Topological sequence entropy of continuous maps on topological spaces

Abstract

In this paper we propose a new definition of topological sequence entropy for continuous maps on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new sequence entropy, and compare the new sequence entropy with the existing ones. The defined sequence entropy generates that of Goodman. Yet, it holds various basic properties of Goodman’s sequence entropy, e.g., the sequence entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same sequence entropy, the sequence entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new sequence entropy coincides with Goodman’s sequence entropy for compact systems.