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

TL;DR

This paper investigates K{ u}rka's dichotomy in cellular automata on groups, establishing its validity on virtually-Z groups and its failure on free groups with multiple generators, with implications for subgroup structures.

Contribution

It proves the validity of K{ u}rka's dichotomy on virtually-Z groups and demonstrates its failure on free groups with two or more generators, also analyzing subgroup inheritance.

Findings

K{ u}rka's dichotomy holds on virtually-Z groups.

The dichotomy does not hold on free groups with ≥2 generators.

If the dichotomy holds on a group, it holds on all its subgroups.

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.