Cellular Automata on Group Sets and the Uniform Curtis-Hedlund-Lyndon Theorem

TL;DR

This paper extends the theory of cellular automata to left homogeneous spaces, establishing uniform and topological versions of the Curtis-Hedlund-Lyndon theorem with diverse geometric and algebraic examples.

Contribution

It introduces cellular automata on left homogeneous spaces and proves new variants of the Curtis-Hedlund-Lyndon theorem applicable to these spaces.

Findings

Established a uniform Curtis-Hedlund-Lyndon theorem for cellular automata on homogeneous spaces.

Provided examples including spheres, Euclidean, hyperbolic spaces, and Cayley graphs.

Extended the automata theory to new geometric and algebraic contexts.

Abstract

We introduce cellular automata whose cell spaces are left homogeneous spaces and prove a uniform as well as a topological variant of the Curtis-Hedlund-Lyndon theorem. Examples of left homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations.