Substreetutions and more on trees

A. Baraviera; R. Leplaideur

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arXiv · 2112.05242

Substreetutions and more on trees

A. Baraviera·R. Leplaideur

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Abstract

We define a notion of substitution on colored binary trees that we call substreetution. We show that a fixed point by a substreetution may be (or not) almost periodic, thus the closure of the orbit under $\mathbb{F}_2^+$-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 measure on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

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A. Baraviera, R. Leplaideur. 2021-12-09. Substreetutions and more on trees. https://arxiv.org/abs/2112.05242

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