Combinatorial properties of infinite words associated with cut-and-project sequences
Louis-Sébastien Guimond, Zuzana Masáková, Edita Pelantová · Journal de Théorie des Nombres de Bordeaux · 2003
The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite word) in two or three letters. In fact, these sequences can be constructed also by a coding of a 2 - or 3 -interval exchange transformation. According to the complexity the cut-and-project construction includes words with complexity n + 1 , n + const. and 2 n + 1 . The words on two letter alphabet have complexity n + 1 and thus are Sturmian. The ternary words associated to the cut-and-project sequences have complexity n + const. or 2 n + 1 . A cut-and-project scheme has three parameters, two of them specifying the projection subspaces, the third one determining the cutting strip. We classify the triples that correspond to combinatorially equivalent infinite words.