Homoclinically Expansive Actions and a Garden of Eden Theorem for Harmonic Models.

Let Γ be a countable Abelian group and f ∈ Z [ Γ ] , where Z [ Γ ] denotes the integral group ring of Γ . Consider the Pontryagin dual Xf of the cyclic Z [ Γ ] -module Z [ Γ ] / Z [ Γ ] f and suppose that f is weakly expansive (e.g., f is invertible in ℓ 1 (Γ) , or, when Γ is not virtually Z or Z 2 , f is well-balanced) and that Xf is connected. We prove that if τ : X f → X f is a Γ -equivariant continuous map, then τ is surjective if and only if the restriction of τ to each Γ -homoclinicity class is injective. We also show that this equivalence remains valid in the case when Γ = Z d and f ∈ Z [ Γ ] = Z [ u 1 , u 1 - 1 , ... , u d , u d - 1 ] is an irreducible atoral polynomial whose zero-set Z(f) satisfies some suitable finiteness conditions (e.g., when d ≥ 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 Γ . [ABSTRACT FROM AUTHOR]