Surjunctivity for cellular automata in Besicovitch spaces

Surjunctivity and reversibility of cellular automata over concrete categories

Following ideas developed by Misha Gromov, we investigate surjunctivity and reversibility properties of cellular automata defined over certain concrete categories.

关于非均匀元胞自动机的满射性与伊甸园定理

On the surjunctivity and the Garden of Eden theorem for non-uniform cellular automata

本文研究非均匀元胞自动机（NUCA）的满射性与伊甸园定理，证明均匀回归或一维回归分布下定理成立，非回归分布存在反例，且渐近回归分布定义满射NUCA。

关于非均匀细胞自动机的戈茨克兰猜想

On Gottschalk's surjunctivity conjecture for non-uniform cellular automata

本文研究了非均匀细胞自动机在surjunctive群宇宙中的surjunctivity性质，证明了可逆的非均匀细胞自动机必须可逆，并扩展了Capobianco等人关于细胞自动机的对偶surjunctivity定理。

Geometry and Dynamics of the Besicovitch and Weyl Spaces

We study the geometric properties of Cantor subshifts in the Besicovitch space, proving that sofic shifts occupy exactly the homotopy classes of simplicial complexes. In addition, we study canonical projections into subshifts, characterize the cellular automata that are contracting or isometric in the Besicovitch or Weyl spaces, study continuous functions that locally look like cellular automata, and present a new proof for the nonexistence of transitive cellular automata in the Besicovitch space.

Cellular automata and substitutions in topological spaces defined via edit distances

The Besicovitch pseudo-metric is a shift-invariant pseudo-metric on the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. They correspond to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudo-metric, the Feldman pseudo-…

On symbolic group varieties and dual surjunctivity

Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of rational points of $X$. We study algebraic group cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M$ is a finite memory. When $G$ is sofic and $K$ is uncountable, we show that if $τ$ is post-surjective then it is weakly pre-injective. Our result extends the dual version of Gottschalk's Conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When $G$ is amenable, we prove that …