Extensibility and denseness of periodic semigroup actions

TL;DR

This paper investigates the properties of periodic points and invariant measures in continuous semigroup actions, establishing connections between measure extensibility and denseness, with applications to shift actions on the full shift.

Contribution

It introduces new notions of periodicity in topological and measure-theoretical contexts and links measure extensibility to the denseness of finitely supported invariant measures for certain semigroups.

Findings

Finitely supported invariant measures are dense for residually finite amenable semigroups.

A relationship between measure extensibility and denseness is established for embeddable semigroups.

The results apply to shift actions on the full shift for specific classes of semigroups.

Abstract

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.