Sequence entropies and mean sequence dimension for amenable group actions

Abstract

Let G be an infinite discrete countable amenable group acting continuously on a compact metrizable space X and Ω be a sequence in G. By using open covers of X, a topological invariant, the topological sequence entropy, is defined. It is shown that the topological sequence entropy can also be defined through relative spanning set or relative separate set. We prove that the topological sequence entropy equals the topological entropy multiplying a constant K(Ω) depending only on sequence Ω. The notion of the measure-theoretic sequence entropy is defined. The two sequence entropies are related by the variational principle. The mean topological sequence dimension is defined. It will be nonzero only if the topological sequence entropy is infinite.