Topological Pressure of Generic Points for Amenable Group Actions

Abstract

Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure $\mu$ and any continuous map $\varphi$,

$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$

where $\{F_n\}$ is a Følner sequence, $X_{\mu }$ is the set of generic points of $\mu$ with respect to (w.r.t. for short) $\{F_n\}$, $P(X_{\mu },\varphi ,\{F_n\})$ is the topological pressure of $X_{\mu }$ for $\varphi$ w.r.t. $\{F_n\}$ and $h_{\mu }(X)$ is the measure-theoretic entropy.

Appendices

Appendix A: Proof of Proposition 2.5

Before proceeding with the proof, we first need to prove a lemma.

Lemma 5.1

Let $\delta >0$, $A,B,S,E \subset F(G)$ and $P=(S\cup \{e\})(S\cup \{e\})^{-1}$,

• (1) If $PA \cap B =\emptyset$, then $PB\cap A=\emptyset$.

• (2) If E is $(P,\delta )$-invariant and set $\tilde{E}=\{c \in E: Pc \subset E\}$, then $|E {\setminus } \tilde{E}|<\delta |E|$.

Proof

(1). Assume that $PB \cap A \ne \emptyset$, then there exist $a \in A, b\in B,s_1,s_2 \in S\cup \{e\}$ with $s_1(s_2)^{-1}b=a$, thus $s_2(s_1)^{-1}a=b$ which implies $PA \cap B \ne \emptyset$, a contradiction.

(2). Note that $e \in P$, then $E {\setminus } \tilde{E} \subset \partial _{P}E$ and hence $|E {\setminus } \tilde{E}|\le |\partial _{P}E| <\delta |E|$. $\square$

Proof of Proposition 2.5

The proof is similar to Proposition 2.1 in [13]. Let g be a mistake-density function and define $H: (0,1) \rightarrow F(G)$ as follows

$$\begin{aligned} H(\varepsilon )=(K\cup \{e\})(K\cup \{e\})^{-1}, \end{aligned}$$

where $K \subset F(G)$ as in Definition 2.4 w.r.t. $\varepsilon$.

Let $\Delta _j^{'}=2^{-j}, j\in \mathbb {N}$, and we define two maps $M:(0,1) \rightarrow F(G)$ and $\delta :(0,1) \rightarrow (0,1)$ as follows:

$$\begin{aligned} M(\varepsilon )=H(\Delta _j^{'}), \delta (\varepsilon )=g(\Delta _j^{'})\,\, \text {when}\,\, 2\Delta _j^{'} \le \varepsilon <2\Delta _{j-1}^{'}. \end{aligned}$$

Then we define the map $m:(0,1) \rightarrow F(G)\times (0,1)$ as $m(\varepsilon )=(M(\varepsilon ),\delta (\varepsilon ))$, next we will show this map is we need as in Definition 2.2.

For any $\varepsilon _1,\varepsilon _2,\dots ,\varepsilon _p>0$ and any $x_1,x_2,\dots ,x_p \in X$, it is sufficient to prove for $\varepsilon _i$ of the form $2\Delta _{j_i}^{'}, i=1,2,\dots ,k$. We rewrite $\{\varepsilon _1,\varepsilon _2,\dots ,\varepsilon _p\}$ and $\{x_1,x_2,\dots ,x_p\}$ as $\{\varepsilon _1^{'}>\varepsilon _2^{'}>\dots >\varepsilon _k^{'}\}$ and $\{y_{i,j}:1 \le j \le l_i, 1 \le i \le k\}$, where $\varepsilon _i^{'}=2\Delta _{t_i}^{'}, i=1,2,\dots , k$. We just need to show that when $F_{i,j}\in F(G)$ is $m(\varepsilon _i^{'})$-invariant and $\{F_{i,j}: 1 \le j \le l_i, 1\le i \le k\}$ are pairwise disjoint, then

$$\begin{aligned} \bigcap _{i=1}^{k}\bigcap _{j=1}^{l_i}B(g;F_{i,j},y_{i,j},\varepsilon _i^{'})\ne \emptyset . \end{aligned}$$

(*)

The $(*)$ can be proved in k-steps.

Step 1 Note that $m(\varepsilon _1^{'})=(M(\varepsilon _1^{'}),\delta (\varepsilon _1^{'})) =(H(\varepsilon _{1}^{'}/2),g(\varepsilon _1^{'}/2))$. For $1 \le j \le l_1$, $F_{1,j}$ is $m(\varepsilon _1^{'})$-invariant. We set $\tilde{F}_{1,j}:=\{c\in F_{1,j}: H(\varepsilon _1^{'}/2)c \subset F_{1,j}\}$, then

$$\begin{aligned} H(\varepsilon _1^{'}/2)\tilde{F}_{1,i} \cap \tilde{F}_{1,j} =\emptyset , 1 \le i \ne j \le l_1. \end{aligned}$$

By weak specification property, there exists $z_1 \in X$ such that

$$\begin{aligned} d(hz_1,hy_{1,j}) \le \frac{\varepsilon _1^{'}}{2}, \ \ \forall h \in \tilde{F}_{1,j}, \ \ j=1,2,\dots l_1, \end{aligned}$$

and by Lemma 5.1 (2),

$$\begin{aligned} |F_{1,j} {\setminus } \tilde{F}_{1,j}| <g(\varepsilon _1^{'}/2) |F_{1,j}|, 1 \le j \le l_1, \end{aligned}$$

which implies

$$\begin{aligned} z_1 \in \bigcap _{j=1}^{l_1}B(g;F_{1,j},y_{1,j},\varepsilon _1^{'}/2)\subset \bigcap _{j=1}^{l_1}B(g;F_{1,j},y_{1,j},\varepsilon _1^{'}). \end{aligned}$$

Step 2 Note that $m(\varepsilon _2^{'})=(M(\varepsilon _2^{'}),\delta (\varepsilon _2^{'})) =(H(\varepsilon _{2}^{'}/2),g(\varepsilon _2^{'}/2))$. For $1 \le j \le l_2$, $F_{2,j}$ is $m(\varepsilon _2^{'})$-invariant. We set $\tilde{F}_{2,j}:=\{c\in F_{2,j}: H(\varepsilon _2^{'}/2)c \subset F_{2,j}\}$, then

$$\begin{aligned} H(\varepsilon _2^{'}/2)\tilde{F}_{2,i} \cap \tilde{F}_{2,j} =\emptyset , 1 \le i \ne j \le l_2, \end{aligned}$$

and

$$\begin{aligned} H(\varepsilon _2^{'}/2)\tilde{F}_{2,i} \cap \bigcup _{j=1}^{l_1}\tilde{F}_{1,j} =\emptyset , 1 \le i \le l_2, \end{aligned}$$

by Lemma 5.1 (1) we also have

$$\begin{aligned} H(\varepsilon _2^{'}/2)\left( \bigcup _{j=1}^{l_1}\tilde{F}_{1,j}\right) \cap \tilde{F}_{2,i} =\emptyset , 1 \le i \le l_2. \end{aligned}$$

Then by weak specification property, there exists $z_2 \in X$ such that

$$\begin{aligned} d(hz_2,hy_{2,j}) \le \frac{\varepsilon _2^{'}}{2}, \ \ \forall h \in \tilde{F}_{2,j}, \ \ j=1,2,\dots l_2, \end{aligned}$$

and

$$\begin{aligned} d(hz_2,hz_1) \le \frac{\varepsilon _2^{'}}{2}, \ \ \forall h \in \bigcup _{j=1}^{l_1}\tilde{F}_{1,j}. \end{aligned}$$

Hence

$$\begin{aligned} d(hz_2,hy_{1,j}) \le d(hz_2,hz_1)+d(hz_1,hy_{1,j}) \le \frac{\varepsilon _1^{'}}{2}+\frac{\varepsilon _2^{'}}{2} <\varepsilon _1^{'}, \ \ \forall h \in \tilde{F}_{1,j}, 1 \le j \le l_1. \end{aligned}$$

By Lemma 5.1 (2),

$$\begin{aligned} |F_{2,j} {\setminus } \tilde{F}_{2,j}| <g(\varepsilon _2^{'}/2) |F_{2,j}|, 1 \le j \le l_2, \end{aligned}$$

hence

$$\begin{aligned} z_2 \in \left( \bigcap _{j=1}^{l_1}B\left( g;F_{1,j},y_{1,j}, \frac{\varepsilon _1^{'}}{2}+\frac{\varepsilon _2^{'}}{2}\right) \right) \cap \left( \bigcap _{j=1}^{l_2}B\left( g;F_{2,j},y_{2,j}, \frac{\varepsilon _2^{'}}{2}\right) \right) \end{aligned}$$

Continuing in this way, after $k-1$ steps, we have $\tilde{F}_{i,j} \subset F_{i,j}$ with

$$\begin{aligned} |F_{i,j} {\setminus } \tilde{F}_{i,j}|<g(\varepsilon _i^{'}/2)|F_{i,j}| \le g(\varepsilon _i^{'})|F_{i,j}|, 1 \le j \le l_i, 1 \le i \le k-1 \end{aligned}$$

and $z_{k-1} \in X$ such that

$$\begin{aligned} z_{k-1} \in \bigcap _{i=1}^{k-1}\bigcap _{j=1}^{l_i}B\left( g;F_{i,j},y_{i,j}, \sum _{j=i}^{k-1}\frac{\varepsilon _j^{'}}{2}\right) . \end{aligned}$$

Step k Note that $m(\varepsilon _k^{'})=(M(\varepsilon _k^{'}),\delta (\varepsilon _k^{'}))=(H(\varepsilon _{k}^{'}/2),g(\varepsilon _k^{'}/2))$. For $1 \le j \le l_k$, $F_{k,j}$ is $m(\varepsilon _k^{'})$-invariant. We set $\tilde{F}_{k,j}:=\{c\in F_{k,j}: H(\varepsilon _k^{'}/2)c \subset F_{k,j}\}$, then

$$\begin{aligned} H(\varepsilon _k^{'}/2)\tilde{F}_{k,i} \cap \tilde{F}_{k,j} =\emptyset , 1 \le i \ne j \le l_k, \end{aligned}$$

and

$$\begin{aligned} H(\varepsilon _k^{'}/2)\tilde{F}_{k,i} \cap \left( \bigcup _{t=1}^{k-1}\bigcup _{j=1}^{l_t}\tilde{F}_{t,j}\right) =\emptyset , 1 \le i \le l_k, \end{aligned}$$

by Lemma 5.1 (1) we also have

$$\begin{aligned} H(\varepsilon _k^{'}/2)\left( \bigcup _{t=1}^{k-1}\bigcup _{j=1}^{l_t}\tilde{F}_{t,j}\right) \cap \tilde{F}_{k,i} =\emptyset , 1 \le i \le l_k. \end{aligned}$$

Then by weak specification property, there exists $z_k \in X$ such that

$$\begin{aligned} d(hz_k,hy_{k,j}) \le \frac{\varepsilon _k^{'}}{2}, \ \ \forall h \in \tilde{F}_{k,j}, \ \ j=1,2,\dots l_k, \end{aligned}$$

and

$$\begin{aligned} d(hz_k,hz_{k-1}) \le \frac{\varepsilon _k^{'}}{2}, \ \ \forall h \in \bigcup _{t=1}^{k-1}\bigcup _{j=1}^{l_t}\tilde{F}_{t,j} . \end{aligned}$$

Hence $\forall h \in \tilde{F}_{i,j}, 1 \le j \le l_i, 1\le i \le k-1$,

$$\begin{aligned} d(hz_k,hy_{i,j}) \le d(hz_k,hz_{k-1})+d(hz_{k-1},hy_{i,j}) \le \frac{\varepsilon _k^{'}}{2}+\sum _{t=i}^{k-1}\frac{\varepsilon _t^{'}}{2}=\sum _{t=i}^{k}\frac{\varepsilon _t^{'}}{2}<\varepsilon _i^{'}. \end{aligned}$$

By Lemma 5.1 (2),

$$\begin{aligned} |F_{k,j} {\setminus } \tilde{F}_{k,j}| <g(\varepsilon _k^{'}/2) |F_{k,j}| \le g(\varepsilon _k^{'}) |F_{k,j}|, 1 \le j \le l_k, \end{aligned}$$

hence

$$\begin{aligned} z_k \in \bigcap _{i=1}^{k}\bigcap _{j=1}^{l_i}B(g;F_{i,j},y_{i,j}, \varepsilon _i^{'}), \end{aligned}$$

and the proposition is proved. $\square$

Appendix B: Proof of Lemma 2.7

Here is the amenable version of Proposition 2.1 in [12],

Lemma 6.1

Let (X, G) be a t.d.s., $\{K_n\}$ be a Følner sequence and $\mu \in E(X,G)$. Let $h^* < h_{\mu }(X)$. Then there exists $\delta ^*>0$ and $\varepsilon ^*>0$ such that for each neighbourhood $F \subset M(X)$ of $\mu$, there exists $n_F^* \in \mathbb {N}$ such that for any $n \ge n_F^*$, there exists $\Gamma _n \subset X_{K_n,F}$ which is $(K_{n},\delta ^*, \varepsilon ^*)$-separated and satisfies

$$\begin{aligned} |\Gamma _n|\ge e^{|K_n|h^*}. \end{aligned}$$

Proof

The proof is the basically the same as Proposition 2.1 in [12]. The only difference is that in the proof of Proposition 2.1 in [12], for the case of zero entropy, the authors applied the pointwise ergodic theorem to ensure that $X_{K_n,F} \ne \emptyset$. Here we can use the mean ergodic theorem (see Theorem 3.23 in [8] for example) to show that $X_{K_n,F} \ne \emptyset$. So we do not need to assume the Følner sequence to be tempered. $\square$

With the above Lemma and Lemma 4.2, we have the following lemma.

Lemma 6.2

Let (X, G) be a t.d.s., $\{K_n\}$ be a Følner sequence and $\mu \in E(X,G)$, then

$$\begin{aligned} h_{\mu }(X)=h_{\mu }(X,\{K_n\})=\lim _{\varepsilon \rightarrow 0}\lim _{\delta \rightarrow 0}\underline{s}(\mu ; \delta , \varepsilon ,\{K_n\}). \end{aligned}$$

Proof

The lemma is proved by Lemmas 6.1 and 4.2. $\square$

Now we can prove Lemma 2.7.

Proof of Lemma 2.7

Let $\mu =\int _{E(X,G)}\tau d \pi (\tau )$ be the ergodic decomposition of $\mu$, where $\pi$ is the corresponding Borel probability measure on E(X, G). Then by the Monotone-convergence theorem and Lemma 6.2 we have

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\lim _{\delta \rightarrow 0}\int _{E(X,G)} \underline{s}(\tau ;\delta ,\varepsilon ,\{K_n\})d\pi (\tau )\\&\quad =\int _{E(X,G)} \lim _{\varepsilon \rightarrow 0}\lim _{\delta \rightarrow 0} \underline{s}(\tau ;\delta ,\varepsilon ,\{K_n\})d\pi (\tau )\\&\quad =\int _{E(X,G)} h_{\tau }(X)d\pi (\tau )=h_{\mu }(X). \end{aligned}$$

Then there exist $\delta ^*>0,\varepsilon ^*>0$ such that

$$\begin{aligned} \int _{E(X,G)} \underline{s}(\tau ; \delta ^*,\varepsilon ^*)d\pi (\tau )>h_{\mu }(X)-\frac{\eta }{3}, \end{aligned}$$

where $\eta :=h_{\mu }(X)-h>0$.

Let $\{A_1,A_2,\dots ,A_p\}$ be a partition of M(X, G) with diameter less than t / 2 and $a_i=\pi (A_i)$, choose $\mu _i \in A_i\cap E(X,G)$ such that

$$\begin{aligned} \underline{s}(\mu _i;\delta ^*,\varepsilon ^*)>\sup _{\tau \in A_i \cap E(X,G)}\underline{s}(\mu _i;\delta ^*,\varepsilon ^*)-\frac{\eta }{3}. \end{aligned}$$

Then we have

$$\begin{aligned} D\left( \mu ,\sum _{i=1}^{p}a_i\mu _i\right) <t/2 \,\,\text {and }\,\, \sum _{i=1}^{p}a_i\underline{s}(\mu _i;\delta ^*,\varepsilon ^*)>h_{\mu }(X)-\frac{2}{3}\eta . \end{aligned}$$

Note that $D(\mu ,\sum _{i=1}^{p}a_i\mu _i)$ and $\sum _{i=1}^{p}a_i\underline{s}(\mu _i;\delta ^*,\varepsilon ^*)$ vary continuously on $\{(a_1,a_2,\dots ,a_p){:}\sum _{i=1}^{p}a_i=1\}$, the lemma is proved. $\square$