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 Folner 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.