Sequence entropy and mean sequence dimension for non-compact metric spaces

Abstract

In this paper, we introduce an extension of topological sequence entropy and d-sequence entropy for a dynamical system on a non-compact metric space. Then we give some elementary properties of those sequence entropy. In particular, we establish a variational principle which states that for a class of sequences, the supremum of the measure-theoretic sequence entropies and the minimum of the d-sequence entropies always coincide, and they are not less than the topological sequence entropy. Finally, we introduce the concepts of mean sequence dimension and maximal pattern mean topological dimension and give some properties of them.