The Garden of Eden Theorem for Cellular Automata on Group Sets

TL;DR

This paper extends the Garden of Eden theorem to cellular automata on group sets, establishing a fundamental equivalence between surjectivity and pre-injectivity for automata with finite states and neighborhoods on specific homogeneous spaces.

Contribution

It generalizes the Garden of Eden theorem to cellular automata on right amenable left homogeneous spaces with finite stabilizers, broadening its applicability.

Findings

Proves the equivalence of surjectivity and pre-injectivity for the specified automata.

Extends classical results to automata on group sets with finite neighborhoods.

Provides a theoretical foundation for cellular automata on complex geometric structures.

Abstract

We prove the Garden of Eden theorem for cellular automata with finite set of states and finite neighbourhood on right amenable left homogeneous spaces with finite stabilisers. It states that the global transition function of such an automaton is surjective if and only if it is pre-injective. 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.