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On K{\r u}rka's dichotomy for cellular automata on groups

Jade Angela Hope Audouard, Guillaume Theyssier

Abstract

This paper is about topological dynamics of cellular automata on finitely generated groups. We tackle the problem of determining for which group sensitivity to initial conditions is equivalent to the absence of equicontinuity points (so-called K{\r u}rka's dichotomy). We show that the dichotomy holds on any virtually-Z group but not on free groups with 2 or more generators. We also show that if it holds on some group, it must hold on any of its subgroups.

On K{\r u}rka's dichotomy for cellular automata on groups

Abstract

This paper is about topological dynamics of cellular automata on finitely generated groups. We tackle the problem of determining for which group sensitivity to initial conditions is equivalent to the absence of equicontinuity points (so-called K{\r u}rka's dichotomy). We show that the dichotomy holds on any virtually-Z group but not on free groups with 2 or more generators. We also show that if it holds on some group, it must hold on any of its subgroups.
Paper Structure (5 sections, 17 theorems, 44 equations)

This paper contains 5 sections, 17 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Formal definitions and basic facts
  3. Kůrka's dichotomy on virtually $\mathbb{Z}$ groups
  4. Lifting a counter-example from a subgroup
  5. Free group with 2 or more generators

Key Result

Lemma 1

Let $E_1,E_2$ be two finite generating sets of the same group $G$. Denote by $d^1$ and $d^2$ the associated Cantor distances over $A^G$. Furthermore, let $B^i(x,r)$ denote the ball of center $x\in A^G$ and radius $r$ for distance $d^i$ (for $i=1,2$). It holds:

Theorems & Definitions (35)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Definition 1: Blocking word
  • Proposition 2
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • ...and 25 more