Continuous orbit equivalence of semigroup actions

TL;DR

This paper introduces and characterizes continuous orbit equivalence for semigroup actions on compact spaces, extending concepts from group actions and analyzing their relations via groupoids.

Contribution

It defines continuous orbit equivalence for semigroup actions, characterizes it using semi-groupoids, and relates it to group actions in the case of homeomorphisms.

Findings

Characterization of continuous orbit equivalence via semi-groupoids

Extension of orbit equivalence concepts from groups to semigroups

Relation between semigroup and group actions in orbit equivalence

Abstract

In this paper, we consider semigroup actions of discrete countable semigroups on compact spaces by surjective local homeomorphisms. We introduce notions of continuous one-sided orbit equivalence and continuous orbit equivalence for semigroup actions, and characterize them in terms of the corresponding semi-groupoids and transformation groupoids respectively. Finally, we consider the case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of group actions.