Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules

Abstract

In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin.<br> First we show a representation result for classes of permutative cellular automata: those with associative type local rule are the product of a group cellular automaton with a translation map, and if they satisfy a scaling condition, they are the product of an affine cellular automaton (the alphabet is an Abelian group) with a translation map.<br> For cellular automata of this type with an Abelian factor group, and starting from a translation invariant probability measure with complete connections and summable decay, it is shown that the Cesàro mean of the iteration of this measure by the cellular automaton converges to the product of the uniform Bernoulli measure with a shift invariant measure.<br> Finally, the following characterization is shown for affine cellular automaton whose alphabet is a group of prime order: the uniform Bernoulli measure is the unique invariant probability measure which has positive entropy for the automaton, and is either ergodic for the shift or ergodic for the $\mathbb Z^2$-action induced by the shift and the automaton, together with a condition on the rational eigenvalues of the automaton.<br>