Abstract
The applicability of Bowen’s equation to an arbitrary subset Z of a compact metric space has been well-studied by Climenhaga (Ergod Theory Dyn Syst 31(4):1163–1182, 2011). This paper aims to generalize the main results obtained by Climenhaga to free semigroup actions. To this end, we adopt the Carathéodory–Pesin structure (C–P structure) and introduce the notions of the topological pressure and lower and upper capacity topological pressures of a free semigroup action for an arbitrary subset. Some properties of these notions follow and we obtain the following three main results. First, by Bowen’s equation, we characterize the Hausdorff dimension of an arbitrary subset, where the points of the subset have the positive lower Lyapunov exponents and satisfy a tempered contraction condition. Second, we give an estimation of the topological pressure of a free semigroup action on an arbitrary subset. Finally, we analyze the relationship between the upper capacity topological pressure of a skew-product transformation and a free semigroup action in an arbitrary subset.
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1 Introduction
As an application of topological pressure, Bowen’s equation plays an important role in dynamical systems. Bowen [7] first studied the connection between topological pressure and Hausdorff dimension. He proved that, for certain compact sets \(J \subset {\mathbb {C}}\) which arise as invariant sets of fractional linear transformations f of the Riemann sphere, the Hausdorff dimension \(t=dim_{H} J\) is the unique root of the equation (namely, Bowen’s equation)
where \(P_{J}\) is the topological pressure of the map \(f: J \rightarrow J\), and \(\varphi (z)=\log |f'(z)|\). Ruelle [31] showed that (1.1) yields the Hausdorff dimension of J as f is a \(C^{1+\varepsilon }\) conformal map on a Riemann manifold and J is a repeller. Gatzouras and Peres [17] further extended the result to the case in which f is \(C^{1}\). When X is a metric space (not necessarily a manifold), Rugh [32] redefined the conformal map and confirmed that the result remains valid. By the Bowen’s equation, there exists extensive literature that studied the settings of Julia sets and non-conformal repellers (see, for example, [1, 9, 15, 22, 23, 28, 29, 34,35,36]).
For a given map f, we conclude the above results derive the Hausdorff dimension of some dynamically significant set J through Bowen’s equation. This leads to an important question of how to find the Hausdorff dimension of subset \(Z \subset J\).
For certain subsets, results in this direction are given by the multifractal analysis (see, for example, [2, 26, 40]).
By Carathéodory structure, Pesin [24] (or see [25])gave a new characterization of topological pressure for non-compact sets, which we referred to as the Carathéodory–Pesin structure or C–P structure for short. This extended the earlier definition of topological entropy for non-compact sets by Bowen [6]. Using the C–P structure introduced in [24], Barreira and Schmeling [3] introduced the notion of the \(u-dimension \) \(dim_{u} Z\) for positive functions u, and showed that \(dim_{u} Z\) is the unique number t such that \(P_{Z}(-tu)=0\). They also confirmed that for a subset Z of a conformal repeller J, where \(u=\log \Vert Df\Vert >0\), \(dim_{u} Z=dim_{H} Z\) holds. This implies that the Hausdorff dimension of any subset \(Z \subset J\) can be given by Bowen’s equation, regardless of compact or invariant Z. Moreover, Climenhaga [14] examined that the applicability of Bowen’s equation to an arbitrary subset Z of a compact metric space. The paper needs positive lower Lyapunov exponents on Z with a tempered contraction condition only, and generalizes the results of the uniformly expanding case.
Definition 1.1
Let X be a compact metric space with metric d and \(f: X \longrightarrow X\) a continuous map. We say that \(f: X \longrightarrow X\) is conformal with factor a(x) if for every \(x \in X\), we have
where \(a:X\longrightarrow [0,\infty )\) is continuous.
Denote the Birkhoff sums of \(\log a\) by
the lower and upper limits of this sequence are then the lower and upper Lyapunov exponent, respectively:
If the limit exists, then their common value is the Lyapunov exponent:
We say that a point \(x\in X\) satisfies tempered contraction condition ([14]), if
Denote \({\mathcal {B}}\) by the set of all points in X which satisfy (1.2). Given \(E \subset {\mathbb {R}}\) and let \({\mathcal {A}}(E)\) be the set of points along whose orbits all the asymptotic exponential expansion rates of the map f lie in E:
and \({\mathcal {A}}(\alpha )={\mathcal {A}}(\{\alpha \})\).
More precisely, Climenhaga [14] proved the following theorem ([13, Theorem 2.4]) :
Theorem 1.1
Let X be a compact metric space and \(f: X\rightarrow X\) be continuous and conformal with factor a(x). Suppose that f has no critical points and no singularities-that is, that \(0<a(x)<\infty \) for all \(x \in X\). Consider \(Z \subset {\mathcal {A}}((0,\infty ))\bigcap {\mathcal {B}}\). Then the Hausdorff dimension of Z is given by
Furthermore, if \(Z \subset {\mathcal {A}}((\alpha ,\infty ))\bigcap {\mathcal {B}}\) for some \(\alpha >0\), then \(t^{*}\) is the unique root of Bowen’s equation
Finally, if \(Z \subset {\mathcal {A}}(\alpha )\) for some \(\alpha >0\), then \(P_{Z}(f,-t \log a)=h_{Z}(f)-t\alpha \), and hence
Here \(P_{Z}(f,-t \log a)\) denotes the topological pressure of f with respect to \(-t \log a\) on Z, \(h_{Z}(f)\) is the topological entropy on Z (see [24])and \(dim_{H}(Z)\) is the Hausdorff dimension of Z.
The above result focuses on a single map. In the past two decades, researchers are growing to study the case of free semigroup actions. Related studies include [4, 8, 10,11,12, 18,19,20, 30, 37, 38]. Naturally, we wonder if the result of Theorem 1.1 remains valid in the case of free semigroup actions. To answer this question and generalize the result of Climenhaga [14], we introduce the notions of new topological pressure by using C–P structure.
This paper is organized as follows. In Sect. 2, we give our main results. In Sect. 3, we give some preliminaries. In Sect. 4, by using the C–P structure we give the new definitions of the topological pressure and lower and upper capacity topological pressures of free semigroup actions. Several of their properties are provided. In Sect. 5, we give other two definitions of the topological pressure and lower and upper capacity topological pressures of free semigroup actions on arbitrary subset of X, which are both equivalent to the definitions in Sect. 4. In Sects. 6, 7 and 8, we give the proofs of the main results, respectively.
2 Statement of Main Results
Let (X, d) be a compact metric space, denote G the free semigroup with m generators \(f_0,f_1,\ldots ,f_{m-1}\) acting on X and let \(f_{i}: X \longrightarrow X\) be continuous and conformal with factor \(a_{i}(x)~(i=0,1,\ldots ,m-1)\). Let \(\Phi =\{\log a_{0},\log a_{1},\ldots , \log a_{m-1}\}\). For \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+}\), where \(F_{m}^{+}\) denotes the set of all finite words of symbols \(0,1,\ldots ,m-1\), denote
and
where
and
For any \(\omega =(i_{0},i_{1},\ldots )\in \Sigma _m^+\), where \(\Sigma _m^+\) denotes the one-side symbol space, denote \(\omega |_{[0,n-1]}:=i_{0}i_{1}\cdots i_{n-1}\) and set
and
then we call \({\underline{\lambda }}_{\omega }(x)\) and \({\overline{\lambda }}_{\omega }(x)\) the lower and upper Lyapunov exponents of the free semigroup G relative to \(\omega \in \Sigma _m^+\) at x. If the two are equal (i.e., if the limit exists), then their common value is the Lyapunov exponent relative to \(\omega \in \Sigma _m^+\) at x:
Our first main result relates the Hausdorff dimension of Z to the topological pressure of \(\Phi \) on Z, provided every point in Z has \({\underline{\lambda }}_{\omega }(x)>0\) and satisfies the following so-called tempered contraction condition:
Denote \({\mathcal {B}}\) as the set of all points in X which satisfy (2.1) .
Remark 2.1
-
(1)
Proposition 6.2 shows that if for any the \(\omega \in \Sigma _m^+\), the Lyapunov exponent of x exists and is positive, that is, if \({\underline{\lambda }}_{\omega }(x)={\overline{\lambda }}_{\omega }(x)>0\) for every \(\omega \in \Sigma _m^+\), then x satisfies (2.1).
-
(2)
We observe that if \(a_{i}(x) \ge 1\) for any \(x \in X\) and \(i=0,1,\ldots ,m-1\), then \({\mathcal {B}}=X\).
-
(3)
We say that x has bounded contraction if \(\inf \{ S_{w}\Phi (x)-S_{w'}\Phi (x): w\in F_{m}^{+}, \overline{w'}\le {\overline{w}}\}>-\infty .\) Any point which has bounded contraction satisfies (2.1).
Given \(E \subset {\mathbb {R}}\) and let \({\mathcal {A}}(E)\) be the set of points along whose orbits all the asymptotic exponential expansion rates of the G relative to \(\omega \in \Sigma _m^+\) lie in E:
In particular, \({\mathcal {A}}((0,\infty ))\) is the set of all points for which \({\underline{\lambda }}_{\omega }(x)>0\) for every \(\omega \in \Sigma _m^+\) and \({\mathcal {A}}(\alpha )={\mathcal {A}}(\{\alpha \})\). The first main result deals with subsets \(Z \subset X\) that lie in both \({\mathcal {A}}((0,\infty ))\) and \({\mathcal {B}}\).
Example 2.1
Let \(X=S^{1}\) be the unit circle, \(f_{0}: X \rightarrow X, f_{0}(x)=x^{2}\) and \(f_{1}: X \rightarrow X, f_{1}(x)=x^{3}\). Then for any \(\omega \in \Sigma _{2}^{+}\) and \(x \in X , [{\underline{\lambda }}_{\omega }(x),{\overline{\lambda }}_{\omega }(x)] \subset [\log 2,\log 3]\). Moreover, for any \(x\in X\), x satisfies tempered contraction condition. Thus, \({\mathcal {A}}((0,\infty ))\) \(={\mathcal {B}}=X\).
Theorem 2.2
Let (X, d) be a compact metric space, G be the free semigroup with m generators \(f_0,f_1,\ldots ,f_{m-1}\) acting on X and let \(f_{i}: X\longrightarrow X\) be continuous and conformal with factor \(a_{i}(x)\). Assume \(f_{i}\) has no critical points and singularities, that is, \(0< a_{i}(x)< \infty \) for all \(x \in X\) and \(i \in \{0,1,2,\ldots ,m-1\}\). Consider \(Z \subset {\mathcal {A}}((0,\infty ))\bigcap {\mathcal {B}}\) and \(\Phi =\{\log a_{0},\log a_{1},\ldots , \log a_{m-1}\}\). Then the Hausdorff dimension of Z is given by
Furthermore, if \(Z \subset {\mathcal {A}}((\alpha ,\infty ))\bigcap {\mathcal {B}}\) for some \(\alpha >0\), then \(t^{*}\) is the unique root of Bowen’s equation
Finally, if \(Z \subset {\mathcal {A}}(\alpha )\) for some \(\alpha >0\), then \(P_{Z}(G,-t \Phi )=h_{Z}(G)-t\alpha \), and hence
Here \(h_{Z}(G)\) is the topological entropy on Z defined by Ju et al in [18] and \( P_{Z}(G,-t \Phi )\) denotes the topological pressure of G with respect to \(-t\Phi \) on Z (see Sect.4).
Remark 2.2
When \(m=1,\) i.e., \(G=\{f\},~\Phi =\{\log a\}\), the result coincides with [14].
Next, we give our second main result. Considering a compact metric space (X, d), a free semigroup \(G=\{f_0, f_1, \ldots , f_{m-1}\}\) acting on X, where \(f_{i}(i=0,1,\ldots ,m-1)\) is continuous transformation from X to itself and \(\varphi _{0}, \varphi _{1}, \ldots , \varphi _{m-1} \in C(X,{\mathbb {R}})\), where \(C(X,{\mathbb {R}})\) denotes the Banach algebra of real-valued continuous functions of X equipped with the supremum norm. Denote \(\Phi =\{\varphi _{0}, \varphi _{1}, \ldots , \varphi _{m-1}\} \). For \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+}\), \(B_{w}(x,r)\) is the (w, r)-Bowen ball at x and denote
Let \(\mu \) be a Borel probability measure on X and \(w\in F_m^{+}\). Denote
and
Now we give two estimations about topological pressure on \(Z \subset X\).
Theorem 2.3
Let \(\mu \) denote a Borel probability measure on X, Z be a Borel subset of X and \( s \in {\mathbb {R}}\).
-
(1)
If \({\underline{P}}(G, \Phi , x)\ge s\) for all \(x\in Z\) and \(\mu (Z)>0\) then \(P_{Z}(G,\Phi ) \ge s\).
-
(2)
If \({\overline{P}}(G, \Phi , x)\le s\) for all \(x\in Z\) then \(P_{Z}(G,\Phi )\le s\).
Remark 2.3
When \(\varphi _{0}=\varphi _{1}=\cdots =\varphi _{m-1}=0\), it coincides with the result in [18]. Moreover, if \( m=1 \), then the above theorem coincides with the main results that Ma and Wen proved in [21]. A similar result of topological entropy were obtained by Biś for pseudogroups in [5].
Finally, the third result describes the relationship between the upper capacity topological pressure of a free semigroup action and the upper capacity topological pressure of a skew-product transformation. Let X be a compact metric space with metric d and suppose a free semigroup \(G=\{ f_0, f_1, \ldots , f_{m-1}\}\) with m generators acting on X, \(\varphi \in C(X,{\mathbb {R}})\), where \(f_{0},f_{1},\ldots , f_{m-1}\) are continuous transformations from X to itself. Let \(F: \Sigma _m \times X \rightarrow \Sigma _m \times X\) be a skew-product transformation and \(g: \Sigma _m \times X \rightarrow {\mathbb {R}}\) be defined by the formula \(g(\omega , x) =c+\varphi (x)\), where \(\Sigma _m\) denotes the two-side symbol space and c is a constant. \({\overline{CP}}_{ \Sigma _m \times Z}(F,g)\) denotes the upper capacity topological pressure of F with respect to g on \(\Sigma _m \times Z\) (see [24]), \({\overline{CP}}_{Z}(G,\varphi )\) the upper capacity topological pressure of G with respect to \(\varphi \) on Z (see Sect. 4), \(P_{ \Sigma _m \times X}(F,g)\) the topological pressure of F with respect to g on \(\Sigma _m \times X\) (see [39]), \(P_{X}(G,\varphi )\) the topological pressure defined by Lin et al in [19], \(h_{\Sigma _m \times X}(F)\) the topological entropy (see [39]) and \(h_{X}(G)\) the topological entropy of \(G=\{ f_{0},f_{1},\ldots ,f_{m-1}\}\) (see [8]). Then we have the following theorem:
Theorem 2.4
For any set \( Z \subset X \), we have
Remark 2.4
-
(1)
If \(g(\omega , x) =\varphi (x)\equiv 0\), it is \({\overline{Ch}}_{ \Sigma _m \times Z}(F) = \log m + {\overline{Ch}}_{Z}(G)\), which coincides with the result in [18]. When \(Z=X\), we have \({\overline{CP}}_{ \Sigma _m \times X}(F,g)= P_{ \Sigma _m \times X}(F,g),~{\overline{CP}}_{X}(G,\varphi )= P_{X}(G,\varphi )\), then we can get \(P_{ \Sigma _m \times X}(F,g)=\log m + P_{X}(G,\varphi )+c\), which has been proved by Lin et al in [19]. Further, if \(g(\omega , x) =\varphi (x)\equiv 0\), it is easy to get \(h_{\Sigma _m \times X}(F)=\log m +h_{X}(G)\), which has been proved by Bufetov in [8].
-
(2)
This result also retains valid for one-side shift, i.e.
$$\begin{aligned} {\overline{CP}}_{ \Sigma ^{+}_m \times Z}(F,g) = \log m + {\overline{CP}}_{Z}(G,\varphi )+c. \end{aligned}$$
3 Preliminaries
3.1 Carathéodory–Pesin structure
Let X and \(\mathcal {{\mathcal {S}}}\) be arbitrary sets and \({\mathcal {F}}=\{U_{s}: s\in {\mathcal {S}}\}\) a collection of subsets in X. From Pesin [24], we assume that there exist two functions \(\eta ,\psi :\mathcal {{\mathcal {S}}}\rightarrow {{\mathbb {R}}^{+}}\) satisfying the following conditions:
(1) there exists \(s_{0}\in \mathcal {{\mathcal {S}}}\) such that \(U_{s_{0}}=\emptyset ;\) if \(U_{s}=\emptyset \) then \(\eta (s)=\psi (s)=0\); if \(U_{s}\not =\emptyset \) then \(\eta (s)>0\) and \(\psi (s)>0\);
(2) for any \(\delta >0\) one can find \(\varepsilon >0\) such that \(\eta (s)\le \delta \) for any \(s\in \mathcal {{\mathcal {S}}}\) with \(\psi (s)\le \varepsilon \);
(3) for any \(\varepsilon >0\) there exists a finite or countable subcollection \({\mathcal {G}}\subset \mathcal {{\mathcal {S}}}\) which covers X (i.e., \(\cup _{s\in {\mathcal {G}}}U_{s}\supset X\)) and \(\psi ({\mathcal {G}})\):=sup{\(\psi (s):s\in \mathcal {{\mathcal {G}}}\)} \(\le \varepsilon \).
Let \(\xi :\mathcal {{\mathcal {S}}}\rightarrow {{\mathbb {R}}^{+}}\) be a function, we say that the set \(\mathcal {{\mathcal {S}}}\), collection of subsets \({\mathcal {F}}\), and the set functions \(\xi ,\eta ,\psi \) satisfying conditions (1), (2) and (3), form the Carathéodory–Pesin structure or C–P structure \(\tau \) on X and write \(\tau =(\mathcal {{\mathcal {S}}},{\mathcal {F}},\xi ,\eta ,\psi )\).
Given a subsetset Z of X, \(\alpha \in {\mathbb {R}}\), and \(\varepsilon >0\), we define
where the infimum is taken over all finite or countable subcollections \({\mathcal {G}} \subset \mathcal {{\mathcal {S}}}\) covering Z with \(\psi ({\mathcal {G}})\le \varepsilon \). By condition (3) the function \(M(Z,\alpha ,\varepsilon )\) is correctly defined. It is non-decreasing as \(\varepsilon \) decreases. Therefore, the following limit exists:
It was shown in [24] that there exists a critical value \(\alpha _{C}\in [-\infty ,\infty ]\) such that
The number \(\alpha _{C}\) is called the Carathéodory–Pesin dimension of the set Z.
Now we assume that the following condition holds:
\((3^{'})\) there exists \( \epsilon >0\) such that for any \(0< \varepsilon \le \epsilon \) there exists a finite or countable subcollection \({\mathcal {G}} \subset \mathcal {{\mathcal {S}}}\) covering X such that \(\psi (s)=\varepsilon \) for any \(s\in {\mathcal {G}}\).
Given \(\alpha \in {\mathbb {R}}\) and \(\varepsilon >0\), for any subset \(Z\subset X\), define
where the infimum is taken over all finite or countable subcollections \({\mathcal {G}} \subset {\mathcal {S}}\) covering Z such that \(\psi (s)=\varepsilon \) for any \( s\in {\mathcal {G}} \). Set
It was shown in [24] that there exist \({\underline{\alpha }}_{C},{\overline{\alpha }}_{C}\in {\mathbb {R}}\) such that
The numbers \({\underline{\alpha }}_{C}\) and \({\overline{\alpha }}_{C}\) are called the lower and upper Carathéodory–Pesin capacities of the set Z respectively.
For any \( \varepsilon >0 \) and subset \( Z \subset X \), put
where the infimum is taken over all finite or countable subcollections \({\mathcal {G}} \subset {\mathcal {S}}\) covering Z such that \(\psi (s)=\varepsilon \) for any \( s\in {\mathcal {G}} \).
Assume that the function \(\eta \) satisfies the following condition:
(4) \(\eta (s_{1})=\eta (s_{2})\) for any \(s_{1}, s_{2}\in \mathcal {{\mathcal {S}}}\) for which \(\psi (s_{1})=\psi (s_{2}) \).
It was shown in [24] that if the function \(\eta \) satisfies condition (4) then for any subset \(Z \subset X\),
3.2 Words and Sequences
Let \(F_m^+\) be the set of all finite words of symbols \(0,1,\ldots ,m-1\). For any \(w \in F_m^+\), |w| stands for the length of w, that is, the digits of symbols in w. Denote \(F_m^+(n)=\{w\in F_m^+: |w|=n, n \in {\mathbb {N}}\}\). Obviously, \(F_m^+\) with respect to the law of composition is a free semigroup with m generators. We write \({w}^{\prime }\le {w}\) if there exists a word \(w^{\prime \prime }\in {F_m^+}\) such that \(w=w^{\prime \prime }w^{\prime }\). For \(w=i_1\cdots i_k \in F_m^{+}\), denote \({\overline{w}}=i_k\cdots i_1\).
Let \(\Sigma _m\) be the set of all two-side infinite sequences of symbols \(0,1,\ldots ,m-1\), that is,
The metric on \(\Sigma _{m}\) is defined by
Obviously, \(\Sigma _m\) is compact with respect to this metric. The Bernoulli shift \(\sigma _{m}:\Sigma _m\rightarrow \Sigma _m\) is a homeomorphism of \(\Sigma _m\) given by the formula
Suppose that \(\omega \in \Sigma _{m},w\in {F_{m}^{+}},a,b\) are integers, and \(a\le {b}\). We write \(\omega |_{[a,b]}=w\) if \(w=i_{a}i_{a+1} \cdots i_{b-1}i_{b}\).
Let \(\Sigma _m^+\) be the set of all one-side infinite sequences of symbols \(0,1,\ldots ,m-1\):
3.3 Hausdorff Dimension
Let (X, d) be a metric space and \({\mathcal {D}}(Z,r)\) denote the collection of countable open covers \(\{U_{i}\}_{i=1}^{\infty }\) of Z for which \(diam U_{i} < r\) for all i. Given a subset \( Z \subset X, t\ge 0\), define
When r decreases, \(m_{H}(Z,t, r)\) increases. Therefore there exists the limit
which is called the t-dimensional Hausdorff measure of Z. The number
is called the Hausdorff dimension of Z.
One may equivalently define Hausdorff dimension of using covers by open balls rather than arbitrary open sets. Let \({\mathcal {D}}^{b}(Z,r)\) denote the collection of countable open balls covers \(\{B(x_{i},r_{i})\}_{i=1}^{\infty }\) of with \(x_{i} \in Z\) and \(r_{i} < r\) for all i, and then define
Finally, define \(m^{b}_{H}(Z,t)\) and \(dim_{H}^{b}Z\) by the same procedure as above. Moreover, from [13, 16] we know \(dim_{H}^{b}Z=dim_{H}Z\).
4 Topological Pressure and Lower and Upper Capacity Topological Pressures of a Free Semigroup Action and Their Properties
In this section, we introduce the definitions of topological pressure and lower and upper capacity topological pressures of a free semigroup action by using C–P structure and provide some properties of them.
4.1 Topological Pressure and Lower and Upper Capacity Topological Pressures
Let X be a compact metric space with metric d, \(f_0,f_1,\ldots ,f_{m-1}\) continuous transformations from X to itself. Suppose that a free semigroup \(G=\{f_0,f_1,\ldots ,f_{m-1}\}\) with m generators acting on X. Given \(\varphi _{0},\varphi _{1},\ldots ,\varphi _{m-1} \in C(X,{\mathbb {R}})\), denote \(\Phi =\{\varphi _{0},\varphi _{1},\ldots ,\varphi _{m-1} \}\). Let \(w=i_1i_2 \ldots i_n \in F_m^+\), where \(i_j=0,1,\ldots ,m-1\) for all \(j=1,\ldots ,n\) and \(f_w=f_{i_1}\circ f_{i_2}\circ \cdots \circ f_{i_n}\). Obviously, \(f_{ww'}=f_wf_{w'}\). For \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+}\), denote
Considering a finite open cover \({\mathcal {U}}\) of X, let
where \({\mathcal {U}}^{n+1}=\prod \limits _{i=1}^{n+1}{\mathcal {U}}\) and \(n \ge 0\). For any string \(\mathbf{U }\in \mathcal {{\mathcal {S}}}_{n+1}({\mathcal {U}})\), define the length of \(\mathbf{U }\) as \(m(\mathbf{U }):=n+1\). We put \({\mathcal {S}}={\mathcal {S}}({\mathcal {U}})= \cup _{n\ge 1} {\mathcal {S}}_{n}({\mathcal {U}})\). For any \(\omega =(i_{1},i_{2},\ldots ,i_{n},\ldots )\in \Sigma _m^+\), \(n\ge 1\), and a given string \(\mathbf{U }=(U_0,U_1,\ldots ,U_{n}) \in {\mathcal {S}}_{n+1}({\mathcal {U}})\), we associate the set
If \(w_{\mathbf{U }}=\omega |_{[0,n-1]}=i_{1}i_{2}\cdots i_{n} \in F_m^+\), we also denote \( X_{\omega }(\mathbf{U })\) by \(X_{w_{\mathbf{U }}}(\mathbf{U })\) for the sake of convenience. Define
and three functions \(\xi ,\eta ,\psi :{\mathcal {S}}\rightarrow {\mathbb {R}}^+\) as follows
It is easy to verify that the set \(\mathcal {{\mathcal {S}}}\), collection of subsets \( {\mathcal {F}},\) and the functions \(\xi , \eta ,\) and \(\psi \) satisfy the conditions (1), (2) and (3) in Sect. 3.1 and hence they determine a C–P structure \(\tau =(\mathcal {{\mathcal {S}}},{\mathcal {F}},\xi ,\eta ,\psi )\) on X.
Given \(w \in F_m^{+}, |w|=N, Z\subset X\) and \(\alpha \in {\mathbb {R}}\), we define
where the infimum is taken over all finite or countable collections of strings \({\mathcal {G}}_{w} \subset \mathcal {{\mathcal {S}}}({\mathcal {U}})\) such that \(m(\mathbf{U })\ge N+1\) for all \(\mathbf{U }\in {\mathcal {G}}_{w}\) and \( {\mathcal {G}}_{w}\) covers Z (i.e., for any \(\mathbf{U }\in {\mathcal {G}}_{w}\), there exists \( w_{\mathbf{U }} \in F^+_m \) such that \( {\overline{w}}\le \overline{w_{\mathbf{U }}}\) and \( \bigcup \limits _{\mathbf{U }\in {\mathcal {G}}_{w}}X_{w_{\mathbf{U }}}(\mathbf{U })\supset Z\)).
Let
We can easily verify that the function \(M(Z,\alpha ,\Phi ,{\mathcal {U}},N)\) is non-decreasing as N increases. Therefore there exists the limit
Furthermore, given \( w \in F_m^+ \) and \( |w|=N \), by the condition \((3^{'})\) in Sect. 3.1, we can define
where \(\Lambda _{w}(Z,\Phi ,{\mathcal {U}},N)=\inf \limits _{{\mathcal {G}}_{w}}\left\{ \sum \limits _{\mathbf{U }\in {\mathcal {G}}_{w}}\exp \left( \sup \limits _{x \in X_{w}(\mathbf{U })} S_{w}\Phi (x)\right) \right\} ,\) the infimum is taken over all finite or countable collections of strings \({\mathcal {G}}_{w} \subset \mathcal {{\mathcal {S}}}({\mathcal {U}})\) such that \(m(\mathbf{U })=N+1\) for all \(\mathbf{U }\in {\mathcal {G}}_{w}\) and \( {\mathcal {G}}_{w}\) covers Z (i.e., for any \(\mathbf{U }\in {\mathcal {G}}_{w}\), there exists \( w_{\mathbf{U }} \in F^+_m \) such that \( w_{\mathbf{U }}=w \) and \( \bigcup \limits _{\mathbf{U }\in {\mathcal {G}}_{w}}X_{w_{\mathbf{U }}}(\mathbf{U })\supset Z\)).
Let
and
It is easy to see that
Set
The C–P structure \(\tau \) generates the Carathéodory–Pesin dimension of Z and the lower and upper Carathéodory–Pesin capacities of Z with respect to G. We denote them by \(P_{Z}(G,\Phi ,{\mathcal {U}}), {\underline{CP}}_{Z}(G,\Phi ,{\mathcal {U}})\) and \({\overline{CP}}_{Z}(G,\Phi ,{\mathcal {U}})\) respectively. We have
and
Theorem 4.1
For any set \(Z\subset X\), the following limits exist:
Proof
We use the analogous method as that of [24]. Let \({\mathcal {V}}\) be a finite open cover of X with diameter smaller than the Lebesgue number of \({\mathcal {U}}\). Then each element \(V\in {\mathcal {V}}\) is contained in some element \(U(V ) \in {\mathcal {U}}\). For any \(\mathbf{V }= (V_{0}, V_{1},\ldots , V_{k})\in \mathcal {{\mathcal {S}}}_{k+1}({\mathcal {V}}) \subset \mathcal {{\mathcal {S}}}({\mathcal {V}})\), we associate the string \(\mathbf{U }(\mathbf{V })=(U(V_{0}),U(V_{1}),\ldots ,U(V_{k}))\in \mathcal {{\mathcal {S}}}_{k+1}({\mathcal {U}}) \subset \mathcal {{\mathcal {S}}}({\mathcal {U}})\). Let \(w=i_{1}i_{2} \ldots i_{N} \in F_{m}^+.\) If \({\mathcal {G}}_{w}\subset \mathcal {{\mathcal {S}}}({\mathcal {V}})\) covers Z, for each \(\mathbf{V } \in {\mathcal {G}}_{w}, m(\mathbf{V })\ge N+1\) and there is \( w_{\mathbf{V }} \in F^+_m\) such that \( {\overline{w}} \le \overline{w_{\mathbf{V }}}.\) We denote the word that corresponds to \( \mathbf{U }(\mathbf{V }) \) by \( w_{\mathbf{U }(\mathbf{V })}\) such that \(w_{\mathbf{U }(\mathbf{V })}=w_{\mathbf{V }} \), then \(\mathbf{U }({\mathcal {G}}_{w}) = \{\mathbf{U }(\mathbf{V }) : \mathbf{V } \in {\mathcal {G}}_{w}\} \subset \mathcal {{\mathcal {S}}}({\mathcal {U}})\) also covers Z. Let
It follows that
Note that \(m(\mathbf{V })=m(\mathbf{U }(\mathbf{V }))\). Then for any \(\alpha \in {\mathbb {R}}\) and \(N>0\), one can easily see that
Then
Moreover,
This implies that
Since X is compact, it has finite open covers of arbitrarily small diameter. Therefore,
If \({|{\mathcal {U}}| \rightarrow 0}\), then \(\gamma \rightarrow 0\) and hence
This implies the existence of the first limit. The existence of the two other limits can be proved in a similar fashion. \(\square \)
The quantities \(P_{Z}(G,\Phi )\), \({\underline{CP}}_{Z}(G,\Phi ),\) and \({\overline{CP}}_{Z}(G,\Phi )\) are called the topological pressure and lower and upper capacity topological pressures of G with respect to \(\Phi \) on the set Z respectively.
Remark 4.1
-
(1)
It is easy to see that \(P_{Z}(G,\Phi )\le {\underline{CP}}_{Z}(G,\Phi )\le {\overline{CP}}_{Z}(G,\Phi )\).
-
(2)
If \(\varphi _{i}(x)=0, i=0,1,\ldots ,m-1\), we can obtain \(P_{Z}(G, 0)=h_{Z}(G)\), where \(h_{Z}(G)\) is the topological entropy on Z in [18]. Similarly, the lower and upper capacity topological pressures are the corresponding the lower and upper capacity topological entropies in [18].
-
(3)
When \(m=1\), i.e., \(G=\{f\}\) and \(\Phi =\{\varphi \}\), we obtain \(P_{Z}(G,\Phi )=P_{Z}(\varphi )\), \({\underline{CP}}_{Z}(G,\Phi )={\underline{CP}}_{Z}(\varphi )\), \({\overline{CP}}_{Z}(G,\Phi )={\overline{CP}}_{Z}(\varphi )\), for any set \(Z \subset X\), where \(P_{Z}(\varphi )\), \({\underline{CP}}_{Z}(\varphi )\) and \({\overline{CP}}_{Z}(\varphi )\) are the topological pressure and lower and upper capacity topological pressures in [24]. That is to say, this definition agrees with Pesin’s [24]. Moveover, if \(Z=X\), then \(P_{X}(f,\varphi )={\underline{CP}}_{X}(f,\varphi )={\overline{CP}}_{X}(f,\varphi )=P(f,\varphi )\), which is equivalent to the classical topological pressure in [39].
4.2 Properties of Topological Pressure and Lower and Upper Capacity Topological Pressures
Using the basic properties of the Carathéodory–Pesin dimension [24] and definitions, we get the following basic properties of topological pressure and lower and upper capacity topological pressures of free semigroup actions.
Proposition 4.1
-
(1)
\(P_{\emptyset }(G,\Phi )\le 0.\)
-
(2)
\(P_{Z_{1}}(G,\Phi )\le P_{Z_{2}}(G,\Phi )\) if \(Z_{1}\subset Z_{2}\).
-
(3)
\(P_{Z}(G,\Phi )=\sup \limits _{i\ge 1}P_{Z_{i}}(G,\Phi )\) where \(Z=\cup _{i\ge 1}Z_{i}\) and \(Z_{i}\subset X , i=1,2,\ldots \).
Proposition 4.2
-
(1)
\({\underline{CP}}_{\emptyset }(G,\Phi )\le 0,~{\overline{CP}}_{\emptyset }(G,\Phi )\le 0.\)
-
(2)
\({\underline{CP}}_{Z_{1}}(G,\Phi )\le {\underline{CP}}_{Z_{2}}(G,\Phi )\) and \({\overline{CP}}_{Z_{1}}(G,\Phi )\le {\overline{CP}}_{Z_{2}}(G,\Phi )\) if \(Z_{1}\subset Z_{2}\).
-
(3)
\({\underline{CP}}_{Z}(G,\Phi ) \ge \sup \limits _{i\ge 1}{\underline{CP}}_{Z_{i}}(G,\Phi )\) and \({\overline{CP}}_{Z}(G,\Phi ) \ge \sup \limits _{i\ge 1}{\overline{CP}}_{Z_{i}}(G,\Phi )\), where \(Z=\cup _{i\ge 1}Z_{i}\) and \(Z_{i}\subset X , i=1,2,\ldots \).
-
(4)
If \(g:X \rightarrow X\) is a homeomorphism which commutes with G (i.e., \(f_{i}\circ g=g \circ f_{i}\), for all \(f_{i}\in \{f_0,f_1,\ldots ,f_{m-1}\}\) ), then
$$\begin{aligned} P_{Z}(G,\Phi )= & {} P_{g(Z)}(G,\Phi \circ g^{-1}), \\ {\underline{CP}}_{Z}(G,\Phi )= & {} {\underline{CP}}_{g(Z)}(G,\Phi \circ g^{-1}), \\ {\overline{CP}}_{Z}(G,\Phi )= & {} {\overline{CP}}_{g(Z)}(G,\Phi \circ g^{-1}), \end{aligned}$$where \(\Phi \circ g^{-1}=\{\varphi _{0}\circ g^{-1},\varphi _{1}\circ g^{-1},\ldots ,\varphi _{m-1} \circ g^{-1}\}\).
Obviously, the functions \(\eta \) and \(\psi \) satisfy condition (4) in Sect. 3.1. Therefore, similar to the Theorem 2.2 in [24], we obtain the following lemma.
Lemma 4.2
For any open cover \({\mathcal {U}}\) of X and any set \(Z\subset X\), we have
Proof
We will prove the first equality; the second one can be proved in a similar fashion. Put
Given \(\gamma >0,\) choose a sequence \(N_{i}\rightarrow \infty \) such that
It follows that \(R(Z,\alpha +\gamma ,\Phi ,{\mathcal {U}},N_{i})\le 1\) for all sufficiently large i. Therefore, for such i
Moreover,
Therefore,
Hence,
Let us now choose a sequence \(N_{i}^{'} \) such that
We have that
This implies that \(R(Z,\alpha -\gamma ,\Phi ,{\mathcal {U}},N_{i}^{'})\ge 1\) for all sufficiently large i. Therefore, for such i
and hence
Taking the limit as \(i \rightarrow \infty \) we obtain that
and consequently,
Since \(\gamma \) can be chosen arbitrarily small, the inequalities (4.2) and (4.3) imply that \(\alpha =\beta .\) \(\square \)
Remark 4.2
By the Theorem 4.1 and Lemma 4.2, we can obtain
Theorem 4.3
If \(\Phi =\{\varphi _{0},\varphi _{1},\ldots , \varphi _{m-1}\}\) and \(\Psi =\{\psi _{0},\psi _{1},\ldots , \psi _{m-1}\}\), we have
where \(\Vert \cdot \Vert \) denotes the supremum norm in the space of continuous functions on X.
Proof
Notice that for any \(w\in F_m^+\),
Hence
Similarly, we have
Therefore,
Taking limit \(N \rightarrow \infty \) yields
Thus
Let \(|{\mathcal {U}}|\rightarrow 0\), and we obtain
which establishes the first inequality. The proof of the two other inequalities is similar. \(\square \)
For a free semigroup G with m generators acting on X, denoting the maps corresponding to the generators by \(f_0,f_1,\ldots ,f_{m-1}\), a set \(Z\subset X\) is called G-invariant if \(f_{i}^{-1}(Z) = Z\) for all \(f_{i}\in G\). For invariant sets, similar to the lower and upper capacity topological pressures of a single map [24], we have the following theorems.
Theorem 4.4
For any G-invariant set \(Z\subset X\), we have
Moreover, for any open cover \({\mathcal {U}}\) of X, we have
Proof
Let \(Z\subset X\) be a G-invariant set. For any \(w^{(1)}, w^{(2)}\in F_m^+\) where \( |w^{(1)}|=p \) and \(|w^{(2)}|=q\), we choose two collections of strings \({\mathcal {G}}_{w^{(1)}}\subset \mathcal {{\mathcal {S}}}_{p+1}({\mathcal {U}})\) and \({\mathcal {G}}_{w^{(2)}}\subset \mathcal {{\mathcal {S}}}_{q+1}({\mathcal {U}})\) which cover Z. Supposing that \(\mathbf{U }=(U_{0},U_{1}\ldots ,U_{p})\in {\mathcal {G}}_{w^{(1)}}\) and \(\mathbf{V }=(V_{0},V_{1}\ldots ,V_{q})\in {\mathcal {G}}_{w^{(2)}}\), we define
For a fixed \(i\in \{0,1,\ldots ,m-1\}\), we consider
where \(w=w^{(1)}iw^{(2)}\). Then
and \( m(\mathbf{U }\mathbf{V })=m(\mathbf{U })+m(\mathbf{V }).\) Since Z is a G-invariant set, the collection of strings \({\mathcal {G}}_{w}\) also covers Z. By the definition of \(\Lambda _{w}(Z,\Phi ,{\mathcal {U}},p+q+1)\), we have
where \(c=\max _{0\le i \le m-1}e^{\Vert \varphi _{i}\Vert }\), which implies
It follows that
Let \(a_{p}=\log \Lambda (Z,\Phi ,{\mathcal {U}},p)\). Note that \(\Lambda (Z,\Phi ,{\mathcal {U}},p) \ge e^{-p\cdot \max _{0\le i \le m-1}\Vert \varphi _{i}\Vert }\). Therefore, \(\inf \limits _{p>1}\frac{a_{p-1}}{p}>-\infty \). The desired result is now a direct consequence of Remark 4.2 and the following Lemma 4.5. \(\square \)
Lemma 4.5
Let \(\{a_{p}\},~p=1,2,\ldots \) be a sequence of numbers satisfying \(\inf \limits _{p>1}\frac{a_{p-1}}{p}>-\infty \) and \(a_{p+q+1}\le c'+a_{p}+a_{q}\) for all \(p,q > 1\) where \(c'>0\) is a constant. Then the \(\lim \limits _{p\rightarrow \infty }\frac{a_{p}}{p}\) exists and coincides with \(\inf \limits _{p >1}(\frac{a_{p-1}}{p}+\frac{c'}{p})\).
Proof
The proof is similar to the Theorem 4.9 in [39], so we omit the proof. \(\square \)
Remark 4.3
Indeed, when \(Z=X\) and \(\Phi =\{\varphi \}\), i.e., \(\varphi _{0}=\varphi _{1}=\cdots =\varphi _{m-1}=\varphi \), it is easy to get \({\underline{CP}}_{X}(G,\Phi )={\overline{CP}}_{X}(G,\Phi )=P_{X}(G,\varphi )\), where \(P_{X}(G,\varphi )\) is denoted by the topological pressure of free semigroup actions on X in [19].
Example 4.6
Let \(X=[0,1]\), G is generated by \(f_{0}\) and \(f_{1}\), where \(f_{0}\) and \(f_{1}\) are defined by
The images of functions \(f_0\) and \(f_1\) are shown in Fig. 1.
Choose \(Z=(\frac{1}{3}, \frac{2}{3})\), then Z is G-invariant, namely, \(f_{0}^{-1}(Z) = Z\) and \(f_{1}^{-1}(Z) = Z\). Thus, for any \(\Phi =\{\varphi _{0},\varphi _{1}\}\), where \(\varphi _{0}, \varphi _{1} \in C(X, {\mathbb {R}})\), we have
Left: the graph of \(f_{0}\). Right: the graph of \(f_{1}\)
Next, we discuss the relationship between the topological pressure and upper capacity topological pressure of a free semigroup action generated by G on Z when Z is a compact G-invariant set. Given a compact G-invariant set \(Z\subset X\) and an open cover \({\mathcal {U}}\) of X, we choose any \(\alpha >P_{Z}(G,\Phi ,{\mathcal {U}})\), then
Since \(M(Z,\alpha ,\Phi ,{\mathcal {U}},N)\) is non-decreasing as N increases and non-negative, it follows that \(M(Z,\alpha ,\Phi ,{\mathcal {U}},N)=0 \) for any N. Therefore, for any \( w \in F_{m}^+ \) and \(|w|=N\), we have \(M_{w}(Z,\alpha ,\Phi ,{\mathcal {U}},N)=0 \). Let \(c:=\max _{0 \le i\le m-1} e^{\Vert \varphi _{i}\Vert },~0<p<1\), such that \(c p < 1\). For \(M_{w}(Z,\alpha ,\Phi ,{\mathcal {U}},2)=0\), there exists \( {\mathcal {A}}_{w} \subset {\mathcal {S}}({\mathcal {U}})\) such that \({\mathcal {A}}_{w} \) covers Z (i.e., for any \(\mathbf{U } \in {\mathcal {A}}_{w}\), there exists \(w_{\mathbf{U }} \in F_{m}^+\) such that \(|w_{\mathbf{U }}|=m(\mathbf{U })-1, {\overline{w}} \le \overline{w_{\mathbf{U }}}\) and \(\bigcup \limits _{\mathbf{U }\in {\mathcal {A}}_{w}}X_{w_{\mathbf{U }}}(\mathbf{U })\supset Z\)) and
Since Z is compact, we can choose \({\mathcal {A}}_{w}\) to be finite and \(K \ge 3\) to be a constant and
For any \(w^{(1)}, w^{(2)} \in F_m^+, |w^{(1)}|=|w^{(2)}|=2\) and \(j \in \{0,1,\ldots , m-1\}\), we can construct
where \( {\mathcal {A}}_{w^{(1)}}, {\mathcal {A}}_{w^{(2)}} \) satisfy (4.4), (4.5). Then
where the word corresponding to \( \mathbf{U }\mathbf{V } \) is \(w_{\mathbf{U }}jw_{\mathbf{V }}\) and \( m(\mathbf{U }\mathbf{V })=m(\mathbf{U })+m(\mathbf{V }) \ge 6\). Since Z is G-invariant, then \({\mathcal {A}}_{w^{(1)}jw^{(2)}}\) covers Z. It is easy to see that
By mathematical induction, for each \(n \in {\mathbb {N}}\) and \(j_1,\ldots ,j_{n-1} \in \{0,1,\ldots , m-1\}\), we can define \({\mathcal {A}}_{w^{(1)}j_1w^{(2)}j_2 \ldots w^{(n-1)}j_{n-1}w^{(n)}}\) which covers Z and satisfies
Let \(\Gamma _{w^{(1)}j_1w^{(2)} \ldots }={\mathcal {A}}_{w^{(1)}}\cup {\mathcal {A}}_{w^{(1)}j_1w^{(2)}}\cup \cdots \). Since Z is G-invariant, then \(\Gamma _{w^{(1)}j_1w^{(2)} \ldots }\) covers Z and
Therefore, for any \(\omega \in \Sigma _{m}^+\), there exists \(\Gamma _{\omega }\) covering Z and \(Q(Z,\alpha ,\Phi ,\Gamma _{\omega }) < \infty .\) Put
Similar to [18], the following condition is given and the following Theorem 4.8 holds under this condition.
Condition 4.7
For any \(N>0\) and any \(w=i_1i_2 \ldots i_N \in F_m^+\), there exists \(\Gamma _{\omega } \in {\mathcal {F}}\) such that for any \( \mathbf{U } \in \Gamma _{\omega } \), \({\overline{w}} \le \overline{w_{\mathbf{U }}}\) and \( N+1 \le m(\mathbf{U }) \le N+K\), where \(w_{\mathbf{U }}\) is the word corresponds to \(\mathbf{U }\) and K is given by (4.5).
Theorem 4.8
Under the condition 4.7, for any compact G-invariant set \(Z\subset X\), we have
Moreover, for any open cover \({\mathcal {U}}\) of X, we have
Proof
Under the condition 4.7, for any \( N >0\) and any \( w=i_1i_2 \ldots i_N \in F_m^+ \), there is \(\Gamma _{\omega }\in {\mathcal {F}}\) covering Z such that for any \(\mathbf{U }\in \Gamma _{\omega }\), the word corresponds to \(\mathbf{U } \) is \( w_{\mathbf{U }} \) and \( {\overline{w}}\le \overline{w_{\mathbf{U }}}.\) Then for any \(x\in Z\), there exists a string \( \mathbf{U }=(U_0,U_1,\ldots , U_N,\ldots ,U_{N+P}) \in \Gamma _{\omega } \) such that \(x\in X_{w_{\mathbf{U }}} (\mathbf{U })\), where \(0\le P<K\). Let \(\mathbf{U }^{*}=(U_0,U_1,\ldots , U_N)\). Then \(X_{w_{\mathbf{U }}} (\mathbf{U }) \subset X_{w} (\mathbf{U }^{*}) \). Using \(\Gamma _{w}^{*}\) denotes the collection of all substrings \(\mathbf{U }^{*}\) constructed above and let
Because of \(\varphi _{i}\in C(X,{\mathbb {R}})\), \(\varphi _{i}\) is uniformly continuous for \(0\le i \le m-1\). Then \(\varepsilon (|{\mathcal {U}}|)\) is finite and \(\lim \limits _{|{\mathcal {U}}|\rightarrow 0}\varepsilon (|{\mathcal {U}}|)=0\). We have
Therefore,
It follows that
Moreover,
By Lemma 4.2
that is,
Letting \(|{\mathcal {U}}|\rightarrow 0\), we get \(P_{Z}(G,\Phi )> {\overline{CP}}_{Z}(G,\Phi )\). \(\square \)
5 Two Equivalent Definitions of Topological Pressure in the Present Paper
Let (X, d) be a compact metric space. Now, we describe two other approaches to redefine the topological pressure and lower and upper capacity topological pressures of \(G=\{f_0,\ldots ,f_{m-1}\}\) on any subset of X and \(\Phi =\{\varphi _{0}, \varphi _{1},\ldots ,\varphi _{m-1}\},\) where \(f_{i}(i=0,1,\ldots ,m-1)\) is continuous and \(\varphi _{0}, \varphi _{1},\ldots ,\varphi _{m-1} \in C(X,{\mathbb {R}})\).
For each \(w \in F_m^{+}\), a new metric \(d_w\) on X (named Bowen metric) is given by
Clearly, if \(\overline{w'} \le \overline{w''}\), then \(d_{w'}(x_1,x_2) \le d_{w''}(x_1,x_2)\) for all \(x_1,x_2 \in X\).
5.1 Definition using Bowen Balls
Fix a number \(\delta > 0\). Given \(w\in F_{m}^+\) and a point \(x\in X\), define the \((w,\delta )\)-Bowen ball at x by
Put \({\mathcal {S}}=X\times F_{m}^{+} \). We define the collection of subsets
and three functions \(\xi ,\eta ,\psi : {\mathcal {S}}\rightarrow {\mathbb {R}} \) as follows
One can easily verify that the collection of subsets \({\mathcal {F}}\) and the functions \(\xi ,\eta ,\psi \) satisfy conditions (1), (2), (3) and \((3')\) in Sect. 3.1. Therefore, they determine a C–P structure \(\tau =({\mathcal {S}},{\mathcal {F}},\xi ,\eta ,\psi )\) on X.
Given \(w \in F_m^{+}, |w|=N, Z\subset X\) and \(\alpha \in {\mathbb {R}}\), we define
where the infimum is taken over all finite or countable subcollections \({\mathcal {G}}_{w} \subset {\mathcal {F}}\) covering Z (i.e., for any \( B_{w'}(x,\delta ) \in {\mathcal {G}}_{w}, {\overline{w}}\le \overline{w'} \) and \( \bigcup \limits _{B_{w'}(x,\delta ) \in {\mathcal {G}}_{w}} B_{w'}(x,\delta ) \supset Z).\)
Let
We can easily verify that the function \({\overline{M}}(Z,\alpha ,\Phi ,\delta ,N)\) is non-decreasing as N increases. Therefore, there exists the limit
Furthermore, by the condition \((3^{'})\) in Sect. 3.1, we can define
where \({\overline{\Lambda }}_{w}(Z,\Phi ,\delta ,N)=\inf \limits _{{\mathcal {G}}_{w}}\left\{ \sum \limits _{B_{w}(x,\delta ) \in {\mathcal {G}}_{w}}\exp \left( \sup \limits _{y \in B_{w}(x,\delta )} S_{w} \Phi (y) \right) \right\} ,\) the infimum is taken over all finite or countable subcollections \({\mathcal {G}}_{w} \subset {\mathcal {F}}\) covering Z and the words corresponding to every ball in \({\mathcal {G}}_{w}\) are all equal.
Let
We set
The C–P structure \(\tau \) generates the Carathéodory–Pesin dimension of Z and the lower and upper Carathéodory–Pesin capacities of Z with respect to G. We denote them by \(P_{Z}(G,\Phi ,\delta ), {\underline{CP}}_{Z}(G,\Phi ,\delta ),\) and \({\overline{CP}}_{Z}(G,\Phi ,\delta )\) respectively. We have that
Theorem 5.1
For any set \(Z \subset X\), the following limits exist:
Proof
Let \({\mathcal {U}}\) be a finite open cover of X, and \(\delta ({\mathcal {U}})\) is the Lebesgue number of \({\mathcal {U}}\). It is easy to see that for any \(x\in X\), if \(x\in X_{\omega }(\mathbf{U })\) for some \(\mathbf{U }\in {\mathcal {S}}_{k+1}({\mathcal {U}})\) and some \(\omega \in \Sigma ^+_m \) then
It follows from Theorem 4.1 that
\(\square \)
Remark 5.1
Similar to Remark 4.2, we have
5.2 Definition Using the Center of Bowen’s Ball
Put \({\mathcal {S}}=X\times F_{m}^+\), define the collection of subsets
We redefine three functions \(\xi ,\eta ,\psi : {\mathcal {S}}\rightarrow {\mathbb {R}} \) as follows
One can easily verify that the collection of subsets \({\mathcal {F}}\) and the functions \(\xi ,\eta ,\psi \) satisfy conditions (1), (2), (3) and \((3')\) in Sect. 3.1. Therefore, they determine a C–P structure \(\tau =({\mathcal {S}},{\mathcal {F}},\xi ,\eta ,\psi )\) on X. Given \(w \in F_m^{+}, |w|=N, Z\subset X\) and \(\alpha \in {\mathbb {R}}\), we define
where the infimum is taken over all finite or countable subcollections \({\mathcal {G}}_{w} \subset {\mathcal {F}}\) covering Z (i.e., for any \( B_{w'}(x,\delta ) \in {\mathcal {G}}_{w},\ {\overline{w}}\le \overline{w'},\ x \in Z \) and \( \bigcup \limits _{B_{w'}(x,\delta ) \in {\mathcal {G}}_{w}} B_{w'}(x,\delta ) \supset Z\)).
Let
We can easily verify that the function \(M'(Z,\alpha ,\Phi ,\delta ,N)\) is non-decreasing as N increases. Therefore, there exists the limit
Furthermore, by the condition \((3^{'})\) in Sect. 3.1, we can define
where \(\Lambda '_{w}(Z,\Phi ,\delta ,N)=\inf \limits _{{\mathcal {G}}_{w}}\left\{ \sum \limits _{B_{w}(x,\delta ) \in {\mathcal {G}}_{w}}\exp \left( S_{w} \Phi (x)\right) \right\} ,\) the infimum is taken over all finite or countable subcollections \({\mathcal {G}}_{w}=\{B_{w}(x,\delta ) : x\in Z\} \subset {\mathcal {F}}\) covering Z and the words corresponding to every ball in \({\mathcal {G}}_{w}\) are all equal.
Let
We set
The C–P structure \(\tau \) generates the Carathéodory–Pesin dimension of Z and the lower and upper Carathéodory–Pesin capacities of Z with respect to G. We denote them by \(P'_{Z}(G,\Phi ,\delta ), \underline{CP'}_{Z}(G,\Phi ,\delta ),\) and \(\overline{CP'}_{Z}(G,\Phi ,\delta )\) respectively. We have that
Theorem 5.2
For any set \(Z \subset X\), the following limits exist:
Proof
We use the analogous method as that of [13]. On the one hand, given \(\delta > 0\), let
and observe that since \(\varphi _{i} \in C(X,{\mathbb {R}})\) and X is compact, \(\varphi _{i}\) is in fact uniformly continuous, hence \(\varepsilon (\delta )\) is finite and \(\lim \limits _{\delta \rightarrow 0}\varepsilon (\delta )=0\). Furthermore, given \(x \in X\), \(w \in F^+_m\) and \(|w|=N\), then for any \(y \in B_{w}(x, \delta )\), we have
Thus
Now fix \(\delta > 0\), choose a finite open cover \({\mathcal {U}}\) of X with \(|{\mathcal {U}}| < \delta \) and let \(\gamma ({\mathcal {U}})\) be the Lebesgue number of \({\mathcal {U}}\). Let \({\mathcal {G}}_{w}=\{B_{w'}(x, \frac{1}{2}\gamma ({\mathcal {U}})): x\in Z, {\overline{w}} \le \overline{w'}\} \) be a open cover of Z, then for each \(B_{w'}(x, \frac{1}{2}\gamma ({\mathcal {U}})) \in {\mathcal {G}}_{w}\) there exists \(\mathbf{U } \in \mathcal {{\mathcal {S}}}_{|w'|+1}({\mathcal {U}})\) such that \(B_{w'}(x, \frac{1}{2}\gamma ({\mathcal {U}})) \subset X_{w'}(\mathbf{U })\). Set
and then
Moreover, we can get
which implies
Taking the limit \(N \rightarrow \infty \) yields
Therefore
and as \(\delta \rightarrow 0\), that is, \(\varepsilon (\delta )\rightarrow 0\), \(|{\mathcal {U}}|\rightarrow 0\), we obtain
On the other hand, fix a cover \({\mathcal {U}}\) of X with \(|{\mathcal {U}}| < \delta \). Given \( w \in F_{m}^{+}, |w|=N\) and \({\mathcal {G}}_{w}\subset {\mathcal {S}}({\mathcal {U}})\) covering Z, we may assume without loss of generality that for every \(\mathbf{U } \in {\mathcal {G}}_{w}\), we have \(X_{w_{\mathbf{U }}}(\mathbf{U }) \cap Z \ne \varnothing \). Then for each \(\mathbf{U } \in {\mathcal {G}}_{w}\), we can choose \(x \in X_{w_{\mathbf{U }}}(\mathbf{U }) \cap Z\). We observe
Using \({\mathcal {F}}_{w}\) denotes the collection of all \((w_{\mathbf{U }},\delta )\)-Bowen ball \(B_{w_{\mathbf{U }}}(x,\delta )\) constructed above and then
It follows
Hence
Therefore
and taking the limit as \(\delta \rightarrow 0\) gives
which completes the proof of the first. The existence of the two other limits can be proved in a similar way. \(\square \)
Remark 5.2
-
(1)
When \(m=1\), i.e., \(G=\{f\},~\Phi =\{\varphi \}\), then \(P_{Z}(G,\Phi )\) coincides with the topological pressure using the centre of Bowen ball which is defined by Climenhaga [14] for every \(Z \subset X\).
-
(2)
Similar to Remark 4.2, we have
$$\begin{aligned} {\underline{CP}}_{Z}(G,\Phi )= & {} \lim \limits _{\delta \rightarrow 0}\liminf _{N\rightarrow \infty }\frac{\log \Lambda '(Z,\Phi ,\delta ,N)}{N}, \\ {\overline{CP}}_{Z}(G,\Phi )= & {} \lim \limits _{\delta \rightarrow 0}\limsup _{N\rightarrow \infty }\frac{\log \Lambda '(Z,\Phi ,\delta ,N)}{N}. \end{aligned}$$
6 The Proof of Theorem 2.2
Let (X, d) be a compact metric space and \(G=\{f_0,f_1,\ldots ,f_{m-1}\}\) be a free semigroup with m generators acting on X, where \(f_0,f_1,\ldots ,f_{m-1}\) are continuous transformations from X to itself. In this section, as the application of the topological pressure, we give the connection between topological pressure and Hausdorff dimension on some Z in the form of Bowen’s equation, which extends the results of Climenhaga [14]. Before proving the Theorem 2.2, we give some relevant results.
Proposition 6.1
Let \(f_{i}: X\longrightarrow X\) be as in Theorem 2.2. Fix \(0< \alpha \le \beta < \infty \) and \(Z \subset {\mathcal {A}}([\alpha ,\beta ])\), then
-
(1)
for any \(t \in {\mathbb {R}}\) and \(h > 0\), we have
$$\begin{aligned} P_{Z}(G,-t\Phi )-\beta h \le P_{Z}(G,-(t+h)\Phi )\le P_{Z}(G,-t\Phi )-\alpha h. \end{aligned}$$(6.1) -
(2)
The equation \(P_{Z}(G,-t\Phi )=0\) has a unique root \(t^{*}\) and
$$\begin{aligned} \frac{h_{Z}(G)}{\beta } \le t^{*} \le \frac{h_{Z}(G)}{\alpha }. \end{aligned}$$ -
(3)
If \(\alpha =\beta \), then \( t^{*} = \frac{h_{Z}(G)}{\alpha }\).
Here \(h_{Z}(G)\) is the topological entropy in Ju et al [18], \(\Phi =\{\log a_{0},\log a_{1},\ldots , \log a_{m-1}\}\) and \(t \Phi =\{t\cdot \log a_{0},t\cdot \log a_{1},\ldots , t\cdot \log a_{m-1}\}\).
Proof
(1) Given arbitrary \(\varepsilon >0\) and \(k \ge 1\), let
and notice that \(Z= \cup _{k=1}^{\infty } Z_{k}\). Now fix \(t \in {\mathbb {R}},~h > 0\), \(w=i_{1}i_{2}\cdots i_{N} \in F_{m}^{+}\) and \(N \ge k\). It follows that for any \(\delta >0\), \(s \in {\mathbb {R}}\),
where \({\mathcal {G}}_{w}= \{B_{w'}(x,\delta ): {\overline{w}} \le \overline{w'}, x \in Z_{k}\} \subset {\mathcal {F}}\) covers \(Z_{k}\). It follows that
and then
Then we have
Letting \(\delta \rightarrow 0\), it follows that
Using the Proposition 4.1 and taking the supremum over all \(k\ge 1\), we can get
Since \(\varepsilon > 0\) is arbitrary, this establishes the right of inequality (6.1).
Using the similar computation, we obtain
It follows that
and then
Hence,
Letting \(\delta \rightarrow 0\), it follows that
Take the supremum over all \(k\ge 1\), and by the Proposition 4.1 we can get
Since \(\varepsilon > 0\) is arbitrary, this establishes the left of inequality (6.1). This complete the proof of the inequality (6.1).
(2) We observe that the map \(t \mapsto P_{Z}(G,-t\Phi )\) is continuous and strictly decreasing. First applied the left of (6.1) with \(t=0\) and \(h=\frac{h_{Z}(G)}{\beta }\), we have
and second applied the right of (6.1) with \(t=0\) and \(h=\frac{h_{Z}(G)}{\alpha }\), we have
Thus it get the desired result by Intermediate Value Theorem.
(3) It follows from (2) immediately. \(\square \)
Similar to [14], we also have the following proposition. The proof of this proposition is similar to that of [14]. Therefore, we omit the proof.
Proposition 6.2
Let \(f_{i}: X\longrightarrow X\) be as in Theorem 2.2 and suppose that for any \(\omega \in \Sigma _{m}^{+}, ~\lambda _{\omega }(x)\) exists and is positive. Then \(x \in {\mathcal {B}}\) .
Before proving Theorem 2.2, similar to [14], we give the following two lemmas.
Lemma 6.1
Let \(f_{i}: X\longrightarrow X\) be as in Theorem 2.2. Then given any \(x \in {\mathcal {B}}\) and \(\varepsilon > 0\), there exist \(\delta _{0}=\delta _{0}(\varepsilon )>0\) and \(\eta =\eta (x,\varepsilon )>0\) such that for each \(w \in F_{m}^{+}\) and \(0< \delta < \delta _{0}\),
Proof
Since \(f_{i}\) is conformal with factor \(a_{i}(x)>0\), for each \(i \in \{0,1,2\ldots ,m-1\}\) we have
Since \(a_{i}(x)>0\) everywhere, we can take logarithms and get
It can be extended the continuous function \(\zeta _{i}: X \times X\longrightarrow {\mathbb {R}}\)
Because \(X \times X\) is compact, \(\zeta _{i}\) is uniformly continuous, \(i \in \{0,1,\ldots ,m-1\}\). Hence for \(\varepsilon >0\), there exists \(\delta _{0}=\delta _{0}(\varepsilon )>0\) such that for every \(0< \delta < \delta _{0}\) and \((x,y),(x',y') \in X \times X\) with
we have \(|\zeta _{i}(x,y)-\zeta _{i}(x',y')| <\varepsilon \). In particular, for \(x,y \in X\) with \(d(x,y)<\delta \), we have \((d\times d)((x,y),(x,x))<\delta \). Therefore,
that is,
and taking exponentials, we obtain
whenever the middle quantity is less than \(\delta \). Given \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+}\). Now we show the second half of (6.2). Let \(\Phi =\{\log a_{0},\log a_{1},\ldots , \log a_{m-1}\}\). Suppose \(y \in B_{w}(x,\delta )\), then \(d(f_{w'}(x),f_{w'}(y)) < \delta \) for all \(w' \le {\overline{w}}\). Then repeated application of the second inequality in (6.3) yields
Then
Now we prove the first inclusion in (6.2). We observe that if \(d(x,y)<\delta \), then by the first inequality in (6.3) we get
Then if \(d(x,y)<\delta e^{-(\log a_{i_{1}}(x)+\varepsilon )}\), we have \(d(f_{i_1}(x),f_{i_1}(y))<\delta \) and so
Continuing in this method, we can obtain that if
for each \(\overline{w'}\le {\overline{w}}\), we have \(d(f_{\overline{w'}}(x),f_{\overline{w'}}(y))<\delta \), and hence \(y \in B_{w}(x,\delta )\). Therefore
which is almost what we wanted. If the minimum was always achieved at \(w'=w\), we would be done; however, this may not be the case. Therefore, now we find what \(\eta \) should be for any \(\overline{w'}\le {\overline{w}}\), and we observe that
Since x satisfies the tempered contraction condition, there exists \(\eta =\eta (x,\varepsilon )>0\) such that
for arbitrary \(w \in F_{m}^{+},~\overline{w'}\le {\overline{w}}\), and hence
Then for such \(w,w'\), we have
and combining with (6.4)
Taking \(\delta _{0}=\delta _{0}(\varepsilon /2)\) gives that desired result. \(\square \)
Remark 6.1
-
(1)
If x has bounded contraction, then \(\eta =\eta (x)\) may be chosen independently of \(\varepsilon \).
-
(2)
Furthermore, if \(a_{i}(x) \ge 1, ~i \in \{0,1,\ldots ,m-1\}\) for any \(x \in X,\) then \(\eta =1\) suffices.
Lemma 6.2
Let \(f_{i}: X\longrightarrow X\) satisfy the conditions of Theorem 2.2 and fix \(Z \subset {\mathcal {A}}((\alpha ,\infty ))\bigcap {\mathcal {B}},\) where \(0<\alpha <\infty \). Let \(t^{*}\) be the unique real number with \(P_{Z}(G,-t^{*} \Phi )=0\), whose existence and uniqueness are guaranteed by Proposition 6.1, where \(\Phi =\{\log a_{0},\log a_{1},\ldots ,~\log a_{m-1}\}\). Then \(dim_{H} Z=t^{*}\).
Proof
First we prove \(dim_{H} Z \le t^{*}\). Given \(k \ge 1\), and let
and then \(Z=\bigcup _{k=1}^{\infty } Z_{k}\). Consider \(t > t^{*}\), and naturally \(P_{Z}(G,-t \Phi )< 0\). Thus there exists \(\varepsilon >0\) with \(P_{Z}(G,-t \Phi )<-t\varepsilon \) and by Lemma 6.1 there exists \(\delta _{0}=\delta _{0}(\varepsilon )\) such that for all \(x \in Z_{k},~0<\delta <\delta _{0}\), fixing \(w \in F^{+}_{m}\) and \(|w|\ge k\), we have
Given \(N\ge k\) and \(0< \delta < \delta _{0}\), we have
where \({\mathcal {G}}_{w}= \{ B_{w'}(x,\delta ): {\overline{w}}\le \overline{w'}, x\in Z_{k}\} \subset {\mathcal {F}}\) covers \(Z_{k}\) and \({\mathcal {D}}(Z_{k},2\delta e^{-N(\alpha -\varepsilon )})\) denotes the collection of open covers \(\{ U_{i}\}\) of \(Z_{k}\) for which \(diam U_{i} < 2\delta e^{-N(\alpha -\varepsilon )}\) for all i.
It follows that
Taking the limit as \(N \rightarrow \infty \) gives
Thus
and for sufficiently small \(\delta >0\), we have \(-t \varepsilon > P'_{Z_k}(G,-t \Phi ,\delta )\). Hence \( H^{t}(Z_{k})=0\) by (6.6), which implies \( dim_{H}(Z_{k}) \le t\). Then taking the union over all k gives \(dim_{H}(Z) \le t \) for all \(t>t^{*}\). Therefore \(dim_{H}(Z) \le t^{*}\).
Now we prove the other inequality, \(dim_{H}(Z) \ge t^{*}\). Fix \(t <t^{*}\), and then we show that \(dim_{H}(Z) \ge t\). Suppose \(t>0\), and by the Proposition 6.1\(t^{*}\) is the unique real number such that \(P_{Z}(G,-t^{*} \Phi )=0\). Since the pressure function \(P_{Z}(G,-t\Phi )\) is decreasing, we have \(P_{Z}(G,-t \Phi )>0\). Hence we can choose \(\varepsilon >0\) such that
Set \(\delta _{0}=\delta _{0}(\varepsilon )\) be as in Lemma 6.1. Given \(k\in {\mathbb {N}},~k\ge 1\), consider the set
Obviously, \(Z=\bigcup \limits _{k=1}^{\infty } Z_{k}\) and hence \(P_{Z}(G,-t\Phi )=\sup \limits _{k\ge 1} P_{Z_k}(G,-t \Phi )\). Then there exists \(k\in {\mathbb {N}}\) with \(t\varepsilon <P_{Z_k}(G,-t\Phi )\), and we fix \(0<\delta <\delta _{0}\) such that
Let \(\beta =\max _{i} \{\sup _{x \in X}\log a_{i}(x)\}<\infty \). For any \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+},~x\in X\), denote \(s_{w}(x)=e^{-k} \delta e^{-|w|(\lambda _{w}(x)+\varepsilon )}\), and notice that for any \(i_{n+1} \in \{0,1,\ldots ,m-1\}\)
Moreover, given \(x \in Z_{k}\) and \(r>0\) enough small and for any \(\omega =i_{1}i_{2}\cdots \in \Sigma _{m}^{+}\), there exists \(n=n(x,r,\omega ) \in {\mathbb {N}}\) such that for \(w= i_{1}i_{2} \cdots i_{n}=\omega |_{[0,n-1]}\in F_{m}^{+}(n)\), we have
Moreover, for every \(w' \in F_{m}^{+}\) and \(x \in X\), we have \(\lambda _{w'}(x) \le \beta \) and thus \(s_{w'}(x) \ge \delta e^{-\big (k+|w'|(\beta +\varepsilon )\big )}\). It yields from (6.8) that for \(w=i_{1}i_{2}\cdots i_{n}=\omega |_{[0,n-1]}\), where \(n=n(x,r,\omega )\), we have
and thus
Writing \(N:=N(r,\delta )=\frac{-\log r +\log \delta -k}{\beta +\varepsilon }-1\) and observe that for every fixed \(\delta >0\), we have \(\lim _{r\rightarrow 0}N(r,\delta )=\infty \).
By Lemma 6.1 we have for \(w=i_{1}i_{2}\cdots i_{n}=\omega |_{[0,n-1]},~n=n(x,r,\omega ) \in {\mathbb {N}}\),
Then given any \(\{B(x_{i},r_{i})\}\) with \(Z_{k} \subset \bigcup B(x_{i},r_{i})\), we can get \(Z_{k} \subset \bigcup B_{w_{i}}(x_{i},\delta )\), where \(w_{i}=\omega |_{[0,n_{i}-1]},~n_{i}=n_{i}(x_{i},r_{i},\omega )\) satisfies (6.8).
Thus for all \(r>0,~0<\delta <\delta _{0}\) and fixed \(w=\omega |_{[0,N-1]} \in F^{+}_{m}(N)\), by applying (6.8) we can get
where \({\mathcal {D}}^{b}(Z_{k},r)\) denotes the collection of countable open balls covers \(\{B(x_{i},r_{i}): x_{i}\in Z_{k} \}_{i=1}^{\infty }\) of \(Z_{k}\) with \(r_{i} < r\) for all i and \({\mathcal {G}}'_{w}\) denotes the collection of countable Bowen balls covers \(\{B_{w_{i}}(x_{i},\delta ):x_{i}\in Z_{k}\}_{i=1}^{\infty }\) of \(Z_{k}\) for which \(~{\overline{w}}\le \overline{w_{i}},~w=\omega |_{[0,N-1]}, w_{i}=\omega |_{[0,n_{i}-1]}\) for all i.
It follows that
Letting \(r\rightarrow 0\), we can obtain the quantity on the right goes to \(\infty \) by (6.7), and thus we have \(m^{b}_{H}(Z_{k},t)=\infty \). Therefore,
and since \(t< t^{*}\) was arbitrary, this establishes the Lemma. \(\square \)
Proof of Theorem 2.2
Consider a decreasing sequence of positive numbers \(\alpha _{k}\) which converge to 0, and set \(Z_{k} \subset {\mathcal {A}}((\alpha _{k},\infty ))\bigcap Z\), so that Lemma 6.2 applies to \(Z_{k}\) and we have \(Z=\bigcup _{k=1}^{\infty } Z_{k}\). Let \(t_{k}\) be the unique real number with
for every k and whose existence and uniqueness are guaranteed by Proposition 6.1. Thus by Lemma 6.2 we get
Denote \(t^{*}=\sup _{k} t_{k}\), then \(dim_{H} Z=t^{*}\). Then it remains to prove that
Given \(t\ge 0\), we have
First for any \(t < t^{*}\), there exists \(t_{k}\) with \(t<t_{k}\) and hence \(P_{Z_{k}}(G,-t \Phi )>0\). Then \(t \in \{ t\ge 0: P_{Z}(G,-t \Phi )>0\}\) and thus \(t \le \sup \{ t\ge 0: P_{Z}(G,-t\Phi )>0\}\). Therefore, \(t^{*} \le \sup \{ t\ge 0: P_{Z}(G,-t\Phi )>0\}\).
Next for arbitrary \(t < \sup \{ t\ge 0: P_{Z}(G,-t \Phi )>0\}\), there exists \(t_{j}>t\) with \(P_{Z}(G,-t_{j}\Phi )>0\). Then there exists \(Z_{k}\) such that \(P_{Z_{k}}(G,-t_{j} \Phi )>0\) and thus \(t_{j} < t_{k}\). It follows that \(t<t_{j}<t_{k}<t^{*}\). So \(\sup \{ t\ge 0: P_{Z}(G,-t\Phi )>0\} \le t^{*}\). This establishes (6.9).
Finally, it follows from (6.9) and continuity of \(t \mapsto P_{Z}(G, -t \Phi )\) that \(P_{Z}(G, -t^{*} \Phi )=0\). If \(Z \subset {\mathcal {A}}((\alpha ,\infty ))\) for some \(\alpha >0\), then Proposition 6.1 guarantees that \(t^{*}\) is in fact the unique root of Bowen’s equation. \(\square \)
Example 6.3
Let \(X=[0,1]\) be a closed interval on \({\mathbb {R}}\). Assume G is a free semigroup generated by a series of the Manneville-Pomeau maps (for the study of Manneville-Pomeau map, see, for example [27, 33]), that is,
and \(0<s_{0}<s_{1}<\cdots<s_{m-1}<1.\) Then \(f_{i}~(i=0,1,\ldots ,m-1)\) is continuous conformal and has no critical points and singularities, and \(a_{i}(x)\ge 1\) for any \(x \in X\). The Manneville-Pomeau map is non-uniformly hyperbolic transformation having the most benign type of non-hyperbolicity: an indifferent fixed point at 0, i.e., \(f_{i}(0)=0\), and \(a_{i}(0)=1\), and exhibits intermittent behavior. Moreover, we can verify X, \(G=\{ f_{0},f_{1},\ldots ,f_{m-1}\}\) and \(\Phi =\{\log a_{0},\log a_{1},\ldots ,\log a_{m-1}\}\) satisfy the conditions of Theorem 2.2. What’s more, for any \(\omega \in \Sigma ^{+}_{m}\)
and
Thus \({\mathcal {B}}=X\) and \({\mathcal {A}}((0,\infty ))=X-\{0\}\). Therefore, for any \(Z \subset {\mathcal {A}}((0,\infty ))\), by the Theorem 2.2 we get
If \(Z \subset {\mathcal {A}}((\alpha ,\infty )) \cap {\mathcal {B}}\) for some \(\alpha >0\), then the Hausdorff dimension \(dim_{H} Z\) of Z is the unique root of Bowen’s equation
Moreover, if \(Z \subset {\mathcal {A}}(\alpha )\) for some \(\alpha >0\), then
where \(h_{Z}(G)\) is the topological entropy on Z defined by Ju et al in [18] .
7 The Proof of Theorem 2.3
Proof of Theorem 2.3
(1) Fix an \(\varepsilon > 0\) and for each \( k\ge 1 \). Consider the set
Since \({\underline{P}}(G, \Phi , x)\ge s\) for all \(x\in Z\), the sequence \( \{ Z_k\}_{k=1}^{\infty } \) increases to Z. So by the continuity of the measure, we have
Then select an integer \( k_0 \ge 1 \) with \( \mu (Z_{k_0})>\frac{1}{2}\mu (Z)\). For each \( N\ge 1\), put
Since the sequence \( {\{ Z_{k_0,N}\}}_{N=1}^{\infty } \) increases to \( Z_{k_0}\), we can pick a \( N^{*}\ge 1 \) such that \(\mu ( Z_{k_0,N^{*}})>\frac{1}{2}\mu ( Z_{k_0})\). Write \(Z^{*}=Z_{k_0,N^{*}}\) and \( r^{*}=\frac{1}{k_0} \). Then \( \mu (Z^{*}) >0\), for all \(x \in Z^{*},~0<r \le r^{*}\) and \(n \ge N^{*}\),
For any \(N \ge N^{*}, w \in F_{m}^{+} \) and \(|w|= N\), take a cover of \(Z^{*}\)
such that
For each i, there exists an \(x_i\in Z^{*}\cap B_{\omega _i|_{[0,n_{i}-1]}}(y_i,\frac{r}{2})\). By the triangle inequality
In combination with (7.1), we can get
Therefore, \(M'_{w}(Z^{*},s-\varepsilon ,\Phi ,r,N) \ge \mu (Z^{*})>0\) for all \(N\ge N^{*}\), and we obtain
and consequently
which in turn implies that \(P'_{Z^{*}}(G,\Phi ,r)\ge s-\varepsilon \). Then we have \(P_{Z^{*}}(G,\Phi ) \ge s-\varepsilon \) by letting \( r \rightarrow 0\). It follows that \(P_{Z}(G,\Phi )\ge P_{Z^{*}}(G,\Phi )\ge s-\varepsilon \) and hence \(P_{Z}(G,\Phi ) \ge s\) since \(\varepsilon >0\) is arbitrary.
(2) In order to prove the second result, we need to use the following Lemma.
Lemma 7.1
([18, 21]). Let \(r > 0\) and \({\mathcal {B}}(r) = \{B_w(x,r): x \in X, w \in F_{m}^{+}\}\). For any family \({\mathcal {F}}\subset {\mathcal {B}}(r)\), there exists a (not necessarily countable) subfamily \({\mathcal {G}} \subset {\mathcal {F}}\) consisting of disjoint balls such that
Since \( {\overline{P}}(G,\Phi ,x)\le s\) for all \( x\in Z \), then for any \( \omega \in \Sigma _{m}^+ \) and \( x \in Z \),
For any \( N \ge 1 \) and \( w=i_1i_2 \ldots i_N \in F_{m}^+\). Fix \(\varepsilon > 0\), and put
then we have \(Z=\cup _{k \ge 1}Z_k\). Now fix \(k \ge 1\) and \( 0<r<\frac{1}{3k} \). For each \(x \in Z_k,\) we take \( \omega _{x} \in \Sigma ^+_m \) such that \( \omega _{x}|_{[0,N-1]}=w \), there exists a strictly increasing sequence \(\{n_{j}(x)\}_{j=1}^{\infty }\) such that
for all \(j \ge 1\). So the set \(Z_k\) is contained in the union of the sets in the family
By Lemma 7.1, there exists a sub family \({\mathcal {G}}_{w}=\{B_{\omega _{x_i}|_{[0,n_{i}-1]}}(x_i,r)\}_{i\in I} \subset {\mathcal {F}}_{w}\) consisting of disjoint balls such that for all \(i \in I\),
and
The index set I is at most countable since \(\mu \) is a probability measure and \({\mathcal {G}}_{w}\) is a disjointed family of sets, each of which has positive \(\mu \)-measure. Therefore,
where the disjointness of \(\{B_{\omega _{x_i}|_{[0,n_{i}-1]}}(x_i,r)\}_{i\in I}\) is used in the last inequality. It follows that
and consequently
which in turn implies that \(P'_{Z_k}(G,\Phi ,3r) \le s+\varepsilon \) for any \(0<r<\frac{1}{3k}\). Letting \(r \rightarrow 0\) yields
By Proposition 4.1(3),
Since \(\varepsilon > 0\) is arbitrary, then we can get \(P_{Z}(G,\Phi )\le s\) . \(\square \)
Corollary 7.2
Let \(\mu \) denote a Borel probability measure on X, Z be a Borel subset of X and \( w \in F^{+}_{m},~|w|=n\). Consider the following quantities:
Then \(P_{Z}(G,\Phi ) \le {\overline{P}}\). If in addition \(\mu (Z) > 0\), we have \(P_{Z}(G,\Phi ) \ge {\underline{P}}\).
Proof
-
(1)
It is easy to verify that \({\overline{P}}(G,\Phi ,x) \le {\overline{P}}\) for any \(x \in Z\). Hence, by Theorem 2.2(2), we can get \(P_{Z}(G,\Phi ) \le {\overline{P}}\).
-
(2)
It is obvious to get \({\underline{P}}(G,\Phi ,x) \ge {\underline{P}}\) for any \(x \in Z\). In addition \(\mu (Z) > 0\), then we have \(P_{Z}(G,\Phi ) \ge {\underline{P}}\) by Theorem 2.2(1).
\(\square \)
Remark 7.1
When \(m=1\), i.e., \(G=\{f\},~\Phi =\{\varphi \}\), the Corollary 7.2 coincides with the results that Climenhaga proved in [13].
8 The Proof of Theorem 2.4
Let X be a compact metric space with metric d, suppose a free semigroup \(G=\{ f_0, f_1, \ldots , f_{m-1}\}\) with m generators act on X, the generators are continuous transformations from X to itself and \(\Phi =\{\varphi \}\), i.e., \(\varphi _{0}=\varphi _{1}=\cdots =\varphi _{m-1}=\varphi \in C(X, {\mathbb {R}})\). For \(w=i_{1}i_{2}\cdots i_{n} \in F_{m}^{+},\) denote \((S_{w} \Phi )(x):=(S_{w} \varphi )(x)\).
For any subset \(Z\subset X\), \(w \in F_m^+\) and \(\varepsilon >0\), a subset \(E \subset X\) is said to be a \((w, \varepsilon , Z, f_0, \ldots , f_{m-1})\)-spanning set of Z, if for any \(x \in Z\), there exists \(y \in E\) such that \(d_w(x,y) < \varepsilon \). A subset \(F \subset Z\) is said to be a \((w, \varepsilon , Z, f_0, \ldots , f_{m-1})\)-separated set of Z, if \(x,y \in F\), \(x\ne y\) implies \(d_w(x,y) \ge \varepsilon \).
For any \(w \in F_{m}^{+}\), \(|w|=n\) and \(\varepsilon > 0\), put
and
Set
Similarly, for any \(w \in F_{m}^{+}\), \(|w|=n\) and \(\varepsilon > 0\), set
and
Let
For any \( w \in F^+_m, |w|=n\), since
then \(\Lambda '(Z,\varphi ,\varepsilon ,n)=Q(Z,G,\varphi ,\varepsilon ,n)\). Therefore, by Remark 5.2(2) we can obtain
If \(\delta =\sup \{|\varphi (x)-\varphi (y)|: d(x,y) \le \frac{\varepsilon }{2}\}\), similar to the Lin et al ([18, Remark 3.4(5)]), we have
Moreover, we have
We assign the following skew-product transformation. It’s base is \(\Sigma _m\), it’s fiber is X, and the maps \(F: \Sigma _m \times X \rightarrow \Sigma _m \times X\) and \(g: \Sigma _m \times X \rightarrow {\mathbb {R}}\) are defined by the formula
and
where \(\omega = (\ldots , \omega _{-1}, \omega _0, \omega _1, \ldots ) \in \Sigma _m\) and c is a constant number. Here \(f_{\omega _0}\) stands for \(f_0\) if \(\omega _0=0\), and for \(f_1\) if \(\omega _0=1\), and so on and \(\varphi \in C(X,{\mathbb {R}})\). Obviously, \(g \in C(\Sigma _m \times X ,{\mathbb {R}})\). Then
Moreover, \(S_{n}g(\omega , x)=nc+S_{\overline{\omega |_{[0,n-1]}}}\varphi (x)\). Let \(\omega =(\ldots ,\omega _{-1},\omega _0,\omega _1,\ldots )\in \Sigma _m\), and the metric D on \(\Sigma _m \times X\) is defined as
For \( F: \Sigma _m \times X \rightarrow \Sigma _m \times X\), we can get a C–P structure on \(\Sigma _m \times X\). For any set \( Z \subset X \), from Pesin [24], we denote \({\overline{CP}}_{ \Sigma _m \times Z}(F,g)\) the upper capacity topological pressure of F on the set \(\Sigma _m \times Z\). Our purpose is to find the relationship between the upper capacity topological pressure \({\overline{CP}}_{ \Sigma _m \times Z}(F,g)\) of the skew-product transformation F and the upper capacity topological pressure \({\overline{CP}}_{Z}(G,\varphi )\) of a free semigroup action \( G=\{f_0, \ldots , f_{m-1}\} \) on Z.
To prove this theorem, we give the following two lemmas. The proof of these two lemmas is similar to that of Lin et al [19]. Therefore, we omit the proof.
Lemma 8.1
For any subset Z of X, \(n \ge 1\) and \(0< \varepsilon < \frac{1}{2}\), we have
Lemma 8.2
For any subset Z of X, \(n \ge 1\) and \(\varepsilon >0\), we have
where \(K(\varepsilon )\) is a positive constant that depends only on \(\varepsilon \).
Proof of Theorem 2.4
From Lemma 8.1 we have for any subset Z of X,
whence, taking logarithms and limits, we obtain that
In the same way, from Lemma 8.2, we have
whence
Thus the proof is complete. \(\square \)
Problem 8.3
For any subset \(Z \subset X\), is it true that \(P_{\Sigma _m \times Z}(F,g)= \log m + P_{Z}(G,\varphi )+c\)?
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The authors really appreciate the referees’ valuable remarks and suggestions that helped a lot. The work was supported by National Natural Science Foundation of China (grant no.11771149) and Guangdong Natural Science Foundation 2018B0303110005.
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Xiao, Q., Ma, D. Topological Pressure of Free Semigroup Actions for Non-Compact Sets and Bowen’s Equation, I. J Dyn Diff Equat 35, 199–236 (2023). https://doi.org/10.1007/s10884-021-09983-3
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DOI: https://doi.org/10.1007/s10884-021-09983-3
Keywords
- C–P structure
- Free semigroup actions
- Topological pressure
- Skew-product
- Hausdorff dimension
- Bowen’s equation