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Cesàro mean distribution of group automata starting from measures with summable decay

Published online by Cambridge University Press:  01 December 2000

PABLO A. FERRARI
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05315-970 São Paulo, Brasil (e-mail: pablo@ime.usp.br)
ALEJANDRO MAASS
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Casilla 170-3, Correo 3, Santiago, Chile (e-mail: {amaass, smartine}@dim.uchile.cl)
SERVET MARTÍNEZ
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Casilla 170-3, Correo 3, Santiago, Chile (e-mail: {amaass, smartine}@dim.uchile.cl)
PETER NEY
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA (e-mail: ney@math.wisc.edu)

Abstract

Consider a finite Abelian group $(G,+)$, with $|G|=p^r$, $p$ a prime number, and $\varphi: G^\mathbb{N} \to G^\mathbb{N}$ the cellular automaton given by $(\varphi x)_n=\mu x_n+\nu x_{n+1}$ for any $n\in \mathbb{N}$, where $\mu$ and $\nu$ are integers coprime to $p$. We prove that if $\mathbb{P}$ is a translation invariant probability measure on $G^\mathbb{Z}$ determining a chain with complete connections and summable decay of correlations, then for any ${\underline w}= (w_i:i<0)$the Cesàro mean distribution $${\cal M}_{\mathbb{P}_{\underline w}} =\lim_{M\to\infty} \frac{1}{M} \sum^{M-1}_{m=0}\mathbb{P}_{\underline w}\circ\varphi^{-m},$$ where $\mathbb{P}_{\underline w}$ is the measure induced by $\mathbb{P}$ on $G^\mathbb{N}$ conditioned by $\underline w$, exists and satisfies ${\cal M}_{\mathbb{P}_{\underline w}}=\lambda^\mathbb{N}$, the uniform product measure on $G^\mathbb{N}$. The proof uses a regeneration representation of $\mathbb{P}$.

Type
Research Article
Copyright
© 2000 Cambridge University Press

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