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Measure transfer and S-adic developments for subshifts

Part of: Ergodic theory Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  11 March 2024

NICOLAS BÉDARIDE Open the ORCID record for NICOLAS BÉDARIDE [Opens in a new window] ,
ARNAUD HILION  and
MARTIN LUSTIG
NICOLAS BÉDARIDE*
Affiliation:
Aix Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille, France (e-mail: Martin.Lustig@univ-amu.fr)
ARNAUD HILION
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, UPS F-31062 Toulouse Cedex 9, France (e-mail: arnaud.hilion@math.univ-toulouse.fr)
MARTIN LUSTIG
Affiliation:
Aix Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille, France (e-mail: Martin.Lustig@univ-amu.fr)
*
e-mail: Nicolas.Bedaride@univ-amu.fr
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Abstract

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

Keywords

S-adic developmentmeasure transferergodic measuresletter frequencieseventually recognizable

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37A25: Ergodicity, mixing, rates of mixing 37E25: Maps of trees and graphs
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 35
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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