Limit Measures for Affine Cellular Automata, II

Cellular automata are a famous model of computation, yet it is still a challenging task to assess the computational capacity of a given automaton; especially when it comes to showing negative results. In this paper, we focus on studying this problem via the notion of CA relative simulation. We say that automaton A is simulated by B if each space-time diagram of A can be, after suitable transformations, reproduced by B. We study affine automata - i.e., automata whose local r...

For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.

For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of g...

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, p...

Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local function $\mu :A^S \to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $\tau : A^G \to A^H$, where $H$ is another arbitrary group, via a group homomorphism $\phi : H \to G$. Our definition preserves the essence of CA, as ...

For a general attractive Probabilistic Cellular Automata on S Z d , we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the inuence from the boundary for the invariant measures of the system restricted to nite boxes. For a class of reversible PCA dynamics on {--1, +1} Z d , with a naturally associated Gibbsian poten...

If L=Z^D and A is a finite set, then A^L is a compact space. A cellular automaton (CA) is a continuous transformation F:A^L--> A^L that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the prop...

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 $\text{id}$...

We consider one-dimensional cellular automata $F_{p,q}$ which multiply numbers by $p/q$ in base $pq$ for relatively prime integers $p$ and $q$. By studying the structure of traces with respect to $F_{p,q}$ we show that for $p\geq 2q-1$ (and then as a simple corollary for $p>q>1$) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence $\xi(p/q)^n$, ($n=0,1,2,\dots$) for some $\xi>0$. To the other direction, by studying the mea...

The $\mu$-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to in- finity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as $\mu$-limit set where $\mu$ is the uniform Bernoulli measure.