Geometrical models for a class of reducible Pisot substitutions

TL;DR

This paper develops a geometric framework using Rauzy fractals to analyze the dynamics of reducible Pisot substitutions, revealing their connection to toral translations and providing new combinatorial insights.

Contribution

It introduces a novel geometric theory for reducible Pisot substitutions based on Rauzy fractals, including representations of tilings and a new combinatorial interpretation of their dynamics.

Findings

Geometric representations of stepped surfaces and polygonal tilings.

Identification of self-replicating and periodic Rauzy fractal tilings.

Symbolic systems behave as first returns of toral translations.

Abstract

We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses geometric representations of stepped surfaces and related polygonal tilings, as well as self-replicating and periodic tilings made of Rauzy fractals. We apply our theory to an infinite family of substitutions. For this family, we analyze and interpret in a new combinatorial way the codings of a domain exchange defined on the associated fractal domains. We deduce that the symbolic dynamical systems associated with this family of substitutions behave dynamically as first returns of toral translations.