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Substreetutions and more on trees

Part of: Topological dynamics Graph theory

Published online by Cambridge University Press:  08 April 2024

ALEXANDRE BARAVIERA Open the ORCID record for ALEXANDRE BARAVIERA [Opens in a new window]  and
RENAUD LEPLAIDEUR Open the ORCID record for RENAUD LEPLAIDEUR [Opens in a new window]
ALEXANDRE BARAVIERA
Affiliation:
Instituto de Matemática e Estatística - UFRGS, Avenida Bento Gonçalves, 9500, Porto Alegre CEP 91509-900, RS, Brazil (e-mail: [email protected])
RENAUD LEPLAIDEUR*
Affiliation:
ISEA, Université de la Nouvelle-Calédonie & LMBA CNRS UMR6205, Nouméa, New Caledonia
*
e-mail: [email protected]
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Abstract

We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the $\mathbb {F}_{2}^{+}$-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

Keywords

tree shiftentropygroup actionminimal systems

MSC classification

Primary: 05C05: Trees
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 9 , September 2024 , pp. 2399 - 2453
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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