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

TL;DR

This paper establishes a Garden of Eden theorem analogue for harmonic models on Abelian groups, linking surjectivity of equivariant maps to injectivity on homoclinic classes, with implications for cellular automata.

Contribution

It introduces a new Garden of Eden theorem for harmonic models with weakly expansive functions on Abelian groups, extending classical cellular automata results.

Findings

Surjectivity of maps is equivalent to injectivity on homoclinic classes for certain harmonic models.

Results apply to models with zero-sets contained in hyperplane intersections, including finite zero-sets.

Extends classical Garden of Eden theorem to harmonic models on Abelian groups.

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 $\Gamma$.