Random product of substitutions with the same incidence matrix

TL;DR

This paper studies the geometric and dynamical properties of Rauzy fractals associated with sequences of substitutions sharing the same Pisot matrix, establishing their continuous dependence and potential as models for the underlying symbolic systems.

Contribution

It introduces a method to associate a Rauzy fractal to any sequence of substitutions with the same Pisot matrix and proves its continuous dependence on the sequence.

Findings

Rauzy fractals depend continuously on substitution sequences

The construction provides a geometric model for the symbolic system in some cases

The approach links symbolic dynamics with geometric fractal analysis

Abstract

Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the stepped line associated with an element of the symbolic system on the contracting space of the matrix. We show that this Rauzy fractal depends continuously on the sequence of substitutions, and investigate some of its properties; in some cases, this construction gives a geometric model for the symbolic dynamical system.