A Garden of Eden theorem for principal algebraic actions

TL;DR

This paper extends the Garden of Eden theorem to principal algebraic actions of countable abelian groups, establishing conditions under which equivariant maps are surjective based on their injectivity on homoclinic classes.

Contribution

It proves a new Garden of Eden type theorem for expansive, connected algebraic actions of abelian groups, linking surjectivity to injectivity on homoclinic classes.

Findings

Surjective maps are characterized by injectivity on homoclinic classes.

The theorem generalizes classical cellular automata results to algebraic actions.

Conditions include expansiveness and connectedness of the algebraic action.

Abstract

Let $\Gamma$ be a countable abelian group and $f \in \Z[\Gamma]$, where $\Z[\Gamma]$ denotes the integral group ring of $\Gamma$. Consider the Pontryagin dual $X_f$ of the cyclic $\Z[\Gamma]$-module $\Z[\Gamma]/\Z[\Gamma] f$ and suppose that the natural action of $\Gamma$ on $X_f$ is expansive and that $X_f$ is connected. We prove that if $\tau \colon X_f \to X_f$ is a $\Gamma$-equivariant continuous map, then $\tau$ is surjective if and only if the restriction of $\tau$ to each $\Gamma$-homoclinicity class is injective. This is an analogue of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over $\Gamma$.