Cellular automata and strongly irreducible shifts of finite type

Abstract

If A is a finite alphabet and Γ is a finitely generated amenable group, Ceccherini-Silberstein, Machı̀ and Scarabotti have proved that a local transition function defined on the full shift AΓ is surjective if and only if it is pre-injective; this equivalence is the so-called Garden of Eden theorem. On the other hand, when Γ is the group of the integers, the theorem holds in the case of irreducible shifts of finite type as a consequence of a theorem of Lind and Marcus but it no longer holds in the two-dimensional case.Recently, Gromov has proved a GOE-like theorem in the much more general framework of the spaces of bounded propagation. In this paper we apply Gromov's theorem to our class of spaces proving that all the properties required in the hypotheses of this theorem are satisfied.We give a definition of strong irreducibility that, together with the finite-type condition, it allows us to prove the GOE theorem for the strongly irreducible shifts of finite type in AΓ (provided that Γ is amenable). Finally, we prove that the bounded propagation property for a shift is strictly stronger than the union of strong irreducibility and finite-type condition.