Garden of Eden configurations for cellular automata on Cayley graphs of groups
SIAM Journal on Discrete Mathematics
An introduction to symbolic dynamics and coding
An introduction to symbolic dynamics and coding
Induced Subshifts and Cellular Automata
Language and Automata Theory and Applications
Information and Computation
On Pattern Density and Sliding Block Code Behavior for the Besicovitch and Weyl Pseudo-distances
SOFSEM '10 Proceedings of the 36th Conference on Current Trends in Theory and Practice of Computer Science
Generalized Besicovitch and Weyl spaces: Topology, patterns, and sliding block codes
Theoretical Computer Science
Cellular automata on regular rooted trees
CIAA'12 Proceedings of the 17th international conference on Implementation and Application of Automata
Cellular automata between sofic tree shifts
Theoretical Computer Science
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If A is a finite alphabet and Γ is a finitely generated amenable group, Ceccherini-Silberstein, Machì 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.