# American Institute of Mathematical Sciences

November  2003, 9(6): 1423-1446. doi: 10.3934/dcds.2003.9.1423

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

 1 Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 5 Boulevard Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex, France 2 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071 UCHILE-CNRS, Universidad de Chile, Casilla 170/3 correo 3, Santiago, Chile 3 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071 UCHILE–CNRS, Universidad de Chile, Casilla 170–3, correo 3, Santiago, Chile

Received  May 2002 Revised  May 2003 Published  September 2003

In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin.
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.
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.
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.
Citation: Bernard Host, Alejandro Maass, Servet Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1423-1446. doi: 10.3934/dcds.2003.9.1423
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