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Module shifts and measure rigidity in linear cellular automata

Published online by Cambridge University Press:  16 September 2008

MARCUS PIVATO
MARCUS PIVATO*
Affiliation:
Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, Ontario, Canada K9J 7B8 (email: [email protected])
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Abstract

Suppose ℛ is a finite commutative ring of prime characteristic, 𝒜 is a finite ℛ-module, 𝕄:=ℤD×ℕE, and Φ is an ℛ-linear cellular automaton on 𝒜𝕄. If μ is a Φ-invariant measure which is multiply σ-mixing in a certain way, then we show that μ must be the Haar measure on a coset of some submodule shift of 𝒜𝕄. Under certain conditions, this means that μ must be the uniform Bernoulli measure on 𝒜𝕄.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1945 - 1958
Copyright
Copyright © 2008 Cambridge University Press

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