Substitutions on two letters, cutting segments and their projections

Abstract

Substitutions on two letters, cutting segments and their projections par Sierk W. ROSEMA Résumé. Dans cet article on considère la structure des projections des segments de coupure correspondant aux substitutions unimodulaires sur un alphabet binaire. On montre qu'une telle projection est un bloc de lettres si et seulement si la substitution est sturmienne. Une double application de ce procédé à une substitution de Christoffel donne la substitution originelle. On obtient ainsi une dualité sur l'ensemble des substitutions de Christoffel.

Sturmian substitutions on two letters, cutting paths and their projections

“…For more results on Christoffel substitutions see [3,4,10]. In the present paper we generalize the result from [15] to the set of all Sturmian substitutions that have an incidence matrix with determinant 1.…”

“…Hence jb − ia = 1. It is clear that j ≤ a and it follows from [15] Corollary 4.1 that a ≤ c. Since det M φ = ±1 we get a( jd − ic) = c( jb − ia) ± j = c ± j ≥ 0. If jd − ic = 0, then in w φ (1) after j jumps to the right and i to the left, we are back in our starting position.…”

“…Proof The first congruence follows from Lemma 2.3 of [15]. The proof of the second congruence follows from |u| 0 + |u| 1 = m and gcd(|u| 0 , |u| 1 ) = 1.…”

“…They found that the limit word that is generated in this way is a two-sided Fibonacci word. In [15] the author generalized this property to unimodular substitutions defined over two letters. He showed for a special class of Sturmian substitutions, which he called Christoffel substitutions, that the words v n are generated by another Sturmian substitution.…”

“…Proof Proposition 5.1 of [15] states that D u is a block of integers of length m, if and only if u is strongly balanced. Since P( p 0 ) = 0 we have 0 ∈ D u .…”