Common dynamics of two Pisot substitutions with the same incidence matrix

Abstract

International audienceThe 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"

Balanced pair algorithm for a class of cubic substitutions

“…This common dynamics is done throughout the family of the product automata of the prefix automata associated with the power of substitution σ k 1 and σ k 2 , but these common sets have zero Lebesgue measure. In [17,18], under the Pisot condition, we proved that the intersection of 2 Rauzy fractals associated with 2 unimodular irreducible Pisot substitutions having the same incidence matrix have nonzero Lebesgue measure. We showed that this intersection is substitutive.…”

“…A variant of this algorithm was used later in [17,18] to study the intersection of Rauzy fractals associated with different substitutions having the same incidence matrix. This version of the balanced pair algorithm is used in the present article; we describe it in this section.…”

Symmetric Intersections of Rauzy Fractals

“…We shall assume that the substitutions are primitive. This algorithm was introduced in [17] and [18], in the context of the study of intersection of Rauzy fractals.…”

“…We will assume the following lemma (for the proof see [18]), and we give an idea of the proof of Theorem 3.1.…”

“…The balanced pair algorithm was introduced by Livshits ( [12]) in the context of the Pisot conjecture, it was also used in [24] in the same context. A variant of this algorithm was used later by the first author in [17,18], in the study of the intersection of Rauzy fractals associated with different substitutions having the same incidence matrix. This version of the balanced pair algorithm is used in the present article, we describe it in section 2.1.…”