Conjugacy of local homeomorphisms via groupoids and C*-algebras

TL;DR

This paper characterizes topological conjugacy of local homeomorphism systems using groupoid and C*-algebra isomorphisms, extending previous results to a broader class of dynamical systems.

Contribution

It provides a new, general framework for understanding conjugacy in local homeomorphism systems via groupoid and C*-algebra isomorphisms, broadening prior work.

Findings

Topological conjugacy characterized by groupoid isomorphisms

C*-algebra isomorphisms correspond to system conjugacy

Framework applies to boundary-path spaces of graphs

Abstract

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.