Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

TL;DR

This paper extends the concept of topological entropy to dynamical systems generated by local homeomorphisms, providing new definitions, properties, and computations for infinite graph systems.

Contribution

It introduces a generalized entropy framework for Deaconu-Renault systems, compares different entropy definitions, and computes entropy for infinite graphs and ultragraphs.

Findings

Entropy via separated sets decreases under factor maps.

Entropy via covers provides a lower bound to entropy via separated sets.

Computed entropy for infinite graphs and ultragraphs, comparing with Gurevich's entropy.

Abstract

In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu-Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew's definition of entropy via covers and Bowen's definition of entropy via separated sets. We propose a definition of factor map between Deaconu-Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.