Some Ergodic Properties of Invertible Cellular Automata

TL;DR

This paper investigates the ergodic properties of invertible one-dimensional linear cellular automata over finite rings, proving they are strong mixing and Bernoulli automorphisms under certain conditions.

Contribution

It establishes that specific classes of invertible linear cellular automata are strong mixing and Bernoulli automorphisms without relying on natural extensions.

Findings

Invertible linear CA are strong mixing.

Such CA are Bernoulli automorphisms.

Results apply to CA over rings of the form Z_{p^k}.

Abstract

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 $f:\mathbb{Z}^{r-l+1}_{p^k}\to \mathbb{Z}_{p^k}$, over the ring $\mathbb{Z}_{p^k}$ $(k \geq 2$ and $p$ is a prime number), where $gcd(p,\lambda_r)=1$ and $p| \lambda_i$ for all $i \neq r $ (or $gcd(p, \lambda_l)=1$ and $p|\lambda_i$ for all $i\neq l$). Under some assumptions we prove that any right (left) permutative, invertible one-dimensional linear CA $T_{f[l,r]}$ and its inverse are strong mixing. We also prove that any right(left) permutative, invertible one-dimensional linear CA is Bernoulli automorphism without making use of the natural extension previously used in the literature.