Topological substitutions and Rauzy fractals

TL;DR

This paper explores the connection between fractal-based tilings and topological substitutions, demonstrating that a specific topological substitution can generate the same Rauzy fractal tilings as those from iterated function systems.

Contribution

It establishes a direct link between fractal tilings and topological substitutions by explicitly constructing a substitution that reproduces the Tribonacci Rauzy fractal tilings.

Findings

The topological substitution generates the same tilings as the Rauzy fractal.

A new explicit topological substitution is defined for the Tribonacci case.

The link bridges combinatorial and geometric approaches to tilings.

Abstract

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.