Related papers: A converse to Moore's theorem on cellular automata

We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: "A group $G$ is amenable if and only if every cellular automaton with carrier $G$ that has…

We prove the Garden of Eden theorem for cellular automata with finite set of states and finite neighbourhood on right amenable left homogeneous spaces with finite stabilisers. It states that the global transition function of such an…

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability…

We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the…

We prove that on B-free subshifts, with B satisfying the Erd\"os condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.

Let $G$ be an amenable group and let $X$ be an irreducible complete algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $X$ and let $\tau \colon A^G \to A^G$ be an algebraic cellular automaton…

Moore characterized the amenability of automorphism groups of countable ultrahomogeneous structures by a Ramsey-type property. We extend this result to automorphism groups of metric Fra\"iss\'e structures, which encompass all Polish groups.…

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map T from the underlying space to itself, T is pre-injective if and only if…

We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition…

Let $G$ be an amenable group and let $V$ be a finite-dimensional vector space over an arbitrary field $\K$. We prove that if $X \subset V^G$ is a strongly irreducible linear subshift of finite type and $\tau \colon X \to X$ is a linear…

Let G be a countable group. We proof that there is a model companion for the approximate theory of a Hilbert space with a group G of automorphisms. We show that G is amenable if and only if the structure induced by countable copies of the…

We prove a version of ergodic theorem for an action of an amenable group, where a F{\o} lner sequence needs not to be tempered. Instead, it is assumed that a function satisfies certain mixing condition.

We extend the concept of amenability of a Banach algebra $A$ to the case that there is an extra $\mathfrak A$-module structure on $A$, and show that when $S$ is an inverse semigroup with subsemigroup $E$ of idempotents, then $A=\ell^1(S)$…

The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation…

We extend F{\o}lner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by…

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],\lambda)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely…

Let $\Gamma$ be a countable abelian group and $f \in \Z[\Gamma]$, where $\Z[\Gamma]$ denotes the integral group ring of $\Gamma$. Consider the Pontryagin dual $X_f$ of the cyclic $\Z[\Gamma]$-module $\Z[\Gamma]/\Z[\Gamma] f$ and suppose…

Group cellular automata are continuous, shift-commuting endomorphisms of $G^\mathbb{Z}$, where $G$ is a finite group. We provide an easy-to-check characterization of expansivity for group cellular automata on abelian groups and we prove…