Homoclinically expansive actions and a Garden of Eden theorem for harmonic models

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 $f$ is weakly expansive (e.g., $f$ is invertible in $\ell^1(\Gamma)$, or, when $\Gamma$ is not virtually $\Z$ or $\Z^2$, $f$ is well-balanced) 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. We also show that this equivalence remains valid in the case when $\Gamma = \Z^d$ and $f \in \Z[\Gamma] = \Z[u_1,u_1^{-1}, \ldots, u_d, u_d^{-1}]$ is an irreducible atoral polynomial such that its zero-set $Z(f)$ is contained in the image of the intersection of $[0,1]^d$ and a finite union of hyperplanes in $\R^d$ under the quotient map $\R^d \to \T^d$ (e.g., when $d \geq 2$ such that $Z(f)$ is finite). These two results are analogues of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over $\Gamma$.