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arXiv · 1706.06548

A Garden of Eden theorem for principal algebraic actions

Tullio Ceccherini-Silberstein·Michel Coornaert
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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 $\Gamma$.

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Tullio Ceccherini-Silberstein, Michel Coornaert. 2017-06-20. A Garden of Eden theorem for principal algebraic actions. https://arxiv.org/abs/1706.06548

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