Common dynamics of two Pisot substitutions with the same incidence matrix

TL;DR

This paper investigates the shared dynamical features of two Pisot irreducible substitutions with identical incidence matrices, revealing how common points in their Rauzy fractals can be generated via a substitution on balanced blocks.

Contribution

It demonstrates that common points of the associated Rauzy fractals can be generated through a substitution on balanced blocks when 0 lies in the interior of one fractal.

Findings

Common points can be generated with a substitution on balanced blocks

0 being an interior point of the Rauzy fractal is crucial

Shared dynamics depend on the incidence matrix and fractal properties

Abstract

The matrix of a substitution is not sufficient to completely determine the dynamics associated, even in simplest cases since there are many words with the same abelianization. In this paper we study the common points of the canonical broken lines associated to two different Pisot irreducible substitutions $\sigma_1$ and $\sigma_2$ having the same incidence matrix. We prove that if 0 is inner point to the Rauzy fractal associated to $\sigma_1$ these common points can be generated with a substitution on an alphabet of so-called "balanced blocks".