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Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

Tullio Ceccherini-Silberstein, Michel Coornaert, Hanfeng Li

Abstract

A dynamical system is a pair $(X,G)$, where $X$ is a compact metrizable space and $G$ is a countable group acting by homeomorphisms of $X$. An endomorphism of $(X,G)$ is a continuous selfmap of $X$ which commutes with the action of $G$. One says that a dynamical system $(X,G)$ is surjunctive provided that every injective endomorphism of $(X,G)$ is surjective (and therefore is a homeomorphism). We show that when $G$ is sofic, every expansive dynamical system $(X,G)$ with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.

Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

Abstract

A dynamical system is a pair , where is a compact metrizable space and is a countable group acting by homeomorphisms of . An endomorphism of is a continuous selfmap of which commutes with the action of . One says that a dynamical system is surjunctive provided that every injective endomorphism of is surjective (and therefore is a homeomorphism). We show that when is sofic, every expansive dynamical system with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.

Paper Structure

This paper contains 17 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Actions
  5. Shifts and subshifts
  6. Dynamical systems
  7. Expansivity
  8. The strong topological Markov property
  9. The weak specification property
  10. The pseudo-orbit tracing property
  11. Algebraic dynamical systems
  12. The Hamming distance
  13. Sofic groups
  14. Sofic topological entropy
  15. Proofs
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a countable sofic group and let $(X,G)$ be an expansive dynamical system with the weak specification property and the strong topological Markov property. Assume that $h_\Sigma(X,G) \geq 0$ for some sofic approximation $\Sigma$ of $G$. Then $(X,G)$ is surjunctive.

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2: Uniform expansivity
  • proof
  • Definition 2.3
  • Definition 2.4
  • ...and 20 more