k-Mixing Properties of Multidimensional Cellular Automata

TL;DR

This paper studies the conditions under which multidimensional cellular automata exhibit the k-mixing property, providing an algorithm to verify permutivity at apex points and constructing ergodic automata.

Contribution

It introduces a criterion for k-mixing based on local rule permutivity at apex points and proposes the Mixing Algorithm for verification, with optimal conditions.

Findings

F establishes k-mixing when local rule is permutive at an apex.

The Mixing Algorithm effectively verifies permutivity at apex points.

Conditions for k-mixing are proven to be optimal.

Abstract

This paper investigates the $k$-mixing property of a multidimensional cellular automaton. Suppose $F$ is a cellular automaton with the local rule $f$ defined on a $d$-dimensional convex hull $\mathcal{C}$ which is generated by an apex set $C$. Then $F$ is $k$-mixing with respect to the uniform Bernoulli measure for all positive integer $k$ if $f$ is a permutation at some apex in $C$. An algorithm called the \emph{Mixing Algorithm} is proposed to verify if a local rule $f$ is permutive at some apex in $C$. Moreover, the proposed conditions are optimal. An application of this investigation is to construct a multidimensional ergodic linear cellular automaton.