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.