Local Entropy, Metric Entropy and Topological Entropy for Countable Discrete Amenable Group Actions

Abstract

Let [Formula: see text] be a compact metric space and [Formula: see text] a countable infinite discrete amenable group acting on [Formula: see text]. Like in the [Formula: see text]-action cases we define the notion of local entropy and by it we bound the difference between metric entropy and that of a partition, and bound the difference between topological entropy and that of a separated set, which generalize Theorems 1(1) and 1(2) in [Newhouse, 1989] from [Formula: see text]-actions to amenable group actions. We further prove that the entropy function [Formula: see text] is upper semi-continuous on [Formula: see text] for an asymptotic entropy expansive amenable group action.