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

Let $\mathrm{\Gamma }$ be a countable Abelian group and $f\in \mathbb{Z}[\mathrm{\Gamma }]$, where $\mathbb{Z}[\mathrm{\Gamma }]$ denotes the integral group ring of $\mathrm{\Gamma }$. Consider the Pontryagin dual $X_f$ of the cyclic $\mathbb{Z}[\mathrm{\Gamma }]$-module $\Z[\Gamma]/\Z[\Gamma] f$andsupposethat$f$isweaklyexpansive(e\mathrm{.}g\mathrm{.},$f$isinvertiblein$\ell^1(\Gamma)$,or,when$\Gamma$isnotvirtually$\Z$or$\Z^2$,$f$ is well-balanced) and that $X_f$ is connected. We prove that if $\tau \colon X_f \to X_f$isa$\Gamma$-equivariantcontinuousmap,then$\tau$ is surjective if and only if the restriction of $\tau$ to each $\mathrm{\Gamma }$-homoclinicity class is injective. We also show that this equivalence remains valid in the case when $\mathrm{\Gamma }={\mathbb{Z}}^d$ and $f \in \Z[\Gamma] = \Z[u_1,u_1^{-1}, \ldots, u_d, u_d^{-1}]$isanirreducibleatoralpolynomialsuchthatitszero-set$Z(f)$ is contained in the image of the intersection of $[0,1{]}^d$ and a finite union of hyperplanes in ${\mathbb{R}}^d$ under the quotient map ${\mathbb{R}}^d\to {\text{T}}^d$ (e.g., when $d \geq 2$suchthat$Z(f)$ 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 $\mathrm{\Gamma }$.