Some Ergodic Properties of Invertible Cellular Automata

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 $T_{f[l,r]}:{\mathbb{Z}}_{p^k}^{\mathbb{Z}}\to {\mathbb{Z}}_{p^k}^{\mathbb{Z}}$ which are given by $T_{f[l,r]}(x)=(y_n{)}_{n=-\infty }^{\infty }$, $y_n=f(x_{n+l},...,x_{n+r})=\stackrel{r}{\sum _{i=l}}{\lambda }_i{x}_{n+i}(\text{mod}{p}^k)$, $x=(x_n{)}_{n=-\infty }^{\infty }\in {\mathbb{Z}}_{p^k}^{\mathbb{Z}}$ and $f:{\mathbb{Z}}_{p^k}^{r-l+1}\to {\mathbb{Z}}_{p^k}$, over the ring ${\mathbb{Z}}_{p^k}$ $(k\ge 2$ and $p$ is a prime number), where $gcd(p,{\lambda }_r)=1$ and $p|{\lambda }_i$ for all $i\ne r$ (or $gcd(p,{\lambda }_l)=1$ and $p|{\lambda }_i$ for all $i\ne l$). 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.

9 pages