Packing topological entropy for amenable group actions

Abstract

Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

1 Introduction

In 1973, in a profound and influential paper [3], Bowen introduced a definition of topological entropy of subsets inspired by Hausdorff dimension, which is now known as Bowen topological entropy or dimensional entropy. For dynamical systems over compact Hausdorff spaces, Bowen showed that Bowen topological entropy on the whole space coincides with the Adler–Konheim–McAndrew topological entropy defined through open covers.

Bowen topological entropy can be viewed as dynamically analogous to Hausdorff dimension and has exhibited very deep connections with dimension theory in dynamical system and multifractal analysis ever since its appearance (see, for example, [19]). It is natural to consider the analogous concepts in dynamical systems for other forms of dimensions. For pointwise dimension (of a measure), its dynamical correspondence is the Brin–Katok local entropy [4]. For packing dimension, its dynamical correspondence is the packing topological entropy, which was introduced by Feng and Huang [9]. Applying the methods in geometric measure theory, they also provided variational principles for Bowen topological entropy and packing topological entropy. Other works on packing topological entropy can be found in [27], where the packing topological entropy for certain non-compact subsets was considered.

In this paper we focus on packing topological entropy in the framework of countable discrete amenable group actions.

1.1 Amenable packing entropy and local entropies

Let $(X,G)$ be a G-action topological dynamical system, where X is a compact metric space with metric d and G is a topological group acting continuously on X. Throughout this paper we assume that G is a countable infinite discrete amenable group unless otherwise specified. Recall that a countable discrete group G is amenable if there is a sequence of non-empty finite subsets $\{F_n\}$ of G which are asymptotically invariant, that is,

$$ \begin{align*}\lim_{n\rightarrow+\infty}\frac{|F_n\vartriangle gF_n|}{|F_n|}=0\quad \text{for all } g\in G.\end{align*} $$

Such sequences are called Følner sequences. One may refer to [14, 18] for more details on amenable groups and their actions. A Følner sequence $\{F_n\}$ in G is said to be tempered if there exists a constant $C>0$ which is independent of n such that

(1.1) $$ \begin{align} \bigg|\bigcup_{k<n}F_k^{-1}F_n\bigg|\le C|F_n|\quad\text{for any }n. \end{align} $$

Let $F(G)$ denote the collection of finite subsets of G. For $F\in F(G)$ , let $d_F$ be the metric defined by

$$ \begin{align*} d_F(x,y)=\max_{g\in F}d(gx,gy)\quad\text{for }x,y\in X. \end{align*} $$

Let $\varepsilon>0$ and x in X; we denote

$$ \begin{align*} B_{F}(x,\varepsilon)&=\{y\in X: d_{F}(x,y)<\varepsilon\} \end{align*} $$

and

$$ \begin{align*} \overline{B}_{F}(x,\varepsilon)&=\{y\in X: d_{F}(x,y)\le \varepsilon\}, \end{align*} $$

which are respectively the open and closed (F-)Bowen balls with center x and radius $\varepsilon $ . When we want to clarify the underlying metric d, we also denote the above balls by $B_{F}(x,\varepsilon ,d)$ and $\overline {B}_{F}(x,\varepsilon ,d)$ . For a $\mathbb Z$ -action (or $\mathbb N$ -action) topological dynamical system $(X,T)$ (T is the homeomorphism (or the continuous onto map) on X), let $F=\{0,1,\ldots ,n-1\}:=[0,n-1]$ . The (F-)Bowen balls will be written as $B_n(x,\varepsilon ,T)$ or $B_n(x,\varepsilon ,d)$ .

Definition 1.1 Let $\{F_n\}$ be a sequence of finite subsets of G with $|F_n|\rightarrow \infty $ (which need not be Følner). For $Z\subseteq X, s\ge 0, N\in \mathbb N$ and $\varepsilon> 0$ , define

$$ \begin{align*}P(Z,N,\varepsilon,s,\{F_n\}) = \mathop{\mathrm{sup}}\limits\sum_i\exp(-s|F_{n_i}|),\end{align*} $$

where the supremum is taken over all finite or countable pairwise disjoint families $\{\overline {B}_{F_{n_i}}(x_i,\varepsilon )\}$ such that $x_i\in Z, n_i\ge N$ for all i. The quantity $P(Z,N,\varepsilon ,s,\{F_n\})$ does not increase as N increases, hence the following limit exists:

$$ \begin{align*}P(Z,\varepsilon,s,\{F_n\})=\lim_{N\rightarrow+\infty}P(Z,N,\varepsilon,s,\{F_n\}).\end{align*} $$

Define

$$ \begin{align*} \mathcal{P}(Z,\varepsilon,s,\{F_n\})=\inf \bigg\{\sum_{i=1}^{+\infty}P(Z_i,\varepsilon,s,\{F_n\}): \bigcup_{i=1}^{+\infty}Z_i\supset Z\bigg\}. \end{align*} $$

It is easy to see that if $Z\subseteq \bigcup \nolimits _{i=1}^{+\infty }Z_i$ , then $\mathcal {P}(Z,\varepsilon ,s,\{F_n\})\le \sum \nolimits _{i=1}^{+\infty }\mathcal {P}(Z_i,\varepsilon ,s,\{F_n\})$ . There exists a critical value of the parameter s, which we will denote by $h_{\mathrm {top}}^{P}(Z,\varepsilon , \{F_n\})$ , where $\mathcal {P}(Z,\varepsilon ,s,\{F_n\})$ jumps from $+\infty $ to $0$ , that is,

$$ \begin{align*} \mathcal{P}(Z,\varepsilon,s,\{F_n\})=\begin{cases} 0, \quad& s> h_{\mathrm{top}}^P(Z,\varepsilon, \{F_n\}),\\ +\infty,\quad& s < h_{\mathrm{top}}^P(Z,\varepsilon, \{F_n\}). \end{cases} \end{align*} $$

It is not hard to see that $h_{\mathrm {top}}^P(Z,\varepsilon , \{F_n\})$ increases when $\varepsilon $ decreases. We call

$$ \begin{align*} h_{\mathrm{top}}^P(Z,\{F_n\}):=\lim_{\varepsilon\rightarrow 0} h_{\mathrm{top}}^P(Z,\varepsilon, \{F_n\}) \end{align*} $$

the amenable packing topological entropy (amenable packing entropy or packing entropy, for short) of Z (with respect to the Følner sequence $\{F_n\}$ ).

Let $M(X)$ denote the collection of Borel probability measures on X.

Definition 1.2 Let $\{F_n\}$ be a sequence of finite subsets of G with $|F_n|\rightarrow \infty $ . For $\mu \in M(X)$ and $Z\in \mathcal {B}(X)$ (the Borel $\sigma $ -algebra on X), denote

$$ \begin{align*}\overline h_{\mu}^{\mathrm{loc}}(Z,\{F_n\})=\int_Z\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow +\infty}-\frac{1}{|F_n|}\log \mu(B_{F_n}(x,\varepsilon))\,d{\kern-1pt}{\mu}\end{align*} $$

and

$$ \begin{align*}\underline h_{\mu}^{\mathrm{loc}}(Z,\{F_n\})=\int_Z\lim_{\varepsilon\rightarrow 0}\liminf_{n\rightarrow +\infty}-\frac{1}{|F_n|}\log \mu(B_{F_n}(x,\varepsilon))\,d{\kern-1pt}{\mu},\end{align*} $$

which are called the upper local entropy and the lower local entropy of $\mu $ over Z (with respect to $\{F_n\}$ ), respectively.

In the $\mathbb Z$ -action or $\mathbb N$ -action case, if the sequence $\{F_n\}$ is chosen to be $F_n=[0,n-1]$ (hence $\{F_n\}$ is naturally a Følner sequence), the local entropies of a topological dynamical system $(X,T)$ will be denoted by $\underline h_{\mu }^{\mathrm {loc}}(Z,T)$ and $\overline h_{\mu }^{\mathrm {loc}}(Z,T)$ , respectively.

By the Brin–Katok entropy formula for amenable group actions (see [26]), if $\mu $ is in addition G-invariant and $\{F_n\}$ is a tempered Følner sequence which satisfies the growth condition

(1.2) $$ \begin{align} \lim\limits_{n\rightarrow+\infty}\frac{|F_n|}{\log n}=+\infty, \end{align} $$

then the values of the upper and lower local entropies over the whole space X coincide with the measure-theoretic entropy of the system $(X,G)$ .

We will prove the following variational principle between amenable packing entropy and upper local entropy.

Theorem 1.3 Let $(X,G)$ be a G-action topological dynamical system and G a countable infinite discrete amenable group. Let $\{F_n\}$ be a sequence of finite subsets in G satisfying the growth condition (1.2). Then for any non-empty Borel subset Z of X,

$$ \begin{align*}h^P_{\mathrm{top}}(Z,\{F_n\})=\mathop{\mathrm{sup}}\limits\{\overline h_{\mu}^{\mathrm{loc}}(Z,\{F_n\}):\mu\in M(X),\mu(Z)=1\}.\end{align*} $$

1.2 Amenable packing entropy inequalities via factor maps

Let $(X,G)$ and $(Y,G)$ be two G-action topological dynamical systems. A continuous map $\pi :(X,G)\to (Y,G)$ is called a homomorphism or a factor map from $(X,G)$ to $(Y,G)$ if it is onto and $\pi \circ g=g\circ \pi $ , for all $g\in G$ . We also say that $(X,G)$ is an extension of $(Y,G)$ or $(Y,G)$ is a factor of $(X,G)$ .

For a subset Z of X, we denote by $h_{\mathrm {top}}^{UC}(Z,\{F_n\})$ the upper capacity topological entropy of Z (defined in §2). We will show in §2 that the packing entropy can be estimated via the upper capacity topological entropy with parameters (Proposition 2.7). This is a dynamical version of the fact that the packing dimension can be defined via the upper Minkowski dimension. Applying Proposition 2.7, we can prove the following packing entropy inequalities for factor maps.

Theorem 1.4 Let G be a countable infinite discrete amenable group and $\pi :(X,G)\to (Y,G)$ be a factor map between two G-action topological dynamical systems. Let $\{F_n\}$ be any tempered Følner sequence in G satisfying the growth condition (1.2). Then, for any Borel subset E of X,

(1.3) $$ \begin{align} h^P_{\mathrm{top}}(\pi (E),\{F_n\})&\leq h^P_{\mathrm{top}}( E,\{F_n\})\leq h^P_{\mathrm{top}}(\pi (E),\{F_n\})\nonumber\\&\quad+\mathop{\mathrm{sup}}\limits_{y\in Y}h_{\mathrm{top}}^{UC}(\pi^{-1}(y),\{F_n\}). \end{align} $$

We remark here that for $\mathbb Z$ -actions, the inequalities were proved in [24]. But for amenable group actions, except employing Bowen’s idea in [2] and the quasi-tiling techniques developed by Ornstein and Weiss [18], we need a crucial covering lemma for amenable groups built by Lindenstrauss while proving pointwise theorems for amenable groups in [15].

1.3 Amenable packing entropy for certain subsets

Let $M(X,G)$ and $E(X,G)$ be the collection of G-invariant and ergodic G-invariant Borel probability measures on X, respectively. Since G is amenable, $M(X,G)$ and $E(X,G)$ are both non-empty. For $\mu \in M(X,G)$ , let $h_{\mu }(X,G)$ denote the measure-theoretic entropy of $(X,G)$ with respect to $\mu $ .

For $\mu \in M(X,G)$ and a Følner sequence $\{F_n\}$ in G, let $G_{\mu ,\{F_n\}}$ be the set of generic points for $\mu $ (with respect to $\{F_n\}$ ), which is defined by

$$ \begin{align*} G_{\mu,\{F_n\}}=\bigg\{x\in X: \lim_{n\rightarrow+\infty}\frac{1}{|F_n|}\sum_{g\in F_n}f(gx)=\int_Xf\,d{\kern-1pt}{\mu}, \text{ for any }f\in C(X)\bigg\}. \end{align*} $$

For simplicity, we write $G_{\mu ,\{F_n\}}$ as $G_{\mu }$ when there is no ambiguity over $\{F_n\}$ . But we should note that for different Følner sequence $\{F_n\}$ , the corresponding $G_{\mu }$ may not coincide. When $\mu $ is ergodic and the Følner sequence $\{F_n\}$ is tempered, $G_{\mu }$ has full measure for $\mu $ (see Remark 5.1). But when $\mu $ is not ergodic, the set $G_{\mu }$ could be empty.

For the case $G=\mathbb Z$ , Bowen [3] proved that the Bowen topological entropy of $G_{\mu }$ equals the measure-theoretic entropy of $\mu $ if $\mu $ is ergodic. Pfister and Sullivan [20] extended Bowen’s result to the system with the so-called g-almost product property for invariant Borel probability measure $\mu $ . And the results for the amenable group action version were proved by Zheng and Chen [26] and Zhang [23]. The g-almost product property (see [20]) is an extension of the specification property for $\mathbb Z$ -systems and was generalized to amenable systems in [23] (called the almost specification property there). It was shown in [23] that weak specification implies almost specification for amenable systems. We defer the detailed definitions of almost specification and weak specification to §5.

We will prove for packing entropy the following theorem.

Theorem 1.5 Let $(X,G)$ be a G-action topological dynamical system with G a countable infinite discrete amenable group and let $\mu \in M(X,G)$ and $\{F_n\}$ be a Følner sequence in G satisfying the growth condition (1.2). If either $\mu $ is ergodic and $\{F_n\}$ is tempered or $(X,G)$ satisfies the almost specification property, then

(1.4) $$ \begin{align} h^P_{\mathrm{top}}(G_{\mu}, \{F_n\})=h_{\mu}(X,G). \end{align} $$

In geometric measure theory, a set is said to be regular (or ‘dimension-regular’) if it has equal Hausdorff and packing dimensions [22]. As a counterpart in dynamical systems, we have the following definition.

Definition 1.6 A subset is said to be regular in the sense of dimensional entropy (or regular for short) if it has equal Bowen entropy and packing entropy.

An affirmative task in the study of packing entropy is to compute the dimensional entropies (including packing entropy and its dual, Bowen entropy) for various subsets and to make clear which subsets are regular. From Theorem 1.5 and results in [23, 26] for Bowen entropy, under the conditions in Theorem 1.5, the set $G_{\mu }$ is regular since both of its dimensional entropies equal the measure-theoretic entropy for $\mu $ .

1.4 Organization of the paper

In §2 we give some properties of amenable packing entropy including connections with Bowen entropy and upper capacity entropy. We devote §3 to the proof of Theorem 1.3, the variational principle between amenable packing entropy and upper local entropy. This extends Feng and Huang’s result in [9] from $\mathbb Z$ -actions to amenable group actions. In §4 we prove the packing entropy inequalities for factor maps (Theorem 1.4). Finally, in §5, we give the proof of Theorem 1.5 and provide some examples to discuss the regularity for certain subsets. These examples include subsets of symbolic dynamical system and fibers of the $(T,T^{-1})$ transformation. Some detailed computations on these examples are included in Appendix A.

2 Properties of amenable packing entropy

Due to the definition of packing entropy in §1, it is not hard to prove that the packing entropy has the following properties.

Proposition 2.1 Let $\{F_n\}$ be a sequence of finite subsets in G with $|F_n|\rightarrow \infty $ , and let $Z,Z'$ and $Z_i (i=1,2,\ldots )$ be subsets of X.

1. (1) If $Z\subseteq Z'$ , then

   $$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_n\})\le h_{\mathrm{top}}^P(Z',\{F_n\}).\end{align*} $$

2. (2) If $Z\subseteq \bigcup \nolimits _{i=1}^{+\infty }Z_i$ , then for any $\varepsilon>0$ ,

   $$ \begin{align*}h_{\mathrm{top}}^P(Z,\varepsilon,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{i\ge 1}h_{\mathrm{top}}^P(Z_i,\varepsilon,\{F_n\}).\end{align*} $$
   Hence
   $$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{i\ge 1}h_{\mathrm{top}}^P(Z_i,\{F_n\}).\end{align*} $$

3. (3) If $\{F_{n_k}\}$ is a subsequence of $\{F_n\}$ , then

   $$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_{n_k}\})\le h_{\mathrm{top}}^P(Z,\{F_n\}).\end{align*} $$

In the following we recall the definition of amenable Bowen topological entropy which was introduced in [25].

Let $\{F_n\}$ be a sequence of finite subsets in G with $|F_n|\rightarrow \infty $ . For $Z\subseteq X, s\ge 0, N\in \mathbb N$ and $\varepsilon> 0$ , define

$$ \begin{align*}\mathcal{M}(Z,N,\varepsilon,s,\{F_n\}) = \inf\sum_i\exp(-s|F_{n_i}|),\end{align*} $$

where the infimum is taken over all finite or countable families $\{B_{F_{n_i}}(x_i,\varepsilon )\}$ such that $x_i\in X, n_i\ge N$ and $\bigcup \nolimits _i B_{F_{n_i}}(x_i,\varepsilon )\supseteq Z$ . The quantity $\mathcal {M}(Z,N,\varepsilon ,s,\{F_n\})$ does not decrease as N increases and $\varepsilon $ decreases, hence the following limits exist:

$$ \begin{align*}\mathcal{M}(Z,\varepsilon,s,\{F_n\})=\lim_{N\rightarrow+\infty}\mathcal{M}(Z,N,\varepsilon,s,\{F_n\})\end{align*} $$

and

$$ \begin{align*}\mathcal{M}(Z,s,\{F_n\})=\lim_{\varepsilon\rightarrow0}\mathcal{M}(Z,\varepsilon,s,\{F_n\}).\end{align*} $$

The Bowen topological entropy $h^B_{\mathrm {top}}(Z,\{F_n\})$ is then defined as the critical value of the parameter s, where $\mathcal {M}(Z,s,\{F_n\})$ jumps from $+\infty $ to $0$ , that is,

$$ \begin{align*} \mathcal{M}(Z,s,\{F_n\})=\begin{cases} 0, \quad& s> h^B_{\mathrm{top}}(Z,\{F_n\}),\\ +\infty,\quad& s < h^B_{\mathrm{top}}(Z,\{F_n\}). \end{cases} \end{align*} $$

Next we will compare packing topological entropy with Bowen topological entropy.

Proposition 2.2 Let $\{F_n\}$ be a sequence of finite subsets in G with $|F_n|\rightarrow \infty $ . For any $Z\subseteq X$ ,

$$ \begin{align*}h_{\mathrm{top}}^B(Z,\{F_n\})\le h_{\mathrm{top}}^P(Z,\{F_n\}).\end{align*} $$

Proof. The proof for the case $h_{\mathrm {top}}^B(Z,\{F_n\})=0$ is obvious. Now assume $h^B_{\mathrm {top}} (Z,\{F_n\}){\,>\,}0$ and let $0< s <h^B_{\mathrm {top}}(Z,\{F_n\})$ .

Let $\{Z_i\}_{i=1}^{+\infty }$ be any covering of Z. For each i, for any $n\in \mathbb N$ and $\varepsilon>0$ , let $\{\overline B_{F_n}(x_{i,j}, \varepsilon )\}_{j=1}^{N_i}$ be a disjoint subfamily of $\{\overline B_{F_n}(x, \varepsilon )\}_{x\in Z_i}$ with maximal cardinality $N_i$ . Then

$$ \begin{align*}\bigcup_{i=1}^{N_i}B_{F_n}(x_{i,j},3\varepsilon)\supseteq Z_i.\end{align*} $$

So

$$ \begin{align*}\mathcal M(Z_i,n,3\varepsilon,s,\{F_n\})\le N_ie^{-|F_n|s}\le P(Z_i,n,\varepsilon,s,\{F_n\}),\end{align*} $$

and hence

$$ \begin{align*}\mathcal M(Z_i,3\varepsilon,s,\{F_n\})\le P(Z_i,\varepsilon,s,\{F_n\}).\end{align*} $$

Thus

$$ \begin{align*} \mathcal M(Z,3\varepsilon,s,\{F_n\})\le \sum_{i=1}^{+\infty}M(Z_i,3\varepsilon,s,\{F_n\}) \le \sum_{i=1}^{+\infty} P(Z_i,\varepsilon,s,\{F_n\}), \end{align*} $$

from which we deduce that

$$ \begin{align*}\mathcal M(Z,3\varepsilon,s,\{F_n\})\le \mathcal P(Z,\varepsilon,s,\{F_n\}).\end{align*} $$

Since $s<h^B_{\mathrm {top}}(Z,\{F_n\})$ , we have $\mathcal M(Z,s,\{F_n\})=+\infty $ . So

$$ \begin{align*}1\le \mathcal M(Z,3\varepsilon,s,\{F_n\})\le\mathcal P(Z,\varepsilon,s,\{F_n\})\end{align*} $$

whenever $\varepsilon $ is sufficiently small. This implies that

$$ \begin{align*}h_{\mathrm{top}}^P(Z,\varepsilon,\{F_n\})\ge s.\end{align*} $$

Letting $\varepsilon \rightarrow 0$ , we have $h_{\mathrm {top}}^P(Z,\{F_n\})\ge s$ . Hence

$$ \begin{align*}h_{\mathrm{top}}^B(Z,\{F_n\})\le h_{\mathrm{top}}^P(Z,\{F_n\}).\\[-42pt]\end{align*} $$

Corollary 2.3 Let $\mu \in M(X,G)$ , $Z\subseteq X$ with $\mu (Z)=1$ and $\{F_n\}$ be a Følner sequence in G. Then

$$ \begin{align*}h_{\mu}(X,G)\le h_{\mathrm{top}}^P(Z,\{F_n\}).\end{align*} $$

Proof. Let $\{F_{n_k}\}$ be a subsequence of $\{F_n\}$ which is tempered and satisfies the growth condition (1.2). By [26], $h_{\mu }(X,G)\le h_{\mathrm {top}}^B(Z,\{F_{n_k}\})$ . Together with Proposition 2.2 and (3) of Proposition 2.1, we have

$$ \begin{align*}h_{\mu}(X,G)\le h_{\mathrm{top}}^P(Z,\{F_{n_k}\})\le h_{\mathrm{top}}^P(Z,\{F_{n}\}).\\[-38pt]\end{align*} $$

Let $\varepsilon> 0$ , $Z\subseteq X$ and $F\in F(G)$ . A subset $E\subseteq Z$ is said to be an $(F,\varepsilon )$ -separated set of Z if, for any two distinct points $x, y\in E$ , $d_F(x, y)>\varepsilon $ . Let $s_F(Z, \varepsilon )$ denote the largest cardinality of $(F, \varepsilon )$ -separated sets for Z. A subset $E\subseteq X$ is said to be an $(F,\varepsilon )$ -spanning set of Z if, for any $x\in Z$ , there exists $y\in E$ with $d_{F}(x,y)\le \varepsilon $ . Let $r_F(Z, \varepsilon )$ (sometimes we use $r_F(Z, \varepsilon ,d)$ to indicate the accompanied metric d) denote the smallest cardinality of $(F, \varepsilon )$ -spanning sets for Z. For a sequence of finite subsets $\{F_n\}$ in G with $|F_n|\rightarrow \infty $ , the upper capacity topological entropy of Z is defined as

$$ \begin{align*} h_{\mathrm{top}}^{UC}(Z,\{F_n\})&=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log s_{F_n}(Z, \varepsilon)=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log r_{F_n}(Z, \varepsilon). \end{align*} $$

The second equality comes from the following simple fact:

(2.1) $$ \begin{align} r_{F_n}(Z, 2\varepsilon)\le s_{F_n}(Z, 2\varepsilon) \le r_{F_n}(Z, \varepsilon). \end{align} $$

For convention we denote

$$ \begin{align*}h_{\mathrm{top}}^{UC}(Z,\varepsilon,\{F_n\})=\limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log s_{F_n}(Z, \varepsilon),\end{align*} $$

and then

$$ \begin{align*}h_{\mathrm{top}}^{UC}(Z,\{F_n\})=\lim_{\varepsilon\rightarrow 0}h_{\mathrm{top}}^{UC}(Z,\varepsilon,\{F_n\}).\end{align*} $$

We note here that for the case $Z=X$ and where $\{F_n\}$ is a Følner sequence, the quantity $h_{\mathrm {top}}^{UC}(X,\{F_n\})$ coincides with $h_{\mathrm {top}}(X,G)$ , the topological entropy of $(X,G)$ .

Proposition 2.4 Let $\{F_n\}$ be a sequence of finite subsets in G satisfying the growth condition (1.2). Then for any subset Z of X and any $\varepsilon>0$ ,

$$ \begin{align*}h_{\mathrm{top}}^P(Z,\varepsilon,\{F_n\})\le h_{\mathrm{top}}^{UC}(Z,\varepsilon,\{F_n\}).\end{align*} $$

Hence

$$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_n\})\le h_{\mathrm{top}}^{UC}(Z,\{F_n\}).\end{align*} $$

Proof. Let $\varepsilon>0$ be fixed. The proposition is obvious for the case $h_{\mathrm {top}}^P(Z,\varepsilon ,\{F_n\})=0$ . Assume $h_{\mathrm {top}}^P(Z,\varepsilon , \{F_n\})>0$ and let $0<t<s<h_{\mathrm {top}}^P(Z,\varepsilon ,\{F_n\})$ . Then

$$ \begin{align*}P(Z,\varepsilon,s,\{F_n\})\ge \mathcal P(Z,\varepsilon,s,\{F_n\})=+\infty.\end{align*} $$

Thus for any N, there exists a countable pairwise disjoint family $\{\overline {B}_{F_{n_i}}(x_i, \varepsilon )\}$ with $x_i\in Z$ and $n_i\ge N$ for all i such that $1< \sum \nolimits _ie^{-|F_{n_i}|s}$ . For each k, let $m_k$ be the number of i with $n_i=k$ . Then we have

$$ \begin{align*}\sum_ie^{-|F_{n_i}|s}=\sum_{k\ge N}m_ke^{-|F_k|s}.\end{align*} $$

Since $\{F_n\}$ satisfies the growth condition $\lim \nolimits _{n\rightarrow +\infty }({|F_n|}/{\log n})=+\infty $ , $\sum \nolimits _{k\ge 1}e^{|F_k|(t-s)}$ converges. Let $M=\sum \nolimits _{k\ge 1}e^{|F_k|(t-s)}$ . There must be some $k\ge N$ such that $m_k>({1}/{M})e^{|F_k|t}$ , otherwise the above sum is at most

$$ \begin{align*}\sum_{k\ge 1}\frac{1}{M}e^{|F_k|t}e^{-|F_k|s}=1.\end{align*} $$

So $s_{F_k}(Z, \varepsilon )\ge m_k> ({1}/{M})e^{|F_k|t}$ and hence

$$ \begin{align*}h_{\mathrm{top}}^{UC}(Z,\varepsilon,\{F_n\})=\limsup_{k\rightarrow +\infty}\frac{1}{|F_k|}\log s_{F_k}(Z, \varepsilon)\ge t,\end{align*} $$

from which we deduce that $h_{\mathrm {top}}^P(Z,\varepsilon ,\{F_n\})\le h_{\mathrm {top}}^{UC}(Z,\varepsilon ,\{F_n\})$ .

As a corollary, we have the following result.

Corollary 2.5 If $\{F_n\}$ is a Følner sequence that satisfies the growth condition (1.2), then

$$ \begin{align*}h_{\mathrm{top}}^P(X,\{F_n\})=h_{\mathrm{top}}(X,G).\end{align*} $$

Proof. By Corollary 2.3, for any $\mu \in M(X,G)$ , $h_{\mu }(X,G)\le h_{\mathrm {top}}^P(X,\{F_n\})$ . Applying the variational principle for amenable topological entropy (cf. [17, 21]), we have $h_{\mathrm {top}}^P(X,\{F_n\})\ge h_{\mathrm {top}}(X,G)$ . By Proposition 2.4, we have $h_{\mathrm {top}}^P(X,\{F_n\})\le h_{\mathrm {top}}(X,G)$ .

Remark 2.6

1. (1) By [8] (see also [25]), if $\{F_n\}$ is a tempered Følner sequence and satisfies the growth condition (1.2), then

   $$ \begin{align*}h_{\mathrm{top}}^P(X,\{F_n\})= h_{\mathrm{top}}(X,G)=h_{\mathrm{top}}^B(X,\{F_n\}).\end{align*} $$

2. (2) If we replace X by a G-invariant compact subset, the above equality also holds and hence any G-invariant compact subset is regular (when the Følner sequence $\{F_n\}$ is tempered and satisfies the growth condition (1.2)).

At the end of this section we will give further relations between packing entropy and upper capacity topological entropy.

Proposition 2.7 Let $\varepsilon>0$ , Z be a subset of X and let $\{F_n\}$ be a sequence of finite subsets in G satisfying the growth condition (1.2).

1. (1) We have

   $$ \begin{align*} h^P_{\mathrm{top}}(Z,\varepsilon,\{F_n\})\le \inf\Big\{\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,\varepsilon,\{F_n\}): Z=\bigcup_{i=1}^{\infty}Z_i\Big\}. \end{align*} $$
   Hence
   $$ \begin{align*} h^P_{\mathrm{top}}(Z,\{F_n\})\le\inf\Big\{\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,\{F_n\}): Z=\bigcup_{i=1}^{\infty}Z_i\Big\}. \end{align*} $$

2. (2) For any $\delta>0$ , there exists a cover $\bigcup _{i=1}^{\infty }Z_i=Z$ (which depends on both $\varepsilon $ and $\delta $ ) such that

   $$ \begin{align*} h^P_{\mathrm{top}}(Z,\varepsilon,\{F_n\})+\delta\ge \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,3\varepsilon,\{F_n\}). \end{align*} $$

Proof. For any $Z=\bigcup _{i=1}^{\infty }Z_i$ , by Propositions 2.1 and 2.4,

$$ \begin{align*} h_{\mathrm{top}}^P(Z,\varepsilon,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{i\ge 1}h_{\mathrm{top}}^P(Z_i,\varepsilon,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,\varepsilon,\{F_n\}). \end{align*} $$

Hence

$$ \begin{align*}h^P_{\mathrm{top}}(Z,\varepsilon,\{F_n\})\le \inf\Big\{\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,\varepsilon,\{F_n\}): Z=\bigcup_{i=1}^{\infty}Z_i\Big\}.\end{align*} $$

For the opposite direction, we may assume that $h^P_{\mathrm {top}}(Z,\varepsilon ,\{F_n\})<\infty $ . Let $\delta>0$ be fixed and set $s=h^P_{\mathrm {top}}(Z,\varepsilon ,\{F_n\})+\delta $ . By the definition of the amenable packing entropy, we have that $\mathcal {P}(Z,\varepsilon ,s,\{F_n\})=0$ . Then there exists a cover $\bigcup _{i=1}^{\infty }Z_i\supseteq Z$ such that

$$ \begin{align*}\sum_{i\ge 1}P(Z_i,\varepsilon,s,\{F_n\})<1.\end{align*} $$

For each $Z_i$ , when N is large enough, we have $P(Z_i,N,\varepsilon ,s,\{F_n\})<1$ . Let E be any $(F_N,3\varepsilon )$ -separated subset of $Z_i$ . Noting that the closed Bowen balls $\overline B_{F_N}(x_i,\varepsilon )$ ( $x_i\in E\subset Z_i$ ) are pairwise disjoint, we have

$$ \begin{align*} |E|e^{-s|F_N|}=\sum_{x_i\in E}e^{-s|F_N|}\le P(Z_i,N,\varepsilon,s,\{F_n\})<1. \end{align*} $$

Hence $s_{F_N}(Z_i,3\varepsilon )<e^{s|F_N|}$ , which leads to $h^{UC}_{\mathrm {top}}(Z_i,3\varepsilon ,\{F_n\})\le s$ . Thus we have

$$ \begin{align*} h_{\mathrm{top}}^P(Z,\varepsilon,\{F_n\})+\delta \ge \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,3\varepsilon,\{F_n\}).\\[-42pt] \end{align*} $$

Remark 2.8 Proposition 2.4 is motivated from the equivalent definition of packing dimension through Minkowski dimension in geometric measure theory, which was due to Tricot [22] (see also [16] for reference). It is unclear to us whether

$$ \begin{align*} h^P_{\mathrm{top}}(Z,\{F_n\})= \inf\Big\{\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(Z_i,\{F_n\}): Z=\bigcup_{i=1}^{\infty}Z_i\Big\}. \end{align*} $$

3 A variational principle for amenable packing entropy

In this section we will give the proof of our Theorem 1.3, the variational principle for amenable packing topological entropy. We assume in this section that $\{F_n\}$ is a sequence of finite subsets in G and satisfies growth condition (1.2).

3.1 Lower bound

Proposition 3.1 Let $Z\subseteq X$ be a Borel set. Then

$$ \begin{align*}h^P_{\mathrm{top}}(Z, \{F_n\})\ge\mathop{\mathrm{sup}}\limits\{\overline h_{\mu}^{\mathrm{loc}}(Z,\{F_n\}):\mu\in M(X),\mu(Z)=1\}.\end{align*} $$

For the proof, we need the following classical $5r$ -lemma in geometric measure theory (cf. [16, Theorem 2.1]).

Lemma 3.2 ( $5r$ -lemma)

Let $(X, d)$ be a compact metric space and $\mathcal B =\{B(x_i, r_i)\}_{i\in I}$ be a family of closed (or open) balls in X. Then there exists a finite or countable subfamily $\mathcal B'=\{B(x_i, r_i)\}_{i\in I'}$ of pairwise disjoint balls in $\mathcal B$ such that

$$ \begin{align*}\bigcup_{B\in\mathcal B}B\subseteq \bigcup_{i\in I'}B(x_i,5r_i).\end{align*} $$

We also need the following lemma, which comes directly from the definition of the packing entropy.

Lemma 3.3 Let $E\subset X$ and $s>0$ . Then for any $0<\varepsilon _1<\varepsilon _2$ ,

$$ \begin{align*}P(\overline E, \varepsilon_2, s, \{F_n\})\le P(E, \varepsilon_1, s, \{F_n\}).\end{align*} $$

Proof Proof of Proposition 3.1

Let $\mu \in M(X)$ with $\mu (Z)=1$ and assume $\overline h_{\mu }^{\mathrm {loc}}(Z,\{F_n\})>0$ .

Let $0<s<\overline h_{\mu }^{\mathrm {loc}}(Z,\{F_n\})$ . Then there exist $\varepsilon ,\delta>0$ and a Borel set $A\subset Z$ with $\mu (A)>0$ such that for every $x\in A$ ,

$$ \begin{align*}\overline h_{\mu}(x,\varepsilon,\{F_n\})>s+\delta,\end{align*} $$

where $\overline h_{\mu }(x,\varepsilon ,\{F_n\}):=\limsup \nolimits _{n\rightarrow +\infty }-({1}/{|F_n|})\log \mu (B_{F_n}(x,\varepsilon ))$ .

Let $E\subset A$ be any Borel set with $\mu (E)>0$ . Define

$$ \begin{align*}E_n=\{x\in E:\mu(B_{F_n}(x,\varepsilon))<e^{-|F_n|(s+\delta)}\},\quad n\in\mathbb N.\end{align*} $$

Then $\bigcup _{n=N}^{+\infty }E_n=E$ for any $N\in \mathbb N$ , and hence $\mu (\bigcup _{n=N}^{+\infty }E_n)=\mu (E)$ . Fix $N\in \mathbb N$ . There exists $n\ge N$ such that

$$ \begin{align*}\mu(E_n)\ge \frac{1}{n(n+1)}\mu(E).\end{align*} $$

Fix such n and consider the family of Bowen balls $\{B_{F_n}(x,{\varepsilon }/{5}):x\in E_n\}$ . By the $5r$ -lemma, Lemma 3.2 (we use metric $d_{F_n}$ instead of d), there exists a finite pairwise disjoint family $\{B_{F_n}(x_i,{\varepsilon }/{5})\}$ with $x_i\in E_n$ such that

$$ \begin{align*}\bigcup_{i}B_{F_n}(x_i,\varepsilon)\supset \bigcup_{x\in E_n}B_{F_n}\bigg(x,\frac{\varepsilon}{5}\bigg)\supset E_n.\end{align*} $$

Thus

$$ \begin{align*} P\bigg(E,N,\frac{\varepsilon}{5},s,\{F_n\}\bigg)&\ge P\bigg(E_n,N,\frac{\varepsilon}{5},s,\{F_n\}\bigg)\ge \sum_{i}e^{-|F_n|s}=e^{|F_n|\delta}\sum_{i}e^{-|F_{n}|(s+\delta)}\\ &\ge e^{|F_n|\delta}\sum_{i}\mu(B_{F_n}(x_i,\varepsilon))\ge e^{|F_n|\delta}\mu(E_n)\ge \frac{e^{|F_n|\delta}}{n(n+1)}\mu(E). \end{align*} $$

By the growth condition (1.2) of the sequence $\{F_n\}$ , we have

$$ \begin{align*}\frac{e^{|F_n|\delta}}{n(n+1)}\rightarrow +\infty\quad\text{as }n\rightarrow +\infty.\end{align*} $$

Letting $N\rightarrow +\infty $ , we obtain that

$$ \begin{align*}P\bigg(E,\frac{\varepsilon}{5},s,\{F_n\}\bigg)=+\infty.\end{align*} $$

Note that this equality holds for every Borel set $E\subset A$ with $\mu (E)>0$ .

Let $\{A_i\}_{i=1}^\infty $ be any covering of A. Then by Lemma 3.3,

$$ \begin{align*}\sum_i P\bigg(A_i,\frac{\varepsilon}{10},s,\{F_n\}\bigg)\ge \sum_i P\bigg(\overline A_i,\frac{\varepsilon}{5},s,\{F_n\}\bigg).\end{align*} $$

Since there must exist some $A_i$ such that $\overline A_i\cap A$ (which is a Borel set now) contains a Borel subset $E\subset \overline A_i\cap A$ with $\mu (E)>0$ , we have

$$ \begin{align*}\sum_i P\bigg(A_i,\frac{\varepsilon}{10},s,\{F_n\}\bigg)\ge P\bigg(E,\frac{\varepsilon}{5},s,\{F_n\}\bigg)=+\infty.\end{align*} $$

Thus

$$ \begin{align*}\mathcal P \bigg(Z,\frac{\varepsilon}{10},s,\{F_n\}\bigg)\ge \mathcal P \bigg(A,\frac{\varepsilon}{10},s,\{F_n\}\bigg)=+\infty,\end{align*} $$

from which we deduce that

$$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_n\})\ge h_{\mathrm{top}}^P\bigg(Z,\frac{\varepsilon}{10},\{F_n\}\bigg)\ge s.\end{align*} $$

Since s is chosen arbitrarily in $(0,\overline h_{\mu }(Z,\{F_n\}))$ , we finally show that

$$ \begin{align*}h_{\mathrm{top}}^P(Z,\{F_n\})\ge \overline h_{\mu}^{\mathrm{loc}}(Z,\{F_n\}).\end{align*} $$

This finishes the proof of Proposition 3.1.

3.2 Upper bound

The following proposition is the upper bound part of the variational principle. In fact it is valid for any analytic set Z. Recall that a set in a metric space is said to be analytic if it is a continuous image of the set $\mathcal N$ of infinite sequences of natural numbers. In a Polish space, the collection of analytic subsets contains Borel sets and is closed under countable unions and intersections (cf. [11]).

Proposition 3.4 Let $Z\subseteq X$ be an analytic set with $h^P_{\mathrm {top}}(Z, \{F_n\})>0$ . For any $0<s<h^P_{\mathrm {top}}(Z, \{F_n\})$ , there exist a compact set $K\subseteq Z$ and $\mu \in M(K)$ such that $\overline h_{\mu }^{\mathrm {loc}}(K,\{F_n\})\ge ~s$ .

The following lemma is needed.

Lemma 3.5 Let $Z\subseteq X$ and $s,\varepsilon>0.$ Assume that $\mathcal P (Z,\varepsilon ,s,\{F_n\})=+\infty $ . Then for any given finite interval $(a,b)\subset \mathbb R$ with $a\ge 0$ and any $N\in \mathbb N$ , there exists a finite disjoint collection $\{\overline B_{F_{n_i}}(x_i, \varepsilon )\}$ such that $x_i\in Z, n_i\ge N$ and $\sum _i\exp (-|F_{n_i}|s)\in (a,b)$ .

Proof. See [9, Lemma 4.1].

Proof Proof of Proposition 3.4

Since Z is analytic, there exists a continuous surjective map $\phi : \mathcal {N}\rightarrow Z$ . Let $\Gamma _{n_1,n_2,\ldots ,n_p}=\{(m_1,m_2, \ldots )\in \mathcal {N}: m_1\le n_1, m_2\le n_2, \ldots , m_p\le n_p\}$ and let $Z_{n_1,\ldots ,n_p}=\phi (\Gamma _{n_1,\ldots ,n_p})$ .

For $0<s<h^P_{\mathrm {top}}(Z, \{F_n\})$ , take $\varepsilon>0$ small enough to make $0<s<h^P_{\mathrm {top}}(Z,\varepsilon ,\{F_n\})$ and take $t\in (s, h^P_{\mathrm {top}}(Z,\varepsilon ,\{F_n\}))$ . Following Feng and Huang’s steps (which are inspired by the work of Joyce and Preiss [12] on packing measures), we will construct inductively the following data:

1. (D-1) a sequence of finite sets $(K_i)_{i=1}^{+\infty }$ with $K_i\subset Z$ ;

2. (D-2) a sequence of finite measures $(\mu _i)_{i=1}^{+\infty }$ with each $\mu _i$ supported on $K_i$ ;

3. (D-3) a sequence of integers $(n_i)_{i=1}^{+\infty }$ and a sequence of positive numbers $(\gamma _i)_{i=1}^{+\infty }$ ;

4. (D-4) a sequence of integer-valued functions $(m_i: K_i\rightarrow \mathbb N)_{i=1}^{+\infty }$ .

Moreover, the sequences $(K_i),(\mu _i), (n_i),(\gamma _i)$ and $(m_i(\cdot ))$ will be constructed to satisfy the following conditions.

1. (C-1) For each i, the family $\mathcal V_i:=\{B(x,\gamma _i)\}_{x\in K_i}$ is disjoint. Each element in $\mathcal V_{i+1}$ is a subset of $B(x,\gamma _i/2)$ for some $x\in K_i$ .

2. (C-2) For each i, $K_i\subset Z_{n_1,\ldots ,n_i}$ and $\mu _i=\sum _{y\in K_i}e^{-|F_{m_i(y)}|s}\delta _y$ with $1<\mu _1(K_1)<2$ .

3. (C-3) For each $x\in K_i$ and $z\in B(x,\gamma _i)$ ,

   (3.1) $$ \begin{align} \overline B_{F_{m_i(x)}}(z,\varepsilon)\cap \bigcup_{y\in K_i\setminus \{x\}}B(y,\gamma_i)=\emptyset \end{align} $$
   and
   (3.2) $$ \begin{align} \mu_i(B(x,\gamma_i))=e^{-|F_{m_i(x)}|s}\le \sum_{y\in E_{i+1}(x)}e^{-|F_{m_{i+1}(y)}|s}<(1+2^{-(i+1)})\mu_i(B(x,\gamma_i)), \end{align} $$
   where $E_{i+1}(x)=B(x,\gamma _i)\cap K_{i+1}$ .

We will give the construction later.

Suppose the sequences $(K_i),(\mu _i), (n_i),(\gamma _i)$ and $(m_i(\cdot ))$ have been constructed. By (3.2), for $V_i\in \mathcal V_i$ ,

$$ \begin{align*} \mu_i(V_i)\le \mu_{i+1}(V_i)=\sum_{V\in\mathcal V_{i+1},V\subset V_i}\mu_{i+1}(V)\le (1+2^{-(i+1)})\mu_i(V_i). \end{align*} $$

Repeatedly using the above inequalities, we have for any $j>i$ and any $V_i\in \mathcal V_i$ ,

(3.3) $$ \begin{align} \mu_i(V_i)\le \mu_j(V_i)\le \prod_{n=i+1}^j(1+2^{-n})\mu_i(V_i)\le C\mu_i(V_i), \end{align} $$

where $C:=\prod _{n=1}^{+\infty }(1+2^{-n})<+\infty $ .

Let $\tilde \mu $ be a limit point of $(\mu _i)$ in the weak $^*$ topology. Let

$$ \begin{align*}K=\bigcap_{n=1}^{+\infty}\overline{\bigcup_{i\ge n}K_i}.\end{align*} $$

Then $\tilde \mu $ is supported on K. Furthermore,

$$ \begin{align*}K\subset\bigcap_{p=1}^{+\infty}\overline{Z_{n_1,\ldots,n_p}}.\end{align*} $$

By the continuity of $\phi $ , applying Cantor’s diagonal argument, we can show that $\bigcap _{p=1}^{+\infty }Z_{n_1,\ldots ,n_p}=\bigcap _{p=1}^{+\infty }\overline {Z_{n_1,\ldots ,n_p}}$ . Hence K is a compact subset of Z.

By (3.3), for any $x\in K_i$ ,

$$ \begin{align*}e^{-|F_{m_i(x)}|s}=\mu_i(B(x,\gamma_i))\le \tilde \mu(B(x,\gamma_i))\le C\mu_i(B(x,\gamma_i))=Ce^{-|F_{m_i(x)}|s}.\end{align*} $$

In particular,

$$ \begin{align*}1\le \sum_{x\in K_1}\mu_1(B(x,\gamma_1))\le \tilde \mu(K)\le \sum_{x\in K_1}C\mu_1(B(x,\gamma_1))\le 2C.\end{align*} $$

Note that $K\subset \bigcup _{x\in K_i}B(x,\gamma _i/2)$ . By (3.1), the first part of (C-3), for each $x\in K_i$ and $z\in \overline B(x,\gamma _i)$ , reads

$$ \begin{align*}\tilde \mu(\overline B_{F_{m_i(x)}}(z,\varepsilon))\le\tilde \mu(\overline B(x,\gamma_i/2))\le Ce^{-|F_{m_i(x)}|s}.\end{align*} $$

For each $z\in K$ and $i\in \mathbb N$ , $z\in \overline B(x,\gamma _i/2)$ for some $x\in K_i$ . Hence

$$ \begin{align*}\tilde \mu(B_{F_{m_i(x)}}(z,\varepsilon))\le Ce^{-|F_{m_i(x)}|s}.\end{align*} $$

Define $\mu =\tilde \mu /\tilde \mu (K)$ . Then $\mu \in M(K)$ . For each $z\in K$ , there exists a sequence $k_i\uparrow +\infty $ such that $\mu (B_{F_{k_i}}(z,\varepsilon ))\le Ce^{-|F_{k_i}|s}/\tilde \mu (K)$ . Hence $\overline h_{\mu }^{\mathrm {loc}}(K,\{F_n\})\ge s$ .

Now the only thing left is to give the inductive construction of the data $(K_i),(\mu _i)$ , $(n_i),(\gamma _i)$ and $(m_i(\cdot ))$ . The inductive steps are as follows.

Step 1 Construct the data $K_1,\mu _1, n_1,\gamma _1$ and $m_1(\cdot )$ .

Since $t\in (s, h^P_{\mathrm {top}}(Z,\varepsilon ,\{F_n\}))$ , we have that $\mathcal P (Z,\varepsilon ,t,\{F_n\})=+\infty $ . Let

$$ \begin{align*}H=\bigcup\{U\subset X : U \text{ is open}, \mathcal P (Z\cap U,\varepsilon,t,\{F_n\})=0\}.\end{align*} $$

Then by the separability of X, H is a countable union of the open sets U. Hence $\mathcal P (Z\cap H,\varepsilon ,t,\{F_n\})=0$ . Let $Z'=Z\setminus H=Z\cap (X\setminus H)$ . If $\mathcal P (Z'\cap U,\varepsilon ,t,\{F_n\})=0$ for some open set U, then

$$ \begin{align*} \mathcal P (Z\cap U,\varepsilon,t,\{F_n\})\le \mathcal P (Z'\cap U,\varepsilon,t,\{F_n\})+\mathcal P (Z\cap H,\varepsilon,t,\{F_n\})=0. \end{align*} $$

So $U\subset H$ and then $Z' \cap U=\emptyset $ . Hence for any open set $U\subset X$ , either $Z' \cap U =\emptyset $ or $\mathcal P (Z' \cap U,\varepsilon ,t,\{F_n\})>0$ .

Because $\mathcal P (\kern-0.5pt Z,\varepsilon ,t,\kern-0.5pt\{\kern-0.6pt F_n\})\kern-1.3pt\le\kern-0.5pt \mathcal P (\kern-0.5pt Z',\varepsilon ,t,\{\kern-1ptF_n\})\kern-1pt+\mathcal P (\kern-0.6ptZ\cap H,\varepsilon ,t,\{\kern-1pt F_n\})\kern-1pt=\mathcal P (\kern-0.5ptZ',\varepsilon ,t, \{\kern-1pt F_n\})$ , we have $\mathcal P (Z',\varepsilon ,t,\{F_n\})= \mathcal P (Z,\varepsilon ,t,\{F_n\})=+\infty $ . Then $\mathcal P (Z',\varepsilon ,s,{\{F_n\})=+\infty }$ .

By Lemma 3.5, we can find a finite set $K_1\subset Z' $ , an integer-valued function $m_1(x)$ on $K_1$ such that the collection $\{\overline B_{F_{m_1(x)}}(x,\varepsilon )\}_{x\in K_1}$ is disjoint and

$$ \begin{align*}\sum_{x\in K_1}e^{-|F_{m_1(x)}|s}\in (1,2).\end{align*} $$

Define $\mu _1=\sum _{x\in K_1} e^{-|F_{m_1(x)}|s}\delta _x$ , where $\delta _x$ denotes the Dirac measure at x. Take $\gamma _1>0$ sufficiently small such that for any function $z:K_1\rightarrow X$ with $\max _{x\in K_1}d(x, z(x))\le \gamma _1$ , we have for each $x\in K_1$ ,

(3.4) $$ \begin{align} ( \overline B(z(x),\gamma_1)\cup \overline B_{F_{m_1(x)}}(z(x),\varepsilon) )\cap \bigg(\bigcup_{y\in K_1\setminus \{x\}}\overline B(z(y),\gamma_1)\cup \overline B_{F_{m_1(y)}}(z(y),\varepsilon)\bigg)=\emptyset. \end{align} $$

Since $K_1\subset Z'$ , we have $\mathcal P (Z\cap B(x,\gamma _1/4),\varepsilon ,t,\{F_n\})\ge \mathcal P (Z'\cap B(x,\gamma _1/4),\varepsilon ,t, \{F_n\})>0$ for each $x\in K_1$ . Therefore we can pick a sufficiently large $n_1\in \mathbb N$ so that $Z_{n_1}\supset K_1$ and $\mathcal P (Z_{n_1}\cap B(x,\gamma _1/4),\varepsilon ,t,\{F_n\})>0$ for each $x\in K_1$ .

Step 2 Construct the data $K_2,\mu _2, n_2,\gamma _2$ and $m_2(\cdot )$ .

By (3.4), the family of balls $\{B(x,\gamma _1)\}_{x\in K_1}$ are pairwise disjoint. For each $x\in K_1$ , since $\mathcal P (Z_{n_1}\cap B(x,\gamma _1/4),\varepsilon ,t,\{F_n\})>0$ , similarly to Step 1, we can construct a finite set $E_2(x)\subset Z_{n_1}\cap B(x,\gamma _1/4)$ and an integer-valued function $m_2:E_2(x)\rightarrow \mathbb N\cap [\max \{m_1(y):y\in K_1\},+\infty )$ such that:

1. (2-a) for each open set U with $U\cap E_2(x)\neq \emptyset $ , $\mathcal P (Z_{n_1}\cap U,\varepsilon ,t,\{F_n\})>0$ ;

2. (2-b) the elements in the family $\{\overline B_{F_{m_2(y)}}(y,\varepsilon )\}_{y\in E_2(x)}$ are pairwise disjoint and

   $$ \begin{align*} \mu_1(\{x\})<\sum_{y\in E_2(x)}e^{-|F_{m_2(y)}|s}<(1+2^{-2})\mu_1(\{x\}). \end{align*} $$

To see this, we fix $x\in K_1$ . Denote $V=Z_{n_1}\cap B(x, \gamma _1/4)$ . Let

$$ \begin{align*}H_x:=\bigcup\{U\subset X: U \text{ is open and } \mathcal P (V\cap U,\varepsilon,t,\{F_n\})=0\}.\end{align*} $$

Set $V'=V\setminus H_x$ . Then as in Step 1, we can show that

$$ \begin{align*}\mathcal P (V',\varepsilon,t,\{F_n\})=\mathcal P (V,\varepsilon,t,\{F_n\})>0.\end{align*} $$

Moreover, for any open set $U\subset X$ , either $V'\cap U =\emptyset $ or $\mathcal P (V'\cap U,\varepsilon ,t,\{F_n\})>0$ . Since $s<t$ , we have that $\mathcal P (V',\varepsilon ,s,\{F_n\})=+\infty $ . By Lemma 3.5, we can find a finite set $E_2(x)\subset V'$ and a map $m_2: E_2(x)\rightarrow \mathbb N\cap [\max \{m_1(y):y\in K_1\},+\infty )$ such that (2-b) holds. Notice that if an open set U satisfies $U\cap E_2(x)\neq \emptyset $ , then $U\cap V'\neq \emptyset $ . So $\mathcal P (Z_{n_1}\cap U,\varepsilon ,t,\{F_n\})\ge \mathcal P (V'\cap U,\varepsilon ,t,\{F_n\})>0$ . Hence (2-a) holds.

Since the family $\{B(x,\gamma _1)\}_{x\in K_1}$ is disjoint, so is the family $\{E_2(x)\}_{x\in K_1}$ . Define

$$ \begin{align*}K_2=\bigcup_{x\in K_1}E_2(x)\quad\text{and}\quad\mu_2=\sum_{y\in K_2}e^{-|F_{m_2(y)}|s}\delta_y.\end{align*} $$

By (3.4) and (2-b), the elements in $\{\overline B_{F_{m_2(y)}}(y,\varepsilon )\}_{y\in K_2}$ are pairwise disjoint. Hence we can take $0<\gamma _2<\gamma _1/4$ such that for any function $z:K_2\rightarrow X$ satisfying $\max _{x\in K_{2}}d(x,z(x))< \gamma _2$ , we have for $x\in K_2$ ,

(3.5) $$ \begin{align} ( \overline B(z(x),\gamma_2)\cup \overline B_{F_{m_2(x)}}(z(x),\varepsilon) )\cap \bigg(\bigcup_{y\in K_2\setminus \{x\}}\overline B(z(y),\gamma_2)\cup \overline B_{F_{m_2(y)}}(z(y),\varepsilon)\bigg)=\emptyset. \end{align} $$

Choose a sufficiently large $n_2\in \mathbb N$ so that $Z_{n_1,n_2}\supset K_2$ and

$$ \begin{align*}\mathcal P (Z_{n_1,n_2}\cap B(x,\gamma_2/4),\varepsilon,t,\{F_n\})>0\end{align*} $$

for each $x\in K_2$ .

Step 3 Next we suppose that the data $K_i,\mu _i, n_i,\gamma _i$ and $m_i(\cdot )$ ( $i=1,\ldots ,p$ ) have been constructed. We will construct the data $K_{p+1},\mu _{p+1}, n_{p+1},\gamma _{p+1}$ and $m_{p+1}(\cdot )$ .

Assume that we have constructed $K_i,\mu _i, n_i,\gamma _i$ and $m_i(\cdot )$ for $i=1,\ldots ,p$ . And assume that for any function $z:K_p\rightarrow X$ with $d(x,z(x))<\gamma _p$ for all $x\in K_p$ , we have

(3.6) $$ \begin{align} ( \overline B(z(x),\gamma_p)\cup \overline B_{F_{m_p(x)}}(z(x),\varepsilon) )\cap \bigg(\bigcup_{y\in K_p\setminus \{x\}}\overline B(z(y),\gamma_p)\cup \overline B_{F_{m_p(y)}}(z(y),\varepsilon)\bigg)=\emptyset \end{align} $$

for each $x\in K_p$ ; and $Z_{n_1,\ldots ,n_p}\supset K_p$ and $\mathcal P (Z_{n_1,\ldots ,n_p}\cap B(x,\gamma _p/4),\varepsilon ,t,\{F_n\})>0$ for each $x\in K_p$ .

Note that the family of balls $\{\overline B(x,\gamma _p)\}_{x\in K_p}$ are pairwise disjoint. For each $x\in K_p$ , since $\mathcal P (Z_{n_1,\ldots ,n_p}\cap B(x,\gamma _p/4),\varepsilon ,t,\{F_n\})>0$ , similarly to Step 2, we can construct a finite set $E_{p+1}(x)\subset Z_{n_1,\ldots ,n_p}\cap B(x,\gamma _p/4)$ and an integer-valued function $m_{p+1}:E_{p+1}(x)\rightarrow \mathbb N\cap [\max \{m_p(y):y\in K_p\},+\infty )$ such that:

1. (3-a) for each open set U with $U\cap E_{p+1}(x)\neq \emptyset $ , $\mathcal P (Z_{n_1,\ldots ,n_p}\cap U,\varepsilon ,t,\{F_n\})>0$ ;

2. (3-b) the elements in the family $\{\overline B_{F_{m_{p+1}(y)}}(y,\varepsilon )\}_{y\in E_{p+1}(x)}$ are pairwise disjoint and

   $$ \begin{align*} \mu_p(\{x\})<\sum_{y\in E_{p+1}(x)}e^{-|F_{m_{p+1}(y)}|s}<(1+2^{-(p+1)})\mu_p(\{x\}). \end{align*} $$

By (3.6) and (3-b), the family $\{\overline B_{F_{m_{p+1}(y)}}(y,\varepsilon )\}_{y\in K_{p+1}}$ is disjoint. Hence we can take $0<\gamma _{p+1}< \gamma _p/4$ such that for any function $z:K_{p+1}\rightarrow X$ with $\max _{x\in K_{p+1}}d(x,z(x))<\gamma _{p+1}$ , we have for each $x\in K_{p+1}$ ,

(3.7) $$ \begin{align} ( \overline B(z(x),\gamma_{p+1})\cup &\overline B_{F_{m_{p+1}(x)}}(z(x),\varepsilon) )\nonumber \\ &\cap \bigg(\bigcup_{y\in K_{p+1}\setminus \{x\}}\overline B(z(y),\gamma_{p+1})\cup \overline B_{F_{m_{p+1}(y)}}(z(y),\varepsilon)\bigg)=\emptyset. \end{align} $$

Choose a sufficiently large $n_{p+1}\in \mathbb N$ so that $Z_{n_1,\ldots ,n_{p+1}}\supset K_{p+1}$ and $\mathcal P (Z_{n_1,\ldots ,n_{p+1}}\cap B(x,\gamma _{p+1}/4),\varepsilon ,t,\{F_n\})>0$ for each $x\in K_{p+1}$ .

Then we finish the required construction and complete the proof of the proposition.

4 Packing entropy inequalities for factor maps

In this section we will prove Theorem 1.4. The proof is a combination of Bowen’s method in [2] and quasi-tiling techniques for amenable groups.

4.1 Preliminaries for amenable groups

Let G be a countable infinite discrete amenable group. Let A and K be two non-empty finite subsets of G. Recall that $B(A,K)$ , the K-boundary of A, is defined by

$$ \begin{align*}B(A,K)=\{g\in G: Kg\cap A\neq\emptyset \text { and } Kg\cap(G\setminus A)\neq\emptyset\}.\end{align*} $$

For $\delta>0$ , the set A is said to be $(K, \delta )$ -invariant if

$$ \begin{align*} \frac{|B(A,K)|}{|A|}<\delta. \end{align*} $$

We say a sequence of non-empty finite subsets $\{F_n\}$ of G becomes more and more invariant if, for any $\delta>0$ and any non-empty finite subset K of G, $F_n$ is $(K,\delta )$ -invariant for sufficiently large n. An equivalent condition for the sequence $\{F_n\}$ to be a Følner sequence is that $\{F_n\}$ becomes more and more invariant (see [18]).

Let $\tilde {\mathcal {F}}$ be a collection of finite subsets of G. It is said to be $\delta $ -disjoint if for every $A\in \tilde {\mathcal {F}}$ there exists an $A'\subset A$ such that $|A'|\ge (1-\delta )|A|$ and such that $A'\cap B'=\emptyset $ for every $A\neq B\in \tilde {\mathcal {F}}$ .

The following is a covering lemma for amenable groups by Lindenstrauss.

Lemma 4.1 (Lindenstrauss’s covering lemma, [15, Corollary 2.7])

For any $\delta \in (0,1/100), C>0$ and finite $D\subset G$ , let $M\in \mathbb N$ be sufficiently large (depending only on $\delta ,C$ and D). Let $F_{i,j}$ be an array of finite subsets of G where $i=1,\ldots ,M$ and $j=1,\ldots ,N_i$ , with the following two requirements.

1. (1) For every i, $\bar {F}_{i,*}=\{F_{i,j}\}_{j=1}^{N_i}$ satisfies

   $$ \begin{align*}\bigg|\bigcup_{k'<k}F_{i,k'}^{-1}F_{i,k}\bigg|\le C|F_{i,k}|\quad \text{ for }k=2,\ldots,N_i.\end{align*} $$

2. (2) Denote $F_{i,*}=\bigcup \bar {F}_{i,*}$ . The finite set sequences $F_{i,*}$ satisfy that for every $1<i\le M$ and every $1\le k \le N_i$ ,

   $$ \begin{align*}\bigg|\bigcup_{i'<i}DF_{i',*}^{-1}F_{i,k}\bigg|\le (1+\delta)|F_{i,k}|.\end{align*} $$

Assume that $A_{i,j}$ is another array of finite subsets of G with $F_{i,j}A_{i,j}\subset F$ for some finite subset F of G. Let $A_{i,*}=\bigcup _jA_{i,j}$ and

$$ \begin{align*}\alpha=\frac{\min_{1\le i\le M}|DA_{i,*}|}{|F|}.\end{align*} $$

Then the collection of subsets of F,

$$ \begin{align*}\tilde{\mathcal{F}}=\{F_{i,j}a: 1\le i\le M, 1\le j\le N_i \text{ and }a\in A_{i,j}\},\end{align*} $$

has a subcollection $\mathcal {F}$ that is $10\delta ^{1/4}$ -disjoint such that

$$ \begin{align*}\bigg|\bigcup \mathcal{F}\bigg|\ge (\alpha-\delta^{1/4})|F|.\end{align*} $$

4.2 Proof of Theorem 1.4

Let G, $\{F_n\}$ and $\pi :(X,G)\to (Y,G)$ be as in Theorem 1.4 and $E\subset X$ be a Borel set. Let d and $\rho $ be the compatible metrics on X and Y, respectively.

For any $\varepsilon>0$ , there exists $\delta>0$ such that, for any $x_1,x_2\in X$ with $d(x_1,x_2)\le \delta $ , one has $\rho (\pi (x_1),\pi (x_2))\le \varepsilon $ . Now let $\{y_i\}_{i=1}^{k}\subset \pi (E)$ be any $(F_n,\varepsilon )$ -separated set of $\pi (E)$ and choose for each i a point $x_i\in \pi ^{-1}(y_i)\cap E$ . Then $\{x_i\}_{i=1}^k$ forms an $(F_n,\delta )$ -separated set of E. Hence

$$ \begin{align*}s_{F_n}(\pi(E), \varepsilon)\le s_{F_n}(E, \delta)\end{align*} $$

and

(4.1) $$ \begin{align} h^{UC}_{\mathrm{top}}( \pi (E), \varepsilon,\{F_n\})\leq h^{UC}_{\mathrm{top}}(E, \delta, \{F_n\}). \end{align} $$

By Proposition 2.7(2), for any $\eta>0$ , there exists a cover $\bigcup _{i=1}^{\infty }E_i=E$ such that

$$ \begin{align*}h^P_{\mathrm{top}}(E,\delta/3,\{F_n\})+\eta \ge \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(E_i,\delta,\{F_n\}).\end{align*} $$

Then we have

$$ \begin{align*} h^P_{\mathrm{top}}(\pi (E),\varepsilon,\{F_n\})&\le\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{P}_{\mathrm{top}}(\pi(E_i),\varepsilon,\{F_n\})\quad \text{(by Proposition 2.1(2))}\\[-2pt] &\le\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(\pi(E_i),\varepsilon,\{F_n\})\quad \text{(by Proposition 2.4)}\\[-2pt] &\le \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(E_i,\delta,\{F_n\})\quad \text{(noting that 4 also holds for each}\ E_i)\\[-2pt] &\le h^P_{\mathrm{top}}(E,\delta/3,\{F_n\})+\eta, \end{align*} $$

which implies that

$$ \begin{align*}h^P_{\mathrm{top}}(\pi (E),\{F_n\})\leq h^P_{\mathrm{top}}( E,\{F_n\}).\end{align*} $$

To get the upper bound, we need to prove the following inequality for amenable upper capacity topological entropy, which extends a result by Bowen [2, Theorem 17] to amenable group actions.

Theorem 4.2 Let G be a countable infinite discrete amenable group and $\pi :(X,G)\to (Y,G)$ be a factor map between two G-action topological dynamical systems. Let $\{F_n\}$ be any tempered Følner sequence in G satisfying growth condition (1.2). Then for any subset E of X,

(4.2) $$ \begin{align} h^{UC}_{\mathrm{top}}( E,\{F_n\})\leq h^{UC}_{\mathrm{top}}(\pi (E),\{F_n\})+\mathop{\mathrm{sup}}\limits_{y\in Y}h_{\mathrm{top}}^{UC}(\pi^{-1}(y),\{F_n\}). \end{align} $$

Proof. If $\mathop {\mathrm {sup}}\limits _{y\in Y}h_{\mathrm {top}}^{UC}(\pi ^{-1}(y),\{F_n\})=\infty $ then there is nothing to prove. So we assume that

$$ \begin{align*}a:=\mathop{\mathrm{sup}}\limits_{y\in Y}h_{\mathrm{top}}^{UC}(\pi^{-1}(y),\{F_n\})<\infty.\end{align*} $$

To verify (4.2), we need some preparations in the following three steps.

Step 1 Construct $F_{i,j}$ , the array of subsets of G.

Fix $\tau>0$ . For any $\varepsilon>0$ , let $0<\delta <\min \{\varepsilon ,1/100\}$ be small enough. Let $C>0$ be the constant in the tempered condition (1.1) for the Følner sequence $\{F_n\}$ and let $D=\{e_G\}\subset G$ , where $e_G$ is the identity element of G. Let $M>0$ be large enough as in Lemma 4.1 corresponding to $\delta , C$ and D.

For each $y\in Y$ , choose $m(y)\in \mathbb N$ such that for any $n\ge m(y)$ ,

(4.3) $$ \begin{align} \frac{1}{|F_{n}|}\log r_{F_{n}}(\pi^{-1}(y),\varepsilon,d)\leq h_{\mathrm{top}}^{UC}(\pi^{-1}(y),\{F_n\})+\tau \leq a+\tau. \end{align} $$

Here recall that $r_{F_{n}}(\pi ^{-1}(y),\varepsilon ,d)$ denotes the smallest cardinality of $(F_{n}, \varepsilon )$ -spanning sets for $\pi ^{-1}(y)$ . Let $E_y$ be an $(F_{m(y)},\varepsilon )$ -spanning set of $\pi ^{-1}(y)$ with the smallest cardinality $|E_y|=r_{F_{m(y)}}(\pi ^{-1}(y),\varepsilon ,d).$ Denote

$$ \begin{align*}U_y=\{p\in X: \text{ there exists }q\in E_y \text{ such that }d_{F_{m(y)}}(p,q)<2\varepsilon\},\end{align*} $$

which is an open neighborhood of $\pi ^{-1}(y)$ . Since $\bigcap _{\gamma>0}\pi ^{-1}(\overline {B(y,\gamma ,\rho )})=\pi ^{-1}(y)$ , we have $(X\setminus U_y)\bigcap (\bigcap _{\gamma>0}\pi ^{-1}(\overline {B(y,\gamma ,\rho )}) )=\emptyset $ . Hence by the finite intersection property of compact sets, there is a $W_y:=B(y,\gamma _y,\rho )$ for some $\gamma _y>0$ such that $U_y\supset \pi ^{-1}(W_y).$ Since Y is compact, there exist $y_{1,1},\ldots , y_{1,r_1}$ such that $W_{y_{1,1}},\ldots ,W_{y_{1,r_1}}$ cover Y. List the Følner sets in the collection $\{F_{m(y_{1,k})}: 1\le k\le r_1\}$ by

$$ \begin{align*}F_{n_{1,1}}, F_{n_{1,2}},\ldots, F_{n_{1,N_1}} \quad\text{where }n_{1,1}<n_{1,2}<\cdots<n_{1,N_1}.\end{align*} $$

Note that $N_1=\#\{F_{m(y_{1,k})}: 1\le k\le r_1\}\le r_1$ .

For each $y\in Y$ , choose $m(y)>n_{1,N_1}$ such that (4.3) holds for any $n\ge m(y)$ . Repeating the above process, we can obtain $y_{2,1},\ldots , y_{2,r_2}\in Y$ such that $W_{y_{2,1}},\ldots ,W_{y_{2,r_2}}$ cover Y. We then list the Følner sets in the collection $\{F_{m(y_{2,k})}: 1\le k\le r_2\}$ :

$$ \begin{align*}F_{n_{2,1}}, F_{n_{2,2}},\ldots, F_{n_{2,N_2}} \quad\text{where }n_{2,1}<n_{2,2}<\cdots<n_{2,N_2}.\end{align*} $$

Note that $N_2=\#\{F_{m(y_{2,k})}: 1\le k\le r_2\}\le r_2$ .

Repeating the above steps inductively, we can obtain, for each $1\le i\le M$ :

1. (1) a collection of points $y_{i,1},\ldots , y_{i,r_i}\in Y$ such that $W_{y_{i,1}},\ldots ,W_{y_{i,r_i}}$ cover Y;

2. (2) a collection of Følner sets $\{F_{n_{i,1}}, F_{n_{2,2}},\ldots , F_{n_{i,N_i}}\}$ ( $=\{F_{m(y_{i,k})}: 1\le k\le r_i\}$ ) with $n_{i,1}<n_{i,2}<\cdots <n_{i,N_i}$ and $N_i\le r_i$ .

From the above construction, for each $1\le i\le M-1$ , we have $n_{i,N_i}<n_{i+1,1}$ . Moreover, $n_{i+1,1}$ can be chosen sufficiently large compared with $n_{i,N_i}$ such that, for every $1<i\le M$ and every $1\le k \le N_i$ ,

(4.4) $$ \begin{align} \bigg|\bigcup_{i'<i}F_{n_{i'},*}^{-1}F_{n_i,k}\bigg|\le (1+\delta)|F_{n_i,k}|. \end{align} $$

For simplification, we denote $F_{i,j}=F_{n_{i,j}}$ for each $1\le i\le M$ and $1\le j\le N_i$ , which is the array of G we required.

Step 2 Produce quasi-tilings from $F_{i,j}$ .

Let $\eta _1$ be a common Lebesgue number of the family of open covers $\{W_{y_{i,1}},\ldots ,W_{y_{i,r_i}}\}$ with respect to the metric $\rho $ . Denote $\eta =({\eta _1}/{2})$ .

Let N be large enough such that, for every $n>N$ , $F_n$ is $(F_{i,*}\cup \{e_G\},\delta )$ -invariant for all $1\le i\le M$ .

For each $y\in Y$ and $n>N$ , let

$$ \begin{align*} A_{i,j}&=\{a\in F_n: F_{i,j}a\subset F_n \text{ and there exists }1\le k\le r_i \text{ such that } F_{m(y_{i,k})}=F_{i,j} \\ &\quad \ \text{ and }\overline{B}(ay,\eta,\rho)\subseteq W_{y_{i,k}} \}. \end{align*} $$

We note here that $A_{i,j}$ depends on y.

For any $g\in F_n\setminus B(F_n,F_{i,*}\cup \{e_G\})$ , we have $F_{i,*}g\subset F_n$ . Since $\eta _1$ ( $=2\eta $ ) is a Lebesgue number of $\{W_{y_{i,1}},\ldots ,W_{y_{i,r_i}}\}$ , $\overline {B}(gy,\eta ,\rho )$ is contained in some $W_{y_{i,k}}$ and then $g\in A_{i,*}$ . Hence

$$ \begin{align*}A_{i,*}\supseteq F_n\setminus B(F_n,F_{i,*}\cup\{e_G\})\end{align*} $$

for each $1\le i \le M$ , and

$$ \begin{align*}\alpha=\frac{\min_{1\le i\le M}|DA_{i,*}|}{|F_n|}=\frac{\min_{1\le i\le M}|A_{i,*}|}{|F_n|}>1-\delta.\end{align*} $$

We are now able to apply Lemma 4.1: the temperedness assumption for $\{F_n\}$ makes requirement (1) of that lemma hold and (4.4) makes requirement (2) hold. From the collection of subsets of $F_n$ ,

$$ \begin{align*}\tilde{\mathcal{F}}=\{F_{i,j}a: 1\le i\le M, 1\le j\le N_i \text{ and }a\in A_{i,j}\},\end{align*} $$

we can find by Lemma 4.1 a subcollection $\mathcal {F}$ which is $10\delta ^{1/4}$ -disjoint and

(4.5) $$ \begin{align} \left|\bigcup \mathcal{F}\right|\ge (\alpha-\delta^{1/4})|F_n|\ge (1-\delta-\delta^{1/4})|F_n|. \end{align} $$

In fact, this subcollection $\mathcal {F}$ is a $10\delta ^{1/4}$ quasi-tiling of $F_n$ subordinate to $y\in Y$ (see, for example, [14] for the detail definition of quasi-tiling).

There may exist overlaps between elements in $\mathcal {F}$ . Since $\mathcal {F}$ is $10\delta ^{1/4}$ -disjoint, there exits $T'\subset T$ for each $T\in \mathcal {F}$ such that $|T'|/|T|\ge 1-10\delta ^{1/4}$ and the collection $\{T': T\in \mathcal {F}\}$ is disjoint. Denote this new collection by $\mathcal {F}'$ . By (4.5),

(4.6) $$ \begin{align} \left|\bigcup \mathcal{F}\right|\le\sum_{T\in\mathcal{F}}|T|\le \frac{1}{1-10\delta^{1/4}}\sum_{T\in\mathcal{F}}|T'|\le \frac{1}{1-10\delta^{1/4}}|F_n| \end{align} $$

and

(4.7) $$ \begin{align} \left|\bigcup\mathcal{F}'\right|= \sum_{T\in\mathcal{F}}|T'|\ge (1-10\delta^{1/4})(1-\delta-\delta^{1/4})|F_n|. \end{align} $$

Step 3 Cover $\pi ^{-1}B_{F_n}(y,\eta ,\rho )$ through $F_n$ -Bowen balls in $(X,G)$ .

Claim. When $\delta $ is sufficiently small, for any $y\in Y$ and $n>N$ , there exist $l(y)>0$ and $v_1(y),v_2(y),\ldots , v_{l(y)}(y)\in X$ such that

$$ \begin{align*}\bigcup_{i=1}^{l(y)}B_{F_n}(v_i(y),4\varepsilon,d)\supseteq \pi^{-1}(B_{F_n}(y,\eta,\rho))\end{align*} $$

and

$$ \begin{align*}l(y)\le\exp((a+2\tau)|F_n| ).\end{align*} $$

Proof Proof of the Claim

For each $T=F_{i,j}a\in \mathcal {F}$ , since $a\in A_{i,j}$ , by the construction of $A_{i,j}$ , there exists some point in $\{y_{i,1}, y_{i,2}, \ldots , y_{i,r_i}\}$ , denoted by $y_T$ , such that $\overline {B}(ay,\eta ,\rho )\subseteq W_{y_T}$ and $F_{m(y_T)}=F_{i,j}$ .

In the following we will recover the $F_n$ -orbits of $(X,G)$ from the T-orbits.

Let $E\subset X$ be any finite $\varepsilon $ -spanning set under the metric d. For any sequence of points $\{z_T\}_{T\in \mathcal {F}}$ with each $z_T\in E_{y_{T}}$ and any sequence of points $\{z_g\}_{g\in F_n\setminus \cup \mathcal {F}'}$ with each $z_g\in E$ , let

$$ \begin{align*} V(y;\{z_T\}, \{z_g\}):=\Big\{&u\in\ X: d_{T'}(u,a^{-1}z_T)<2\varepsilon \text{ for all }T=F_{i,j}a\in \mathcal{F}, \\ &d(gu,z_g)<2\varepsilon \text{ for all }g\in F_n\setminus \bigcup\mathcal {F}'\Big\}. \end{align*} $$

It is not hard to verify that

$$ \begin{align*} \bigcup_{\{z_T\}, \{z_g\}}V(y;\{z_T\}, \{z_g\})\supseteq\pi^{-1}(B_{F_n}(y,\eta,\rho)), \end{align*} $$

that is, the family $\{V(y;\{z_T\}, \{z_g\}):z_T\in E_{y_{T}}, T\in \mathcal {F}, z_g\in E, g\in F_n\setminus \bigcup \mathcal {F}'\}$ forms an open cover of $\pi ^{-1}(B_{F_n}(y,\eta ,\rho )).$ We also note that some of the $V(y;\{z_T\}, \{z_g\})$ may be empty.

We pick any point $v(\{z_T\}, \{z_g\})$ in each non-empty $V(y;\{z_T\}, \{z_g\})$ . Then

$$ \begin{align*}B_{F_n}(v(\{z_T\}, \{z_g\}), 4\varepsilon, d)\supseteq V(y;\{z_T\}, \{z_g\}).\end{align*} $$

Enumerate these $v(\{z_T\}, \{z_g\})$ by $y_1,y_2,\ldots ,y_{l(y)}$ . We then obtain

(4.8) $$ \begin{align} \bigcup_{i=1}^{l(y)}B_{F_n}(y_i, 4\varepsilon, d)\supseteq\pi^{-1}(B_{F_n}(y,\eta,\rho)). \end{align} $$

Now the only thing left is to estimate $l(y)$ . Clearly,

$$ \begin{align*} l(y)&\le \prod_{T\in\mathcal{F}}|E_{y_T}|\cdot\prod_{g\in F_n\setminus \cup\mathcal {F}'}|E| = \prod_{T\in\mathcal{F}}r_{F_{m(y_T)}}(\pi^{-1}(y),\varepsilon,d)\cdot |E|^{|F_n|-|\bigcup\mathcal {F}'|}\\ &\le \exp\bigg(\sum_{T\in\mathcal{F}}|F_{m(y_T)}|(a+\tau)+\left(|F_n|-\left|\bigcup\mathcal {F}'\right|\right)\log|E|\bigg)\\ &\le \exp\bigg( \bigg(\frac{1}{1-10\delta^{1/4}}(a+\tau)+(1-(1-10\delta^{1/4})(1-\delta-\delta^{1/4}))\log|E|\bigg)|F_n|\bigg)\\ &\qquad\qquad\qquad\qquad\,\quad\text{(by (4.6) and (4.7))}\\ &\le \exp((a+2\tau)|F_n| )\quad \text{(when }\delta \text{ is sufficiently small). }\\[-3.2pc] \end{align*} $$

We now proceed to prove (4.2). For any subset E of X, let H be an $(F_n,\eta )$ -spanning set of $\pi (E)$ with minimal cardinality $r_{F_n}(\pi (E),\eta ,\rho )$ . Then by the above claim, the set $R=\{v_i(y):1\le i\le l(y), y\in H\}$ forms an $(F_n, 4\varepsilon )$ -spanning set of E, since

$$ \begin{align*} \bigcup_{y\in H}\bigcup_{i=1}^{l(y)}B_{F_n}(v_i(y),4\varepsilon,d)&\supseteq \bigcup_{y\in H}\pi^{-1}(B_{F_n}(y,\eta,\rho))\supseteq \pi^{-1}\pi(E)\supseteq E. \end{align*} $$

Hence

(4.9) $$ \begin{align} r_{F_n}(E,4\varepsilon,d)\le r_{F_n}(\pi(E),\eta,\rho)\cdot\exp((a+2\tau)|F_n| ). \end{align} $$

From this we deduce that

$$ \begin{align*} h^{UC}_{\mathrm{top}}( E,\{F_n\})\leq h^{UC}_{\mathrm{top}}(\pi (E),\{F_n\})+a+2\tau. \end{align*} $$

Letting $\tau $ tend to $0$ , (4.2) is proved.

Proof Proof of the upper bound of $h^P_{\mathrm {top}}( E,\{F_n\})$

What we need in fact is inequality (4.9) in the proof of Theorem 4.2.

From (2.1), we first convert inequality (4.9) into

$$ \begin{align*} s_{F_n}(E,8\varepsilon,d)\le s_{F_n}(\pi(E),\eta,\rho)\cdot\exp((a+2\tau)|F_n| ). \end{align*} $$

This implies that

$$ \begin{align*} h^{UC}_{\mathrm{top}}(E,8\varepsilon,\{F_n\}) \leq h^{UC}_{\mathrm{top}}(\pi(E),\eta,\{F_n\})+a+2\tau. \end{align*} $$

By Proposition 2.7(2) again, for any $\delta>0$ , there exists a cover $\bigcup _{i=1}^{\infty }V_i=\pi (E)$ such that

$$ \begin{align*}h^P_{\mathrm{top}}(\pi(E),\eta/3,\{F_n\})+\delta\ge \mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(V_i,\eta,\{F_n\}).\end{align*} $$

Using an similar argument to that in the proof of the lower bound,

$$ \begin{align*} h^P_{\mathrm{top}}(E,8\varepsilon,\{F_n\})&\le\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{P}_{\mathrm{top}}(\pi^{-1}(V_i),8\varepsilon,\{F_n\})\le\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(\pi^{-1}(V_i),8\varepsilon,\{F_n\})\\ &\le\mathop{\mathrm{sup}}\limits_{i\ge 1} h^{UC}_{\mathrm{top}}(V_i,\eta,\{F_n\})+a+2\tau\\ &\le h^P_{\mathrm{top}}(\pi(E),\eta/3,\{F_n\})+\delta+a+2\tau. \end{align*} $$

Hence

$$ \begin{align*} h^{P}_{\mathrm{top}}(E,\{F_n\}) \leq h^{P}_{\mathrm{top}}(\pi(E),\{F_n\})+a+2\tau. \end{align*} $$

Since $\tau>0$ is arbitrary, we finally obtain

$$ \begin{align*} h^{P}_{\mathrm{top}}(E,\{F_n\}) \leq h^{P}_{\mathrm{top}}(\pi(E),\{F_n\})+\mathop{\mathrm{sup}}\limits_{y\in Y}h_{\mathrm{top}}^{UC}(\pi^{-1}(y),\{F_n\}).\\[-3.6pc] \end{align*} $$

5 Amenable packing entropy for certain subsets

5.1 The set of generic points

Recall that for $\mu \in M(X,G)$ and a Følner sequence $\{F_n\}$ in G, the set of generic points for $\mu $ (with respect to $\{F_n\}$ ) is defined by

$$ \begin{align*} G_{\mu}=\bigg\{x\in X: \lim_{n\rightarrow+\infty}\frac{1}{|F_n|}\sum_{g\in F_n}f(gx)=\int_Xf\, d{\kern-1pt}{\mu}, \text{ for any }f\in C(X)\bigg\}. \end{align*} $$

Remark 5.1 If $\mu \in E(X,G)$ and $\{F_n\}$ is a tempered Følner sequence then $\mu (G_{\mu })=1$ . To show this, let $\{f_i\}_{i=1}^{\infty }$ be a countable dense subset of $C(X)$ and denote

$$ \begin{align*}X_i=\bigg\{x\in X: \lim_{n\rightarrow+\infty}\frac{1}{|F_n|}\sum_{g\in F_n}f_i(gx)=\int_Xf_i\, d{\kern-1pt}{\mu}\bigg\}.\end{align*} $$

By the pointwise ergodic theorem, $\mu (X_i)=1$ . Hence $G_{\mu }=\bigcap _{i=1}^\infty X_i$ has full measure.

The system $(X,G)$ is said to have the almost specification property if there exists a non-decreasing function $g: (0,1)\rightarrow (0,1)$ with $\lim _{r\rightarrow 0}g(r)=0$ (a mistake-density function) and a map $m: (0,1)\rightarrow F(G)\times (0,1)$ such that for any $k\in \mathbb N$ , any $\varepsilon _1, \varepsilon _2,\ldots , \varepsilon _k\in (0,1)$ , and any $x_1, x_2,\ldots ,x_k\in X$ , if $F_i$ is $m(\varepsilon _i)$ -invariant, $i=1, 2,\ldots , k$ , and $\{F_i\}_{i=1}^k$ are pairwise disjoint, then

$$ \begin{align*}\bigcap_{1\le i\le m}B(g; F_i,x_i,\varepsilon_i)\neq\emptyset,\end{align*} $$

where $B(g; F,x,\varepsilon ):=\{y\in X: |\{h\in F: d(hx,hy)>\varepsilon \}|\le g(\varepsilon )|F|\}$ , the Bowen ball allowing a mistake with density $g(\varepsilon )$ .

Remark 5.2 It was shown in [23] that the weak specification implies the almost specification. Recall that in [6] the system $(X,G)$ (the group G need not be amenable) has weak specification if for any $\varepsilon>0$ there exists a non-empty finite subset F of G with the following property: for any finite collection $F_1, \ldots , F_m$ of finite subsets G with

$$ \begin{align*}FF_i\cap F_j=\emptyset \quad\text{for } 1\le i,j\le m,i\neq j,\end{align*} $$

and for any collection of points $x_1,\ldots ,x_m \in X$ , there exists a point $y\in X$ such that

$$ \begin{align*}d(gx_i,gy)\le \varepsilon \quad\text{for all } g\in F_i, 1\le i\le m,\end{align*} $$

that is,

$$ \begin{align*}\bigcap_{1\le i\le m}\overline{B}_{F_i}(x_i,\varepsilon)\neq\emptyset.\end{align*} $$

In this section, we will prove Theorem 1.5, that is, if $\mu \in M(X,G)$ , the Følner sequence $\{F_n\}$ satisfies the growth condition (1.2) and either $\mu $ is ergodic and $\{F_n\}$ is tempered or $(X,G)$ has almost specification, then

(5.1) $$ \begin{align} h^P_{\mathrm{top}}(G_{\mu}, \{F_n\})=h_{\mu}(X,G). \end{align} $$

The idea of the proof comes from Pfister and Sullivan [20] (see also [23, 26] for amenable group actions).

5.1.1 Upper bound for $h_{\mathrm {top}}^P(G_{\mu },\{F_n\})$

In the following we are going to prove $h_{\mathrm {top}}^P(G_{\mu },\{F_n\})\leq h_{\mu }(X,G)$ assuming that the Følner sequence $\{F_n\}$ satisfies the growth condition (1.2).

For $\mu \in M(X,G)$ , let $\{K_m\}_{m\in \mathbb N}$ be a decreasing sequence of closed convex neighborhoods of $\mu $ in $M(X)$ such that $\bigcap _{m\in \mathbb N}K_m=\{\mu \}$ . Let

$$ \begin{align*}A_{n,m}=\bigg\{x\in X:\frac{1}{|F_n|}\sum_{g\in F_n}\delta_x\circ g^{-1}\in K_m\bigg\}\quad\text{for }m,n\in\mathbb N,\end{align*} $$

and

$$ \begin{align*}R_{N,m}=\bigg\{x\in X:\text{ for any }n>N, \frac{1}{|F_n|}\sum_{g\in F_n}\delta_x\circ g^{-1}\in K_m\bigg\}\quad\text{for }m,N\in\mathbb N.\end{align*} $$

Then for any $m,N\ge 1$ ,

$$ \begin{align*}R_{N,m}=\bigcap_{n>N}A_{n,m} \quad\text{and}\quad G_{\mu}\subseteq \bigcup_{k> N}R_{k,m}.\end{align*} $$

For $\varepsilon>0$ and $Z\subseteq X$ , recall that $s_{F_n}(Z,\varepsilon )$ denotes the maximal cardinality of any $(F_n,\varepsilon )$ -separated subset of Z. Then we have

(5.2) $$ \begin{align} \limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log s_{F_n}(R_{N,m},\varepsilon) \le \limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log s_{F_n}(A_{n,m},\varepsilon) \quad\text{for any }m,N\ge 1. \end{align} $$

By the claim in [26, p. 878] (we note that it also works for non-ergodic invariant measures),

$$ \begin{align*} \lim_{\varepsilon\rightarrow 0}\lim_{m\rightarrow+\infty}\limsup_{n\rightarrow+\infty}\frac{1}{|F_n|}\log s_{F_n}(A_{n,m},\varepsilon)\le h_{\mu}(X,G). \end{align*} $$

Hence for any $\eta>0$ , there exists $0<\varepsilon _1$ such that, for any $0<\varepsilon <\varepsilon _1$ , there exists $M=M(\varepsilon )\in \mathbb N$ such that

$$ \begin{align*} \limsup_{n\rightarrow+\infty}\frac{1}{|F_n|}\log s_{F_n}(A_{n,m},\varepsilon)< h_{\mu}(X,G)+\eta, \end{align*} $$

whenever $m\ge M$ . Especially,

(5.3) $$ \begin{align} \limsup_{n\rightarrow+\infty}\frac{1}{|F_n|}\log s_{F_n}(A_{n,M},\varepsilon)< h_{\mu}(X,G)+\eta. \end{align} $$

Taking (5.2) and (5.3) together, for any $0<\varepsilon <\varepsilon _1$ , we have that for any $N\in \mathbb N$ ,

$$ \begin{align*}\limsup_{n\rightarrow +\infty}\frac{1}{|F_n|}\log s_{F_n}(R_{N,M},\varepsilon)< h_{\mu}(X,G)+\eta.\end{align*} $$

Since for any $N'\in \mathbb N$ , $G_{\mu }\subseteq \bigcup _{N\ge N'}R_{N,M}$ , by Propositions 2.1 and 2.4,

$$ \begin{align*}h_{\mathrm{top}}^P(G_{\mu},\varepsilon,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{N\ge N'}h_{\mathrm{top}}^P(R_{N,M},\varepsilon,\{F_n\})\le \mathop{\mathrm{sup}}\limits_{N\ge N'}\limsup_{n\rightarrow+\infty}\frac{1}{|F_n|}\log s_{F_n}(R_{N,M},\varepsilon),\end{align*} $$

from which it follows that

$$ \begin{align*}h_{\mathrm{top}}^P(G_{\mu},\varepsilon,\{F_n\})\le h_{\mu}(X,G)+\eta.\end{align*} $$

Letting $\varepsilon \rightarrow 0$ and then $\eta \rightarrow 0$ , we obtain that $h_{\mathrm {top}}^P(G_{\mu },\{F_n\})\le h_{\mu }(X,G)$ .

5.1.2 Lower bound for $h_{\mathrm {top}}^P(G_{\mu },\{F_n\})$

For the case where $\mu $ is ergodic and $\{F_n\}$ is tempered, since $\mu (G_{\mu })=1$ , Corollary 2.3 gives the lower bound.

For the case when $\mu \in M(X,G)$ and the system $(X,G)$ has almost specification property, the proof of the lower bound becomes rather complicated because of the quasi-tiling techniques for amenable groups. But it was shown in [23] that $h_{\mathrm {top}}^B(G_{\mu },\{F_n\})= h_{\mu }(X,G)$ . Hence by Proposition 2.2, we obtain $h_{\mathrm {top}}^P(G_{\mu },\{F_n\})\ge h_{\mu }(X,G)$ .

5.2 G-symbolic dynamical system

Let A be a finite set with cardinality $|A|\ge 2$ and let $A^G=\{(x_g)_{g\in G}: x_g\in A\}$ be the G-symbolic space over A. Consider the left action of G on $A^G$ :

$$ \begin{align*}g'(x_g)_{g\in G}=(x_{gg'})_{g\in G} \quad\text{for all }g'\in G\text{ and }(x_g)_{g\in G}\in A^G.\end{align*} $$

$(A^G,G)$ forms a G-symbolic dynamical system or a G-acting full shift (over A). For any non-empty closed G-invariant subset X of $A^G$ , the subsystem $(X,G)$ is called a subshift. For $x=(x_g)_{g\in G}$ and a finite subset $F\subset G$ , denote by $x|_F=(x_g)_{g\in F}\in A^F$ the restriction of x to F and denote $[x|_F]=\{\omega \in A^G: \omega _g=x_g \text { for all }g\in F\}$ (which is called a cylinder).

Fix any tempered Følner sequence $\{F_n\}$ of G with $F_0=\{e_G\}\subsetneq F_1\subsetneq F_2\subsetneq \ldots $ and $\bigcup _{n}F_n=G$ . Note that $\{F_n\}$ satisfies the growth condition (1.2) automatically. We can then define a metric d on $A^G$ associated to $\{F_n\}$ as follows:

(5.4) $$ \begin{align} d(x,y)=\begin{cases} 1\quad& \text{if }x \text{ and }y \text{ are not equal on } F_0,\\ e^{-|F_n|}\quad& n=\max\{k: x|_{F_k}=y|_{F_k}\}. \end{cases} \end{align} $$

To discuss the regularity for subsets of $(A^G,G)$ , we need to consider the relation between Bowen entropy (packing entropy) and the corresponding Hausdorff dimension (packing dimension). Before that, we recall the definitions of Hausdorff dimension and packing dimension (cf. [16]).

Definition 5.3 Let $(X,d)$ be a compact metric space. Let $0\le s<\infty $ . For $E\subset X$ and $\varepsilon>0$ , put

$$ \begin{align*}P_{\varepsilon}^s(E)=\mathop{\mathrm{sup}}\limits\sum_i(\text{diam} B_i)^s\end{align*} $$

where the supremum is taken over all disjoint families of closed balls $\{\overline B_i\}$ such that $\text {diam} B_i \le \varepsilon $ and the centers of the $B_i$ are in E.

Then set $P^s(E)=\lim \nolimits _{\varepsilon \rightarrow 0}P_{\varepsilon }^s(E)$ (since $P_{\varepsilon }^s(E)$ is non-decreasing on $\varepsilon $ ) and define

$$ \begin{align*}\mathcal P^s(E)=\inf \bigg\{\sum_{i=1}^{\infty}P^s(E_i): E=\bigcup_{i=1}^{\infty}E_i\bigg\}.\end{align*} $$

The packing dimension of E is then defined by

$$ \begin{align*}\mathrm{dim}_P(E)=\inf\{s: \mathcal P^s(E)=0\}=\mathop{\mathrm{sup}}\limits\{s: \mathcal P^s(E)=\infty\}.\end{align*} $$

Let $\mathcal H^s(E)=\lim \nolimits _{\varepsilon \rightarrow 0}\mathcal H_{\varepsilon }^s(E)$ , where

$$ \begin{align*}\mathcal H_{\varepsilon}^s(E)=\inf\bigg\{\sum_{i=1}^\infty(\text{diam} E_i)^s: E\subset \bigcup_{i=1}^\infty E_i, \text{diam} E_i\le \varepsilon\bigg\}.\end{align*} $$

The Hausdorff dimension of E is then defined by

$$ \begin{align*}\mathrm{dim}_H(E)=\inf\{s: \mathcal H^s(E)=0\}=\mathop{\mathrm{sup}}\limits\{s: \mathcal H^s(E)=\infty\}.\end{align*} $$

Comparing with the definitions of dimensional entropies, we have the following proposition.

Proposition 5.4 Let the Følner sequence $\{F_n\}$ satisfy the following two conditions:

1. (1) $F_mF_n\subset F_{m+n}$ for each $m,n\in \mathbb N$ ;

2. (2) $\lim \nolimits _{n\rightarrow \infty } (({F_{n+1}})/{{F_n}})=1$ .

Then for any $E\subset A^G$ ,

$$ \begin{align*} h_{\mathrm{top}}^B(E,\{F_n\})=\mathrm{dim}_HE\quad\text{and}\quad h_{\mathrm{top}}^P(E,\{F_n\})=\mathrm{dim}_PE, \end{align*} $$

where the dimensions $\text {dim}_H$ and $\text {dim}_P$ are both under the metric d defined in (5.4).

Proof. See Appendix A.1.

Remark 5.5

1. (1) Due to [10], the sequence of finite subsets $\{F_n\}$ of G satisfying condition (1) in Proposition 5.4 is called a regular system. If G is a finitely generated group, and letting $F_n$ be the collection of elements in G with word length (with respect to a finite symmetric generating subset) no more than n, then $\{F_n\}$ satisfies condition (1).

2. (2) There are examples of amenable groups which admit Følner sequences satisfying the conditions in Proposition 5.4. An abelian example of the group G is $\mathbb Z^d$ , with $F_n=[-n,n]^d$ . An non-abelian example of the group G is the dihedral group, with $F_n$ chosen to be the collection of elements with word length no more than n (see [5]).

By Proposition 5.4, we have the following result.

Proposition 5.6 Let the Følner sequence $\{F_n\}$ satisfy the conditions in Proposition 5.4. Then any subset $E\subset A^G$ is regular in the sense of dimensional entropy if and only if E is dimension-regular (under the Følner sequence $\{F_n\}$ and metric d).

By Remark 2.6, any non-empty closed G-invariant subset X of $A^G$ is regular in the sense of dimensional entropy. Hence we have the following corollary.

Corollary 5.7 Let the Følner sequence $\{F_n\}$ satisfy the conditions in Proposition 5.4. Then any non-empty closed G-invariant subset X of $A^G$ is both regular in the sense of dimensional entropy and dimension-regular (under the Følner sequence $\{F_n\}$ and metric d).

Let $0\le \alpha < \beta \le 1$ and $A=\{0,1\}$ . Let $H\subset G$ such that

$$ \begin{align*} \liminf_{n\rightarrow+\infty}\frac{|H\cap F_n|}{|F_n|}=\alpha\quad\text{and}\quad\limsup_{n\rightarrow+\infty}\frac{|H\cap F_n|}{|F_n|}=\beta, \end{align*} $$

that is H is a subset of G with lower density $\alpha $ and upper density $\beta $ with respect to $\{F_n\}$ . Now we define $X_{\alpha ,\beta }\subset \{0,1\}^G$ by

$$ \begin{align*} X_{\alpha,\beta}=\{(x_g)_{g\in G}: x_g=0 \text{ if }g\notin H\}. \end{align*} $$

Assume in addition that $\{F_n\}$ satisfies the conditions in Proposition 5.4. Then we have the following proposition.

Proposition 5.8 $h_{\mathrm {top}}^B (X_{\alpha ,\beta },\{F_n\})=\alpha \log 2$ and $h_{\mathrm {top}}^P(X_{\alpha ,\beta },\{F_n\})=\beta \log 2$ . Hence $X_{\alpha ,\beta }$ is not regular.

Proof. See Appendix A.2.

5.3 Fibers of $\{T,T^{-1}\}$ transformation

Random walk in random scenery (RWRS) is a class of stationary random processes which are well studied in both probability theory and ergodic theory. RWRS provides measure-theoretic models with amazingly rich behavior (see [1, 7]). Among the class of RWRS, $\{T,T^{-1}\}$ transformation, although apparently simple, is possibly the best known in the history of ergodic theory since it is a natural example of a K-automorphism that is not Bernoulli [13]. In spite of its measure-theoretic aspect, we will consider the topological model of the $\{T,T^{-1}\}$ transformation and investigate subsets of the topological system.

Definition 5.9 (Topological $\{T,T^{-1}\}$ transformation)

Let $A=\{1,-1\}$ , and by convention we denote the shift map on $A^{\mathbb Z}$ by T, which is defined by

$$ \begin{align*} (T(x))_i=x_{i+1}\quad \text{for any } x=(x_i)_{i\in \mathbb Z}\in A^{\mathbb Z}. \end{align*} $$

The $\{T,T^{-1}\}$ transformation, denoted by S, on $A^{\mathbb Z}\times A^{\mathbb Z}$ is defined by

$$ \begin{align*} S(x,y)=\begin{cases} (T(x), T(y)) \quad&\text{if }y_0=1,\\ (T^{-1}(x), T(y)) \quad&\text{if }y_0=-1. \end{cases} \end{align*} $$

Then $S^n(x,y)=(T^{\omega (y,n)}(x),T^ny)$ for $n\in \mathbb Z$ , where

$$ \begin{align*} \omega(y,n):=\begin{cases} \sum_{j=0}^{n-1}y_j\quad& \text{if }n>0,\\ 0\quad& \text{if }n=0,\\ -\sum_{j=n}^{-1}y_j \quad& \text{if }n<0. \end{cases} \end{align*} $$

Clearly for the system $(A^{\mathbb Z}\times A^{\mathbb Z},S)$ , the acting group G here is the integer group $\mathbb Z$ . The Følner sequence $\{F_n\}$ is chosen naturally to be $F_n=\{0,1,\ldots ,n-1\}:=[0,n-1]$ . Define a metric $\rho $ on $A^{\mathbb Z}\times A^{\mathbb Z}$ by

$$ \begin{align*} \rho((x,y),(x',y'))=\max \{d(x,x'),d(y,y')\}, \end{align*} $$

where d is the metric on $A^{\mathbb Z}$ defined by

$$ \begin{align*} d(x,y)=2^{-n}\quad\text{where }n=\min\{|i|: x_i\neq y_i\}. \end{align*} $$

We note that the metric d here is different from (5.4).

Let $\pi : A^{\mathbb Z}\times A^{\mathbb Z}\rightarrow A^{\mathbb Z}$ be the projection to the second coordinate. Then it induces a factor map between $(A^{\mathbb Z}\times A^{\mathbb Z},S)$ and $(A^{\mathbb Z},T)$ . For any $y\in A^{\mathbb Z}$ , let $E_{y}:=A^{\mathbb Z}\times \{y\}$ denote the fiber of y under the factor map $\pi $ . We denote by

$$ \begin{align*}h_{\mathrm{top}}^P(E_{y},S)=h_{\mathrm{top}}^P(E_{y},\{F_n\})\quad\text{and}\quad h_{\mathrm{top}}^B(E_{y},S)=h_{\mathrm{top}}^B(E_{y},\{F_n\}),\end{align*} $$

the packing and Bowen entropies of $E_y$ for the $\mathbb Z$ -system $(A^{\mathbb Z}\times A^{\mathbb Z},S)$ , respectively.

For $n>0$ , denote

$$ \begin{align*} M(y,n)=\max_{0\le i\le n}\omega(y,i) \quad\text{and}\quad m(y,n)=\min_{0\le i\le n}\omega(y,i). \end{align*} $$

Proposition 5.10 For the packing and Bowen entropies of $E_y$ , we have:

1. (1) $h_{\mathrm {top}}^P(E_{y},S)=\limsup \nolimits _{n\rightarrow +\infty }(({M(y,n)-m(y,n)})/{n})h_{\mathrm {top}}(A^{\mathbb Z},T);$

2. (2) $h_{\mathrm {top}}^B(E_{y},S)=\liminf \nolimits _{n\rightarrow +\infty }(({M(y,n)-m(y,n)})/{n})h_{\mathrm {top}}(A^{\mathbb Z},T)$ .

Here $h_{\mathrm {top}}(A^{\mathbb Z},T)(=\log 2)$ is the topological entropy of the symbolic dynamical system $(A^{\mathbb Z},T)$ . Hence $E_y$ is regular if and only if $\lim \nolimits _{n\rightarrow +\infty }({M(y,n)-m(y,n)})/{n}$ exists.

Proof. See Appendix A.3.