The Moore and the Myhill Property For Strongly Irreducible Subshifts Of Finite Type Over Group Sets

TL;DR

This paper proves the Garden of Eden theorem for strongly irreducible subshifts over certain group sets, establishing a key equivalence between surjectivity and pre-injectivity of cellular automata.

Contribution

It extends the Moore and Myhill properties to strongly irreducible subshifts over right amenable, finitely right generated homogeneous spaces with finite stabilisers.

Findings

Proves the Garden of Eden theorem in this new setting.

Shows the equivalence of surjectivity and pre-injectivity for cellular automata.

Extends classical results to a broader class of group actions.

Abstract

We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular automaton with finite set of states and finite neighbourhood over such a subshift is surjective if and only if it is pre-injective. This statement is known as Garden of Eden theorem. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical.