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Strongly isomorphic symbolic extensions for expansive topological flows

Yonatan Gutman, Ruxi Shi

Abstract

In this paper, we prove that finite-dimensional topological flows without fixed points and having a countable number of periodic orbits, have the small flow boundary property. This enables us to answer positively a question of Bowen and Walters from 1972: Any expansive topological flow has a strongly isomorphic symbolic flow extension, i.e. an extension by a suspension flow over a subshift. Previously Burguet had shown this is true if the flow is assumed to be $C^2$-smooth.

Strongly isomorphic symbolic extensions for expansive topological flows

Abstract

In this paper, we prove that finite-dimensional topological flows without fixed points and having a countable number of periodic orbits, have the small flow boundary property. This enables us to answer positively a question of Bowen and Walters from 1972: Any expansive topological flow has a strongly isomorphic symbolic flow extension, i.e. an extension by a suspension flow over a subshift. Previously Burguet had shown this is true if the flow is assumed to be -smooth.
Paper Structure (25 sections, 38 theorems, 68 equations)

This paper contains 25 sections, 38 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Discrete dynamical systems and topological flows
  5. Cross-sections
  6. Flow boundaries and interiors
  7. Good cross-sections exist
  8. Calculating with flow interiors and boundaries
  9. The small flow boundary property
  10. Suspension flows and symbolic extensions
  11. Some facts from dimension theory
  12. Establishing the small flow boundary property
  13. Notation and assumptions in force in Section 3
  14. Local homeomorphisms between cross-sections
  15. Return time sets and cross-section names
  16. ...and 10 more sections

Key Result

Theorem 2.8

(Whitney) whitney1933regular Let $(X,\Phi)$ be a topological flow without fixed points, then for each $x\in X$ there is a closed cross-section $S$ such that $x\in \text{\rm Int}^{\Phi} S$.

Theorems & Definitions (102)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • proof
  • Theorem 2.9
  • ...and 92 more