Substreetutions and more on trees

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...

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Veröffentlicht in:Ergodic theory and dynamical systems 2024-09, Vol.44 (9), p.2399-2453
Hauptverfasser: BARAVIERA, ALEXANDRE, LEPLAIDEUR, RENAUD
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2023.108