Cellular automata, percolation and dynamical dichotomies

TL;DR

This paper links percolation properties of Cayley graphs to the dynamical behavior of cellular automata on groups, revealing that certain dichotomies hold precisely when the group has specific algebraic and percolation characteristics.

Contribution

It establishes a novel connection between percolation thresholds and dynamical dichotomies of cellular automata on finitely generated groups, characterizing groups where Gilman's dichotomy applies.

Findings

Gilman's dichotomy holds iff the group has a trivial percolation threshold.

A countable group satisfies Gilman's dichotomy iff it is locally virtually cyclic.

The work connects percolation theory with dynamical properties of cellular automata.

Abstract

We establish a connection between percolation on the Cayley graphs of a group and the dynamical diversity of cellular automata on that group. Specifically, we demonstrate that Gilman's dichotomy between equicontinuity and sensitivity with respect to Bernoulli measures holds on a finitely generated group if and only if the group has a trivial percolation threshold. Consequently, we show that a countable group satisfies Gilman's dichotomy if and only if it is locally virtually cyclic.