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Combinatorial properties of infinite words associated with cut-and-project sequences. (English) Zbl 1076.68055

Authors’ abstract: 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 always has 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 also be constructed by a coding of a 2- or 3-interval exchange transformation. According to the complexity, the cut-and-project construction includes words with complexity \(n+1, n+\text{const.}\) and \(2n+1\). The words on a two-letter alphabet have complexity \(n+1\) and thus are Sturmian. The ternary words associated to the cut-and-project sequences have complexity \(n+\text{const.}\) or \(2n+1\). 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.

MSC:

68R15 Combinatorics on words
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