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Invariant measures for substitutions on countable alphabets

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  11 December 2023

WEBERTY DOMINGOS ,
SÉBASTIEN FERENCZI ,
ALI MESSAOUDI  and
GLAUCO VALLE
WEBERTY DOMINGOS
Affiliation:
Departamento de Matemática, Universidade Estadual Paulista, São José do Rio Preto, SP, Brazil (e-mail: domingos.silva@unesp.br)
SÉBASTIEN FERENCZI
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M - UMR 7373, 13453 Marseille, France (e-mail: ferenczi@iml.univ-mrs.fr)
ALI MESSAOUDI*
Affiliation:
Departamento de Matemática, Universidade Estadual Paulista, São José do Rio Preto, SP, Brazil
GLAUCO VALLE
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil (e-mail: glauco.valle@im.ufrj.br)
*
e-mail: ali.messaoudi@unesp.br
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Abstract

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

Keywords

substitutions on infinite alphabetunique ergodicitycountable non-negative matricesshift dynamical systems

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 37A25: Ergodicity, mixing, rates of mixing 37B10: Symbolic dynamics 37A50: Relations with probability theory and stochastic processes
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 32
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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