Limit Measures for Affine Cellular Automata, II

Consider a finite Abelian group (G,+), with |G|=p^r, p a prime number, and F: G^N -> G^N the cellular automaton given by {F(x)}_n= A x_n + B x_{n+1} for any n in N, where A and B are integers relatively primes to p. We prove that if P is a translation invariant probability measure on G^Z determining a chain with complete connections and summable decay of correlations, then for any w= (w_i:i<0) the Cesaro mean distribution of the time iterates of the automaton with initial dis...

Let $(\az,F)$ be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the $\N\times\Z$-action of $F$ and the shift map $\s$ to be the Haar measure on $\gs$, a closed shift-invariant subgroup of the Abelian compact group $\az$. This generalizes simultaneously results of B. Host, A. Maass and S. Mart\'{\i}nez \cite{Host-Maass-Martinez-2003} and M. Pivato \cite{Pivato-2003}. This result is applied to give co...

We revisit the problem of finding the conditions under which synchronous probabilistic cellular automata indexed by the line $\mathbb{Z}$, or the periodic line $\cyl{n}$, depending on 2 neighbours, admit as invariant distribution the law of a space-indexed Markov chain. Our advances concerns PCA defined on a finite alphabet, where most of existing results concern size 2 alphabet. A part of the paper is also devoted to the comparison of different structures ($\mathbb{Z}$, $\cy...

Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.

Consider the cellular automata (CA) of $\mathbb{Z}^{2}$-action $\Phi$ on the space of all doubly infinite sequences with values in a finite set $\mathbb{Z}_{r}$, $r \geq 2$ determined by cellular automata $T_{F[-k, k]}$ with an additive automaton rule $F(x_{n-k},...,x_{n+k})=\sum\limits_{i=-k}^{k}a_{i}x_{n+i}(mod r)$. It is investigated the concept of the measure theoretic directional entropy per unit of length in the direction $\omega_{0}$. It is shown that $h_{\mu}(T_{F[-k,...

In this paper, we investigate some ergodic properties of $Z^{2}$-actions $T_{p,n}$ generated by an additive cellular automata and shift acting on the space of all doubly -infinitive sequences taking values in $Z_{m}$.

In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality $p^k$, i.e. the maps $T_{f[l,r]}:\mathbb{Z}^{\mathbb{Z}}_{p^k}\to\mathbb{Z}^{\mathbb{Z}}_{p^k}$ which are given by $T_{f[l,r]}(x) = (y_n)_{n=-\infty}^{\infty} $, $y_{n} = f(x_{n+l}, ..., x_{n+r}) =\overset{r}{\underset{i=l}{\sum}}\lambda _{i}x_{n+i}(\text{mod} p^k)$, $x=(x_n)_{n=-\infty}^{\infty}\in \mathbb{Z}^{\mathbb{Z}}_{p^k}$ and $f:\mat...

This paper has been withdrawn by the authors, due an error involving the weak* convergence argument in section 2

We add small random perturbations to a cellular automaton and consider the one-parameter family $(F_\epsilon)_{\epsilon>0}$ parameterized by $\epsilon$ where $\epsilon>0$ is the level of noise. The objective of the article is to study the set of limiting invariant distributions as $\epsilon$ tends to zero denoted $\mathcal{M}_0^l$. Some topological obstructions appear, $\mathcal{M}_0^l$ is compact and connected, as well as combinatorial obstructions as the set of cellular aut...

We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass. It is a refinement of the classical notion of omega-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterisation of the persistent lang...