Topological stability of semigroup actions and shadowing

TL;DR

This paper explores the stability and shadowing properties of semigroup actions on compact spaces, characterizing when these actions are expansive and stable, and linking shadowing to finite type subshifts.

Contribution

It provides new characterizations of expansive and shadowing properties for semigroup actions, including conditions for topological stability and finite type subshifts.

Findings

All full shifts are expansive for certain semigroups.

Expansive monoid actions with shadowing are topologically stable.

Finite alphabet subshifts have shadowing iff they are of finite type.

Abstract

We investigate expansiveness, topological stability, and shadowing for continuous actions of semigroups on compact Hausdorff spaces. We characterize semigroups for which all full shifts are expansive. We show that every expansive continuous monoid action on a compact Hausdorff space which has the shadowing property is topologically stable, and that a subshift with finite alphabet over a monoid has the shadowing property if and only if it is of finite type.