Nonwandering sets and the entropy of local homeomorphisms

TL;DR

This paper studies the entropy distribution in local homeomorphisms, showing that metric entropy is concentrated in nonwandering sets and providing a framework for analyzing entropy in such systems with applications to subshifts.

Contribution

It introduces the concepts of nonwandering and wandering sets for local homeomorphisms and establishes how entropy relates to these sets, extending understanding of dynamical complexity.

Findings

Entropy is concentrated in the nonwandering set under mild conditions.

The system's entropy equals the maximum of entropies on the nonwandering and wandering sets.

Applications include examples with subshifts over countable alphabets.

Abstract

A local homeomorphism between open subsets of a locally compact Hausdorff space induces dynamical systems with a wide range of applications, including in C*-algebras. In this paper, we introduce the concepts of nonwandering and wandering sets for such systems and show that, under mild conditions, the metric entropy is concentrated in the nonwandering set. More generally, we demonstrate that the entropy of the system is the maximum of the entropies of the systems restricted to the nonwandering set and the closure of the wandering set. We illustrate these results with several examples, including applications to subshifts over countable alphabets.