Post-surjectivity and balancedness of cellular automata over groups

TL;DR

This paper explores properties of cellular automata over groups, establishing conditions under which they are reversible and balanced, with implications for their measure-preserving behavior and dual concepts like pre-injectivity and post-surjectivity.

Contribution

It proves that pre-injective, post-surjective cellular automata are reversible and that on sofic groups, post-surjectivity alone implies reversibility, advancing understanding of automata over groups.

Findings

Pre-injective, post-surjective automata are reversible.

Post-surjectivity over sofic groups implies reversibility.

Reversible automata preserve the uniform measure.

Abstract

We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space.