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Measure rigidity for algebraic bipermutative cellular automata

Published online by Cambridge University Press:  01 December 2007

MATHIEU SABLIK
MATHIEU SABLIK*
Affiliation:
Institut de Mathématiques de Luminy, UMR 6206-Campus de Luminy, Case 907, 13288 Marseille Cedex 09, France (email: sablik@iml.univ-mrs.fr)
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Abstract

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Let $({\mathcal {A}^{\mathbb {Z}}} ,F)$ be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the $ {\mathbb {N}} \times {\mathbb {Z}} $-action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group $ {\mathcal {A}^{\mathbb {Z}}} $. This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on $ {\mathcal {A}^{\mathbb {N}}} $.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1965 - 1990
Copyright
Copyright © Cambridge University Press 2007

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