Common dynamics of two Pisot substitutions with the same incidence matrix

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".