Strongly Mixing Properties of Multidimensional Cellular Automata

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 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 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.