Topological Dynamics of Groupoid Actions

TL;DR

This paper extends fundamental concepts of topological dynamics to continuous groupoid actions, revealing new phenomena and distinctions from classical group actions, especially when the source map is not open.

Contribution

It introduces the extension of topological dynamics notions to groupoid actions and highlights new phenomena and distinctions from classical group actions.

Findings

Some properties diverge for non-open source maps.

Counterexamples illustrate new phenomena.

Recurrence properties are analyzed in the groupoid context.

Abstract

Some basic notions and results in Topological Dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.