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

摘要

Let X be a compact metric space and G a countable infinite discrete amenable group acting on X. Like in the Z-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 Z-actions to amenable group actions. We further prove that the entropy function h(mu)(G) is upper semi-continuous on M(X, G) for an asymptotic entropy expansive amenable group action.