Billingsley-Type Theorem of Weighted Bowen Topological Entropy for Amenable Group Actions

Let (Xi,di) be a compact metric space with metric di, i=1,2…,k, and G be a discrete infinitely countable amenable group. This paper is based on continuous actions G↷Xi on compact metric spaces (Xi,di). Firstly, we introduce the concept of weighted Bowen balls, and then use the concept of weighted Bowen balls to introduce the corresponding lower (upper) weighted local entropy, as well as propose the concept of weighted Bowen topological entropy defined in terms of Hausdorff dimension by weighted Bowen balls, and prove Billingsley-type theorem between these two types of entropies by using the equivalent definition of weighted Bowen topological entropy.