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Contractible subshifts

Leo Poirier, Ville Salo

Abstract

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only if it is a contractible SFT with a fixed point. For virtually polycyclic groups, contractibility implies dense periodic points. We introduce a ``homotopy theory'' framework for working with this notion, and ``contractibility'' is in fact simply an analog of the usual contractibility in algebraic topology. We also explore the symbolic dynamical analogs of homotopy equivalence and equiconnectedness of subshifts. Contractibility is implied by the map extension property of Meyerovitch, and among SFTs, it implies the finite extension property of Briceño, McGoff and Pavlov. We include thorough comparisons with these classes. We also encounter some new group-geometric notions, in particular a periodic variant of Gromov's asymptotic dimension of a group.

Contractible subshifts

Abstract

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only if it is a contractible SFT with a fixed point. For virtually polycyclic groups, contractibility implies dense periodic points. We introduce a ``homotopy theory'' framework for working with this notion, and ``contractibility'' is in fact simply an analog of the usual contractibility in algebraic topology. We also explore the symbolic dynamical analogs of homotopy equivalence and equiconnectedness of subshifts. Contractibility is implied by the map extension property of Meyerovitch, and among SFTs, it implies the finite extension property of Briceño, McGoff and Pavlov. We include thorough comparisons with these classes. We also encounter some new group-geometric notions, in particular a periodic variant of Gromov's asymptotic dimension of a group.
Paper Structure (26 sections, 83 theorems, 61 equations, 1 figure)

This paper contains 26 sections, 83 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contractibility
  3. Results
  4. Preliminaries
  5. Contractibility in algebraic topology
  6. General and symbolic dynamical definitions
  7. A few words on groups
  8. Two lemmas about SFTs
  9. The natural extension of a contraction homotopy
  10. First examples of contractible SFTs
  11. Characterization of retracts of full shifts
  12. Adaptation to general contractible subshifts
  13. Contractibility and homotopy equivalence
  14. Finite periodic asymptotic dimension and periodic points
  15. Stitching morphisms together
  16. ...and 11 more sections

Key Result

Theorem 1.3

A one-dimensional subshift of finite type is topologically mixing if and only if it is contractible.

Figures (1)

  • Figure 1: Almost-union of two Euclidean balls $A, B$, shown in gray.

Theorems & Definitions (171)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 161 more