On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau \colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $\tau_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology

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

On algebraic cellular automata

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective