Substreetutions and more on trees

TL;DR

This paper introduces a new substitution concept called substreetution on colored binary trees, exploring their periodicity, minimality, and growth properties, and applying these ideas to hyperbolic tilings.

Contribution

It defines substreetution on trees, analyzes conditions for minimality and periodicity, and constructs quasi-periodic hyperbolic tilings using these concepts.

Findings

A fixed point of substreetution can be almost periodic or not.

The minimal case shows exponential growth in preimages.

Examples of periodic trees without invariant measures.

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.