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Noncommutative dynamical systems with two generators and their applications in analysis
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 |
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.
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