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Expansive actions with specification on uniform spaces, topological entropy, and the Myhill property

Tullio Ceccherini-Silberstein, Michel Coornaert

Abstract

We prove that every expansive continuous action with the weak specification property of an amenable group $G$ on a compact Hausdorff space $X$ has the Myhill property, i.e., every pre-injective continuous self-mapping of $X$ commuting with the action of $G$ on $X$ is surjective. This extends a result previously obtained by Hanfeng Li in the case when $X$ is metrizable.

Expansive actions with specification on uniform spaces, topological entropy, and the Myhill property

Abstract

We prove that every expansive continuous action with the weak specification property of an amenable group on a compact Hausdorff space has the Myhill property, i.e., every pre-injective continuous self-mapping of commuting with the action of on is surjective. This extends a result previously obtained by Hanfeng Li in the case when is metrizable.

Paper Structure

This paper contains 19 sections, 39 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. General notation
  4. Subsets of $X \times X$
  5. Uniform spaces
  6. Ultrauniform spaces
  7. Actions
  8. Covers
  9. Shift spaces
  10. Amenable groups
  11. Expansiveness
  12. Topological entropy
  13. Topological entropy
  14. $(F,U)$-separated subsets, $(F,U)$-spanning subsets, and $(F,U)$-covers
  15. Uniform approaches to topological entropy
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ be a compact Hausdorff space equipped with a continuous action of an amenable group $G$. Suppose that $(X,G)$ is expansive and has the weak specification property. Then $(X,G)$ has the Myhill property. In particular, $(X,G)$ is surjunctive.

Theorems & Definitions (81)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Definition 2.3
  • Theorem 2.4: Ornstein-Weiss lemma
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 3.4
  • Lemma 3.5
  • ...and 71 more