On finite monoids of cellular automata.

For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AGτ:AG→AG defined via a finite neighbourhood S⊆GS⊆G (called a memory set of ττ) and a local function μ:AS→Aμ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A)CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A)ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A)CA(G;A). In particular, we prove that CA(G,A)CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A)V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩CA(G;A)=⟨ICA(G;A)∪V⟩

Further results on generalized cellular automata

Given a finite set $A$ and a group homomorphism $\phi : H \to G$, a $\phi$-cellular automaton is a function $\mathcal{T} : A^G \to A^H$ that is continuous with respect to the prodiscrete topologies and $\phi$-equivariant in the sense that $h \cdot \mathcal{T}(x) = \mathcal{T}( \phi(h) \cdot x)$, for all $x \in A^G, h \in H$, where $\cdot$ denotes the shift actions of $G$ and $H$ on $A^G$ and $A^H$, respectively. When $G=H$ and $\phi = \text{id}$, the definition of $\text{id}$-cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of $\phi$-cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a $\phi$-cellular automaton $\mathcal{T} : A^G \to A^H$ has the unique homomorphism property (UHP) if $\mathcal{T}$ is not $\psi$-equivariant for any group homomorphism $\psi : H \to G$, $\psi \neq \phi$. We show that if the difference set $\Delta(\phi, \psi)$ is infinite, then $\mathcal{T}$ is not $\psi$-equivariant; it follows that when $G$ is torsion-free abelian, every non-constant $\mathcal{T}$ has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study $\phi$-cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.Comment: 15 page

On Finite Monoids of Cellular Automata

For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AGτ:AG→AG defined via a finite neighbourhood S⊆GS⊆G (called a memory set of ττ) and a local function μ:AS→Aμ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A)CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A)ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A)CA(G;A). In particular, we prove that CA(G,A)CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A)V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩CA(G;A)=⟨ICA(G;A)∪V⟩

Tropical curves in sandpile models

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to each of its neighbors. In the study of pattern formation in sandpiles, it was experimentally observed by S. Caracciolo, G. Paoletti, and A. Sportiello that the result of the relaxation of a small perturbation of the maximal stable state on $\Omega_h$ contains a clearly visible thin balanced graph in its deviation set. Such graphs are known as tropical curves. In this paper we rigorously formulate (taking a scaling limit for ${ h}\to 0$) this fact and prove it. We rely on the theory of tropical analytic series, which describes the global features of the sandpile dynamic, and on the theory of smoothings of discrete superharmonic functions, which handles local questions.Comment: the text is divided into three parts, became much shorte

On the ergodic theory of cellular automata and two-dimensional Markov shifts generated by them

In this thesis we study measurable and topological dynamics of certain classes of cellular automata and multi-dimensional subshifts. In Chapter 1 we consider one-dimensional cellular automata, i.e. the maps T: PZ -> PZ (P is a finite set with more than one element) which are given by (Tx)i==F(xi+1, ..., xi+r), x=(xi)iEZ E PZ for some integers 1≤r and a mapping F: Pr-1+1 -> P. We prove that if F is right- (left-) permutative (in Hedlund's terminology) and 0≤1<r (resp. 1<r≤0), then the natural extension of the dynamical system (PZ,B,μ,T) is a Bernoulli automorphism (μ stands for the (1/p, ..., 1/p )-Bernoulli measure on the full shift PZ). If r0 and T is surjective, then the natural extension of the system (PZ, B, μ, T) is a Kautomorphism. We also prove that the shift Z2-action on a two-dimensional subshift of finite type canonically associated with the cellular automaton T is mixing, if F is both right and left permutative. Some more results about ergodic properties of surjective cellular automata are obtained Let X be a closed translationally invariant subset of the d-dimensional full shift PZd, where P is a finite set, and suppose that the Zd-action on X by translations has positive topological entropy. Let G be a finitely generated group of polynomial growth. In Chapter 2 we prove that if growth(G)<d, then any G-action on X by homeomorphisms commuting with translations is not expansive. On the other hand, if growth(G) = d, then any G-action on X by homeomorphisms commuting with translations has positive topological entropy. Analogous results hold for semigroups. For a finite abelian group G define the two-dimensional Markov shift XG ={xEGZ2 : x(i,j) + x(i+1,j) + x(i,j+1) = 0 for all (i, j) E Z2 }. Let μG be the Haar measure on the subgroup XG C GZ2. The group Z2 acts on the measure space (XG, μG) by shifts. In Chapter 3 we prove that if G1 and G2 are p-groups and E(G1)≠E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on (XG1, μG1) and (XG2, μG2) are not measure-theoretically isomorphic. We also prove that the shift actions on XG1 and XG2 are topologically conjugate if and only if G1 and G2 are isomorphic. In Chapter 4 we consider the closed shift-invariant subgroups X = = ⊥c (Zp)Z2 defined by the principal ideals c Zp [u±1, v±t] ≃ ((Zp)Z2)^ with f(u, v) = cf(0,0) + cf(1,0)u + cf(0,1)v, cf(i, j) E Zp\{0}, on which Z2 acts by shifts. We give the complete topological classification of these subshifts with respect to measurable isomorphism