which resembles the Hausdorff dimension to construct the topological entropy of f for non-compact sets. In the following, we give the topological entropy of free semigroup actions for non- compact sets by Bowen’s approach [8] and prove that the new definition is equivalent to the definition of topological entropy in section 3.1. Let U be a finite open cover of X. For a set B ⊂ X we write B ≺ U if B is contained in some element of U. + For any ω = (i1, i2,...) ∈ Σm. Let nω,U (B) be the largest non-negative integer n such that B ≺ U and fik ◦ ... ◦ fi1 (B) ≺ U for k = 1, . . . , n − 1. If B ⊀ U then nω,U (B) = 0 and if fik ◦ ... ◦ fi1 (B) ≺ U for all k then nω,U (B) = ∞. Now we set Dω,U (B) = exp − nω,U (B) . 1008 YUJUN JU, DONGKUI MA AND YUPAN WANG

Given Z ⊂ X, ε ∈ (0, 1) and α ≥ 0, let N = [− log(ε)] + 1, where [− log(ε)] is the + integer portion of − log(ε) and w = i1i2 . . . iN ∈ Fm , we define ( ) ( ) X α X µw(Z, α, U, ε) := inf Dω ,U (B) = inf exp − αnω ,U (B) , B B B B B∈B B∈B where the infimum is taken over all covers B of Z such that for any B ∈ B there + exists ωB ∈ Σm satisfied ωB|[0,N−1] = w and DωB ,U (B) < ε. Then we set 1 X µ(Z, α, U, ε) = µ (Z, α, U, ε). mN w |w|=N We can easily verify that the function µ(Z, α, U, ε) is non-decreasing as ε increases. Therefore, there exists the limit µ(Z, α, U) = lim µ(Z, α, U, ε). ε→0 0 µ(Z, α, U) has similar properties as m(Z, α, U), that is, there exists hZ (U,G) such that 0 hZ (U,G) = inf{α : µ(Z, α, U) = 0} = sup{α : µ(Z, α, U) = ∞}. Finally, we set 0 0 hZ (G) = sup{hZ (U,G): U is a finite open cover of Z}, and we have 0 0 hZ (G) = lim hZ (U,G). |U|→0 0 Theorem 4.2. For any set Z ⊂ X, hZ (G) = hZ (G). In order to prove this theorem, we propose the following lemma. Lemma 4.3. Let U be an open cover of X, for any set Z ⊂ X, 0 < ε < 1,N = + [− log(ε)] + 1, w = i1i2 . . . iN ∈ Fm and α ≥ 0, we have: M (Z, α, U,N) w ≤ µ (Z, α, U, ε) ≤ M (Z, α, U,N). 2 w w

Proof. According to the deﬁnition of Mw(Z, α, U,N), we choose collection of strings Gw ⊂ S(U) such that m(U) ≥ N + 1 for all U ∈ Gw and Gw covers Z. For any U = + (Ui0 ,Ui1 ,...,Uim(U)−1 ) ∈ Gw, then there exists ωU ∈ Σm such that ωU|[0,N−1] = w and we associate the set X (U) = U ∩ f −1U ∩ ... ∩ (f ◦ ... ◦ f )−1U . ωU i0 i1 i1 im(U)−1 i1 im(U)−1 Moreover, we have nωU,U XωU (U) ≥ m(U) ≥ N + 1. It follows that exp − nωU,U XωU (U) < ε.

Therefore, we get a cover B = {XωU (U): U ∈ Gw} of Z and DωU,U (XωU (U)) < ε for all XωU (U) ∈ B. For α ≥ 0, this implies that X X exp − αm(U) ≥ exp − αnωU,U Xω(U)

U∈Gw U∈Gw ( ) X ≥ inf exp − αnω ,U (B) . B B B∈B TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1009

Moreover, we have ( ) ( ) X X inf exp − αm(U) ≥ inf exp − αnωB ,U (B) . Gw B U∈Gw B∈B Hence

Mw(Z, α, U,N) ≥ µw(Z, α, U, ε).

On the other hand, for any λ > µw(Z, α, U, ε), there is a cover B of Z such that P α P DωB ,U (B) = exp − αnωB ,U (B) < λ. For every B ∈ B, there exists B∈B B∈B + ωB ∈ Σm such that ωB|[0,N−1] = w and DωB ,U (B) < ε. In the following, we discuss two cases of nωB ,U (B).

Case 1. If nωB ,U (B) < ∞. There exists UB = (Ui0 ,Ui1 ,...,Uin (B)−1 ) such ωB ,U that −1 −1 B ⊂ Ui0 ∩ fi Ui1 ∩ ... ∩ (fin (B)−1 ◦ ... ◦ fi1 ) Uin (B)−1 = XωB (UB). 1 ωB ,U ωB ,U Hence X X exp(−αnωB ,U (B)) = exp(−αm(UB)) < λ.

nωB ,U (B)<∞ nωB ,U (B)<∞

Case 2. If nωB ,U (B) = ∞. For such set B, there are at most countable. We + denote these sets by B1,B2,.... For B1, there exists ωB1 ∈ Σm and UB1 = (1) (1) (1) (Ui ,Ui ,...,Ui ) ∈ S(U) such that ωB1 |[0,N−1] = w and B1 ⊂ XωB 0 1 m(UB )−1 1 1 (UB1 ), m(UB1 ) ≥ N + 1 and exp − αm(UB1 ) < λ/2; For B2, there exists ωB2 ∈ Σ+ and U = (U (2),U (2),...,U (2) ) ∈ S(U) such that ω | = w and m B2 i0 i1 im(U )−1 B2 [0,N−1] B2 B ⊂ X (U ), m(U ) ≥ N + 1 and exp − αm(U ) < λ/22; 2 ωB2 B2 B2 B2 . .

For B , there exists ω ∈ Σ+ and U = (U (n),U (n),...,U (n) ) ∈ S(U) n Bn m Bn i0 i1 im(U )−1 Bn such that ω | = w and B ⊂ X (U ), m(U ) ≥ N + 1 and exp − Bn [0,N−1] n ωBn Bn Bn n αm(UBn ) < λ/2 ; . .

Let GB = {UB : B ∈ B}. Then GB is a collection of strings that covers Z and for any U ∈ GB, we have m(U) ≥ N + 1. X exp − αm(U)

U∈GB X X = exp − αm(UB) + exp − αm(UB)

nωB ,U (B)<∞ nωB ,U (B)=∞ ∞ X λ < λ + = 2λ. 2i i=1 Moreover, we obtain that

Mw(Z, α, U,N) < 2λ, 1010 YUJUN JU, DONGKUI MA AND YUPAN WANG and consequently, M (Z, α, U,N) w ≤ µ (Z, α, U, ε). 2 w

The proof of Theorem 4.2. According to Lemma 4.3, it follows that M(Z, α, U,N) ≤ µ(Z, α, U, ε) ≤ M(Z, α, U,N), 2 Let ε → 0, then N → ∞, we have m(Z, α, U) ≤ µ(Z, α, U) ≤ m(Z, α, U). 2 0 By the deﬁnitions of hZ (G) and hZ (G), we have 0 hZ (U,G) = hZ (U,G). Letting |U| → 0, and the theorem is proved.

0 Remark 4. If G = {f}, then for any Z ⊂ X, hZ (G) is equal to the topological entropy of f on Z deﬁned by Bowen [8].

5. Relationship between the upper capacity topological entropies of a skew-product transformation and a free semigroup action. Let X be a compact metric space with metric d, suppose a free semigroup with m generators acts on X, the generators are continuous transformations G = {f0, f1, . . . , fm−1} of X. + For any subset Z ⊂ X, w ∈ Fm and ε > 0, a subset E ⊂ X is said to be a (w, ε, Z, f0, . . . , fm−1)-spanning set of Z, if for any x ∈ Z, there exists y ∈ E such that dw(x, y) < ε. Deﬁne B(w, ε, Z, f0, . . . , fm−1) to be the minimum cardinality of any (w, ε, Z, f0, . . . , fm−1)-spanning sets of Z. A subset F ⊂ Z is said to be a (w, ε, Z, f0, . . . , fm−1)-separated set of Z, if x, y ∈ F , x 6= y implies dw(x, y) ≥ ε. Let N(w, ε, Z, f0, . . . , fm−1) denotes the maximum cardinality of any (w, ε, Z, f0, . . . , fm−1)-separated sets of Z. For any n ≥ 1, let 1 X B(n, ε, Z, f , . . . , f ) = B(w, ε, Z, f , . . . , f ), 0 m−1 mn 0 m−1 |w|=n 1 X N(n, ε, Z, f , . . . , f ) = N(w, ε, Z, f , . . . , f ). 0 m−1 mn 0 m−1 |w|=n Obviously, ε B(w, , Z, f , . . . , f ) ≥ N(w, ε, Z, f , . . . , f ) ≥ B(w, ε, Z, f , . . . , f ), 2 0 m−1 0 m−1 0 m−1 hence, ε B(n, , Z, f , . . . , f ) ≥ N(n, ε, Z, f , . . . , f ) ≥ B(n, ε, Z, f , . . . , f ). 2 0 m−1 0 m−1 0 m−1 TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1011

+ For any w ∈ Fm , |w| = N, since Λw(Z, ε, N) = B(w, ε, Z, f0, . . . , fm−1), then Λ(Z, ε, N) = B(n, ε, Z, f0, . . . , fm−1). Therefore 1 ChZ (G) = lim lim sup log B(n, ε, Z, f0, . . . , fm−1) ε→0 n→∞ n 1 = lim lim sup log N(n, ε, Z, f0, . . . , fm−1). ε→0 n→∞ n

Remark 5. If Z = X , then ChX (G) = ChX (G) = h(G). Thus upper and lower capacity topological entropies of a free semigroup action on X are as same as the topological entropy of the free semigroup action on X deﬁned by Bufetov [10].

A skew-product transformation F :Σm × X → Σm × X is deﬁned by the formula

F (ω, x) = (σmω, fω0 (x)), where ω = (. . . , ω−1, ω0, ω1,...) ∈ Σm. Here fω0 stands for f0 if ω0 = 0, and for f1 if ω0 = 1, and so on. Let ω = (. . . , ω−1, ω0, ω1,...) ∈ Σm, and the metric D on Σm × X is deﬁned as D(ω, x), (ω0, x) = max d0(ω, ω0), d(x, x0).

For F :Σm × X → Σm × X. Let G = {F }, we can get a C-P structure on

Σm × X. For any set Z ⊂ X, from Pesin [28], we call ChΣm×Z (F ) the upper capacity topological entropy of F on the set Σm × Z. Our purpose is to ﬁnd the relationship between the upper capacity topological entropy ChΣm×Z (F ) of the skew-product transformation F and the upper capacity topological entropy ChZ (G) of a free semigroup action generated by G = {f0, . . . , fm−1}. Theorem 5.1. For any set Z ⊂ X, then

ChΣm×Z (F ) = log m + ChZ (G).

Remark 6. Recall that h(σm) = log m, where h(σm) denotes the topological entropy of σm in symbol space Σm. To prove this theorem, we ﬁrst give the following two lemmas. The proofs of these two lemmas are simliar to that of Bufetov [10]. Therefore, we omit the proof. 1 Lemma 5.2. For any subset Z of X, n ≥ 1 and 0 < ε < 2 , we have X N(n, ε, Σm × Z,F ) ≥ N(w, ε, Z, f0, . . . , fm−1). |w|=n Lemma 5.3. For any subset Z of X, n ≥ 1 and ε > 0, we have X B(n, ε, Σm × Z,F ) ≤ K(ε) B(w, ε, Z, f0, . . . , fm−1) , |w|=n where K(ε) is a positive constant that depends only on ε. Proof of Theorem 5.1. From Lemma 5.2 we have for any subset Z of X, n N(n, ε, Σm × Z,F ) ≥ m · N(n, ε, Z, f0, . . . , fm−1), and then obtain that

ChΣm×Z (F ) ≥ log m + ChZ (G). From Lemma 5.3 we have n B(n, ε, Σm × Z,F ) ≤ K(ε)m · N(n, ε, Z, f0, . . . , fm−1), 1012 YUJUN JU, DONGKUI MA AND YUPAN WANG and hence

ChΣm×Z (F ) ≤ log m + ChZ (G). and the proof is complete.

Remark 7. If Z = X, by Theorem 5.1, we have ChΣm×X (F ) = log m + ChX (G).

Since hΣm×X (F ) = ChΣm×X (F ) and h(f0, . . . , fm−1) = ChX (G), then we get hΣm×X (F ) = log m + h(f0, . . . , fm−1). Therefore, this gives a generalization of the result of Bufetov [10].

Problem. For any subset Z ⊂ X, is it true that hΣm×Z (F ) = log m + hZ (G)? 6. Some estimates of topological entropies of free semigroup actions. Given a compact metric space (X, d), a free semigroup with m generators acts on X, the generators are continuous transformations G = {f0, f1, . . . , fm−1} of X. In this section, we give the concepts of lower and upper local entropies. Moreover, similar as the proof in Ma and Wen [26], we get two estimates. Deﬁnition 6.1. Let µ be a Borel probability measure on X. Then we call

L+ 1 hµ,G(x) = lim lim inf − log min {µ Bw(x, r) } (5) r→0 n→∞ n + 1 |w|=n the L+ lower local entropy of µ at point x with respect to G, while the quantity

L− 1 hµ,G(x) = lim lim inf − log max {µ Bw(x, r) } (6) r→0 n→∞ n + 1 |w|=n is called L− lower local entropy of µ at point x with respect to G.

L+ L− Remark 8. If G = {f}, then hµ,G(x) = hµ,G(x), i.e., the lower local entropy for f deﬁned by Brin and Katok [9]. One can deﬁne the U +(U −) upper local entropy by replacing “ lim inf ” in (5), (6) with “ lim sup ” respectively. n→∞ n→∞ Theorem 6.2. Let µ denote a Borel probability measure on X, Z be a Borel subset of X and s ∈ (0, ∞). If L− hµ,G(x) ≥ s for all x ∈ Z and µ(Z) > 0 then hZ (G) ≥ s. Proof. Fix an ε > 0. For each k ≥ 1, put 1 Zk = x ∈ Z : lim inf − log max {µ(Bw(x, r))} > s − ε for all r ∈ (0, 1/k) . n→∞ n + 1 |w|=n

L− ∞ Since hµ,G(x) ≥ s for all x ∈ Z, the sequence {Zk}k=1 increases to Z. So by the continuity of the measure, we have

lim µ(Zk) = µ(Z) > 0. k→∞ 1 Then select an integer k0 ≥ 1 with µ(Zk0 ) > 2 µ(Z). For each N ≥ 1, put 1 Zk ,N = x ∈ Zk : − log max {µ(Bw(x, r))} > s − ε for all n ≥ N and r ∈ (0, 1/k0) . 0 0 n + 1 |w|=n ∞ ∗ Since the sequence {Zk0,N }N=1 increases to Zk0 , we can pick an N ≥ 1 such that 1 ∗ ∗ 1 ∗ µ(Zk ,N ∗ ) > µ(Zk ). Write Z = Zk ,N ∗ and r = . Then µ(Z ) > 0 and 0 2 0 0 k0 ∗ ∗ ∗ max {µ Bw(x, r) } < exp −(s−ε)·(n+1) for all x ∈ Z , 0 < r ≤ r and n ≥ N . |w|=n (7) TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1013

∗ + ∗ For any N ≥ N , take any w ∈ Fm , |w| = N. Set a cover of Z

n r + o Fw = Bωi| (yi, ): ωi ∈ Σm and ωi|[0,N−1] = w [0,ni−1] 2 which satisﬁes ∗ r ∗ Z ∩ Bω | (yi, ) 6= ∅, ni ≥ N for all i ≥ 1 and 0 < r ≤ r . i [0,ni−1] 2 ∗ r For each i, there exists an xi ∈ Z ∩ Bω | (yi, ). By the triangle inequality i [0,ni−1] 2 r Bω | (yi, ) ⊂ Bω | (xi, r). i [0,ni−1] 2 i [0,ni−1] In combination with (7), we can get

X X ∗ exp − (s − ε) · (ni + 1) ≥ µ Bω | (xi, r) ≥ µ(Z ) > 0. i [0,ni−1] i≥1 i≥1

∗ ∗ ∗ Therefore, M w(Z , s − ε, r, N) ≥ µ(Z ) > 0 for all N ≥ N , and we obtain 1 X M(Z∗, s − ε, r, N) = M (Z∗, s − ε, r, N) ≥ µ(Z∗) > 0, mN w |w|=N and consequently m(Z∗, s − ε, r) = lim M(Z∗, s − ε, r, N) > 0, N→∞ which in turn implies that hZ∗ (r, G) ≥ s − ε. Then we have hZ∗ (G) ≥ s − ε by letting r → 0. It follows that hZ (G) ≥ hZ∗ (G) ≥ s − ε and hence hZ (G) ≥ s since ε > 0 is arbitrary. The proof is completed now.

+ Lemma 6.3. Let r > 0 and B(r) = {Bw(x, r): x ∈ X, w ∈ Fm }. For any family F ⊂ B(r), there exists a (not necessarily countable) subfamily G ⊂ F consisting of disjoint balls such that [ [ ⊂ Bw(x, 3r).

B∈F Bw(x,r)∈G Proof. The proof follows [26] and is omitted.

Theorem 6.4. Let µ denote a Borel probability measure on X, Z be a Borel subset of X and s ∈ (0, ∞). If

L+ hµ,G(x) ≤ s for all x ∈ Z then hZ (G) ≤ s.

L+ + Proof. Since hµ,G(x) ≤ s for all x ∈ Z, then for any ω ∈ Σm and x ∈ Z,

1 L+ lim lim inf − log µ Bω| (x, r) ≤ hµ,G(x) ≤ s. r→0 n→∞ n + 1 [0,n−1] + For any N ≥ 1 and w = i1i2 . . . iN ∈ Fm . Choose ε > 0, we have Z = ∪k≥1Zk, where 1 Zk = x ∈ Z : lim inf − log µ Bω| (x, r) < s + ε n→∞ n + 1 [0,n−1] + for all r ∈ (0, 1/k), ω ∈ Σm and ω|[0,N−1] = w . 1014 YUJUN JU, DONGKUI MA AND YUPAN WANG

1 + Now ﬁx k ≥ 1 and 0 < r < 3k . For each x ∈ Zk, we take ωx ∈ Σm such that ∞ ωx|[0,N−1] = w, there exists a strictly increasing sequence {nj(x)}j=1 such that µ Bω | (x, r) ≥ exp − (s + ε) · (nj(x) + 1) x [0,nj (x)−1] for all j ≥ 1. So, the set Zk is contained in the union of the sets in the family

n + o Fw = Bω | (x, r): x ∈ Zk, ωx ∈ Σ , ωx|[0,N−1] = w and nj(x) ≥ N . x [0,nj (x)−1] m

By Lemma 6.3, there exists a subfamily Gw = {Bω | (xi, r)}i∈I ⊂ Fw consist- xi [0,ni−1] ing of disjoint balls such that for all i ∈ I. [ Zk ⊂ Bω | (xi, 3r), xi [0,ni−1] i∈I and µ Bω | (xi, r) ≥ exp − (s + ε) · (ni + 1) . xi [0,ni−1]

The index set I is at most countable since µ is a probability measure and Gw is a disjointed family of sets, each of which has positive µ-measure. Therefore, X X M w(Zk, s + ε, 3r, N) ≤ exp − (s + ε) · (ni + 1) ≤ µ Bω | (xi, r) ≤ 1, xi [0,ni−1] i∈I i∈I where the disjointness of {Bω | (xi, r)}i∈I is used in the last inequality. It xi [0,ni−1] follows that 1 X M(Z , s + ε, 3r, N) = M (Z , s + ε, 3r, N) ≤ 1 k mN w k |w|=N and consequently

m(Zk, s + ε, 3r) = lim M(Zk, s + ε, r, N) ≤ 1. N→∞

1 which in turn implies that hZk (3r, G) ≤ s + ε for any 0 < r < 3k . Letting r → 0 yields

hZk (G) ≤ s + ε for any k ≥ 1. By Proposition1(3),

h (G) = h ∞ (G) = sup{h (G)} ≤ s + ε. Z ∪k=1Zk Zk k≥1

Therefore, hZ (G) ≤ s since ε > 0 is arbitrary.

Remark 9. For the topological entropy of a pseudogroup introduced in [5], Bi´s proved a similar result to the Theorem 6.2. In [12], the authors obtained a similar result using the skew product.

Remark 10. If m = 1, then the above theorems coincide with the main results that Ma and Wen proved in [26]. TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1015

7. Topological entropy and hausdorff dimension. In dynamical systems, the relation between topological entropy and Hausdorff dimension is a very important problem which many people have studied, such as Dai et al. [16], Misiurewicz [27], Ma and Wu [24], etc. Let X be a nonempty compact metric space. Supposing that a free semigroup with m generators G = {f0, f1, . . . , fm−1} acts on X, we assume that these maps are Lipschitz self-maps with Lipschitz constant Li respectively. In this section, we show that the topological entropy of a free semigroup action generated by G on any subset Z of X is upper bounded by Hausdorff dimension of Z multiplied by the maximum logarithm of {Li}.

Theorem 7.1. Given a free semigroup with m generators G := {f0, . . . , fm−1} acting on a compact metric space X and fi is a Lipschitz continuous map with the Lipschitz constant Li > 1. Then for any Z ⊂ X we get

hZ (G) ≤ HD(Z) · max{log Li}. i In particular, if Z = X, then

hX (G) ≤ HD(X) · max{log Li}. i

Proof. Fix a ﬁnite open cover U of X with a Lebesgue number δ. Let L = max{Li}. i N−1 + If B ⊂ X and diam(B) < δ then B ≺ U. If diam(B) < δ/L then for any w ∈ Fm 0 + and |w| = N, we have diam fw0 (B) < δ for w ≤ w. So for any ω ∈ Σm, we have + nω,U (B) ≥ N. In other words, for every non-negative N and any ω ∈ Σm, if log δ−log diam(B) N < log L + 1 then N ≤ nω,U (B). Hence,

log δ − log diam(B) ≤ n (B). (8) log L ω,U

log δ The number c = log L is a constant, as long as G and U are ﬁxed. We can rewrite (8) as −c 1/ log L Dω,U (B) = exp(−nω,U (B)) ≤ e (diam(B)) , (9)

+ where ω ∈ Σm. + Therefore if B is a cover of Z, then for any λ > 0, w = i1i2 . . . iN ∈ Fm and every + B ∈ B with ωB ∈ Σm such that ωB|[0,N−1] = w, we have

X −cλ X λ/ log L exp(−λnωB ,U (B)) ≤ e (diam(B)) . (10) B∈B B∈B

Choose λ > HD(Z) · log L. By deﬁnition of Hausdorﬀ dimension we can write λ H log L (Z) = 0, so for any ε ∈ (0, 1) there is a cover B of Z such that for any B ∈ B

e−c(diam(B))1/ log L < ε and X e−cλ (diam(B))λ/ log L < ε. B∈B 1016 YUJUN JU, DONGKUI MA AND YUPAN WANG

P For this B we get DωB ,U (B) < ε by (9) and exp(−λnωB ,U (B)) < ε by (10). So B∈B X µw(Z, λ, U, ε) := inf exp(−λnω ,U (B)) : B covers Z,Dω ,U (B) < ε, B B B B∈B + ωB ∈ Σm,N = [− log(ε)] + 1 and ωB|[0,N−1] = w < ε. This implies that 1 X µ(Z, λ, U, ε) = µ (Z, λ, U, ε) < ε. mN w |w|=N Taking the limit as ε → 0 we get that µ(Z, λ, U) = 0 0 and thus hZ (U,G) ≤ λ. Taking supremum over all ﬁnite open covers U of X, we get 0 hZ (G) = hZ (G) ≤ HD(Z) · max{log Li}. i

Remark 11. (1) Let f : X → X be a Lipschitz continuous map with Lipschitz 0 constant L > 1. If G = {f}, for any Z ⊂ X, Misiurewicz [27] proved that hZ (G) ≤ HD(Z) · log L. (2) In the future we will introduce the topological pressure of free semigroup actions for noncompact sets.

Acknowledgments. The authors would like to thank Professor P. Varandas for providing the reference [12] and really appreciate the referees’ valuable remarks and suggestions that helped a lot.

REFERENCES

[1] R. Adler, A. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309–319. [2] L. Barreira, Ya. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7 (1997), 27–38. [3] A. Bi´s, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst, 11 (2004), 639–648. [4] A. Bi´s, Partial variational principle for ﬁnitely generated groups of polynomial growth and some foliated spaces, Colloq. Math., 110 (2008), 431–449. [5] A. Bi´s, An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier., 63 (2013), 839–863. [6] A. Bi´sand M. Urba´nski,Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63–71. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401–414. [8] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125–136. [9] M. Brin and A. Katok, On local entropy, Geometric Dynamics, Springer, Berlin, Heidelberg, 1007 (1983), 30–38. [10] A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Systems, 5 (1999), 137–143. [11] M. Carvalho, F. Rodrigues and P. Varandas, Semigroup actions of expanding maps, J. Stat. Phys., 166 (2017), 114–136. [12] M. Carvalho, F. Rodrigues and P. Varandas, A variational principle for free semigroup actions, Advances in Math., 334 (2018), 450–487. TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS 1017

[13] M. Carvalho, F. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864–886. [14] E. C. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173–1208. [15] V. Climenhaga, Multifractal formalism derived from thermodynamics for general dynamical systems, Electron. Res. Announc. Math. Sci., 17 (2010), 1–11. [16] X. Dai, Z. Zhou and X. Geng, Some relations between Hausdorff-dimensions and entropies, Sci. China Ser. A, 41 (1998), 1068–1075. [17] E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13–16. [18] Y. Dong and X. Tian, Multifractal analysis of the new level sets, arXiv:1510.06514 (2015). [19] S. Friedland, Entropy of graphs, semigroups and groups, London Mathematical Society Lec- ture Note Series, 228. Cambridge University Press, Cambridge, 1996, 319–343. [20] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105–142. [21] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., 4 (1996), 205–233. [22] X. Lin, D. Ma and Y. Wang, On the measure-theoretic entropy and topological pressure of free semigroup actions, Ergodic Theory Dynam. Systems, 38 (2018), 686–716. [23] D. Ma and S. Liu, Some properties of topological pressure of a semigroup of continuous maps, Dyn. Syst, 29 (2014), 1–17. [24] D. Ma and M. Wu, On Hausdorff dimension and topological entropy, Fractals, 18 (2010), 363–370. [25] D. Ma and M. Wu, Topological pressure and topological entropy of a semigroup of maps, Discrete Contin. Dyn. Syst., 31 (2011), 545–557. [26] J. H. Ma and Z. Y. Wen, A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci., Paris, 346 (2008), 503–507. [27] M. Misiurewicz, On Bowen definition of topological entropy, Discrete Contin. Dyn. Syst., 10 (2004), 827–833. [28] Y. Pesin, Dimension Theory in Dynamical Systems, Chicago: The university of Chicago Press, 1997. [29] C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929–956. [30] F. B. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, J. Math. Phys., 57 (2016), 052704, 27 pp. [31] F. Takens and E. Verbitski, Multifractal analysis of local entropies for expansive homeomor- phisms with specification, Comm. Math. Phys, 203 (1999), 593–612. [32] F. Takens and E. Verbitski, Multifractal analysis of dimensions and entropies, Regul. Chaotic Dyn., 5 (2000), 361–382. [33] F. Takens and E. Verbitski, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317–348. [34] Y. Wang and D. Ma, On the topological entropy of a semigroup of continuous maps, J. Math. Anal. Appl., 427 (2015), 1084–1100. [35] Y. Wang, D. Ma and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl., 435 (2016), 1573–1590. [36] P. Waters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982. Received March 2018; revised August 2018. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]