Topological entropy of sets of generic points for actions of amenable groups

Abstract

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered Følner sequence {F _n } in G with ${lim _{n \to + \infty }}\frac{{\left| {{F_n}} \right|}}{{\log n}} = \infty$, we prove the following result:

$$h_{top}^B\left( {{G_\mu },\left\{ {{F_n}} \right\}} \right) = {h_\mu }\left( {X,G} \right),$$

where G _μ is the set of generic points for μ with respect to {F _n } and h_top^B(G _μ ; {F _n }) is the Bowen topological entropy (along {F _n }) on G _μ . This generalizes the classical result of Bowen (1973).