Cellular automata and strongly irreducible shifts of finite type

AbstractIf 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

Statistical Mechanics of Surjective Cellular Automata

Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. We establish a connection between the invariance of Gibbs measures and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the (well-known) invariance of the uniform Bernoulli measure under surjective cellular automata, which corresponds to the conservation of the trivial Hamiltonian. As an application, we obtain results indicating the lack of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic" cellular automata. We discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium, and pose several open questions. As an aside, a shift-invariant pre-image of a Gibbs measure under a pre-injective factor map between shifts of finite type turns out to be always a Gibbs measure. We provide a sufficient condition under which the image of a Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point out a potential application of pre-injective factor maps as a tool in the study of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure

Post-surjectivity and balancedness of cellular automata over groups

We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final version for Discrete Mathematics and Theoretical Computer Science. Prepared according to the editor's request

Cellular automata on regular rooted trees

We study cellular automata on regular rooted trees. This includes the characterization of sofic tree shifts in terms of unrestricted Rabin automata and the decidability of the surjectivity problem for cellular automata between sofic tree shifts

A Garden of Eden theorem for Anosov diffeomorphisms on tori

Let $f$ be an Anosov diffeomorphism of the $n$-dimensional torus ${\mathbb{T}}^n$ and $\tau$ a continuous self-mapping of ${\mathbb{T}}^n$ commuting with $f$. We prove that $\tau$ is surjective if and only if the restriction of $\tau$ to each homoclinicity class of $f$ is injective.Comment: 9 page