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Topological stability of semigroup actions and shadowing

Tullio Ceccherini-Silberstein, Michel Coornaert, Xuan Kien Phung

TL;DR

The paper advances the theory of dynamical stability for semigroup and monoid actions on compact spaces by proving that expansivity plus the pseudo-orbit tracing property implies topological (semi)stability, extending Walters’ classical results beyond group actions. It provides a dynamical characterization of subshifts of finite type over monoids, showing that such subshifts are exactly those with the POTP and hence admit topological stability. It also establishes that equicontinuous, finitely generated actions on Stone spaces possess POTP, while exploring rooted-tree boundary actions to highlight limitations and non-stability phenomena in widely studied groups. These results synthesize expansivity, shadowing, and stability in the broader setting of semigroup dynamics, with implications for symbolic dynamics over non-group alphabets and prodiscrete spaces.

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.

Topological stability of semigroup actions and shadowing

TL;DR

The paper advances the theory of dynamical stability for semigroup and monoid actions on compact spaces by proving that expansivity plus the pseudo-orbit tracing property implies topological (semi)stability, extending Walters’ classical results beyond group actions. It provides a dynamical characterization of subshifts of finite type over monoids, showing that such subshifts are exactly those with the POTP and hence admit topological stability. It also establishes that equicontinuous, finitely generated actions on Stone spaces possess POTP, while exploring rooted-tree boundary actions to highlight limitations and non-stability phenomena in widely studied groups. These results synthesize expansivity, shadowing, and stability in the broader setting of semigroup dynamics, with implications for symbolic dynamics over non-group alphabets and prodiscrete spaces.

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.

Paper Structure

This paper contains 15 sections, 22 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Semigroup actions
  5. Prodiscrete spaces
  6. Stone spaces
  7. The uniform structure of a compact Hausdorff space
  8. Expansivity
  9. The pseudo-orbit tracing property
  10. Topological stability
  11. Dynamical characterization of subshifts of finite type
  12. Shifts and subshifts
  13. Subshifts of finite type
  14. Equicontinuous actions and the pseudo-orbit tracing property
  15. Rooted tree endomorphism semigroups

Key Result

Theorem 1.1

Let $X$ be a compact Hausdorff space equipped with a continuous action of a monoid $M$. Suppose that the dynamical system $(X,M)$ is expansive and has the pseudo-orbit tracing property. Then $(X,M)$ is topologically stable.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Expansivity
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7: Uniform expansivity
  • ...and 58 more