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Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

Part of: Smooth dynamical systems: general theory Topological dynamics Discrete geometry Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  31 May 2024

SEBASTIÁN BARBIERI Open the ORCID record for SEBASTIÁN BARBIERI [Opens in a new window]  and
SÉBASTIEN LABBÉ
SEBASTIÁN BARBIERI*
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile
SÉBASTIEN LABBÉ
Affiliation:
Université de Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

Two asymptotic configurations on a full $\mathbb {Z}^d$-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

Keywords

asymptotic pairsSturmian sequencescomplexitybispecial patternsymbolic dynamicscut and project scheme

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37C29: Homoclinic and heteroclinic orbits 52C23: Quasicrystals, aperiodic tilings 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 2 , February 2025 , pp. 337 - 395
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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