Continuous orbit equivalence for automorphismhi systems of equivalence relations

TL;DR

This paper introduces new concepts of continuous orbit equivalence for automorphism systems of étale equivalence relations, characterizes them via groupoids and $C^*$-algebras, and studies rigidity of expansive automorphism actions.

Contribution

It defines and characterizes continuous orbit equivalence notions for automorphism systems, linking them to groupoid and $C^*$-algebra structures, and explores topological rigidity in this context.

Findings

Characterization of continuous orbit equivalence via groupoids and $C^*$-algebras

Introduction of strong and weak continuous orbit equivalence notions

Analysis of topological rigidity for expansive automorphism actions

Abstract

We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of \'{e}tale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid $C^*$-algebras and the associated $C^*$-automorphism systems of group actions or coactions on them. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups