Operations on Christoffel words. (Opérations sur les mots de Christoffel.) (French) Zbl 1066.11502

The slope \(\rho(f)\) of a finite word on the alphabet \(\{0,1\}\) is defined as the ratio of the number \(f_1\) of 1’s and the number \(f_0\) of 0’s. The definition is naturally extended to the case of infinite words provided the limit of the slopes of finite prefixes exists.
A Lyndon word \(f\in\{0,1\}^*\) is larger than any one of its suffixes (lexicographical order induced by \(0<1\)). The author defines Christoffel words recursively. Let \(f\) and \(g\) be two words; the determinant is \(\det(f,g)=f_0g_1-f_1g_0 (f_0=\text{the number of 0's in} f, \text{etc.})\). 0 and 1 are Christoffel words. If \(f\) and \(g\) are Christoffel words such that \(\det(f,g)=1\), then the concatenated word \(fg\) is a Christoffel word. A Christoffel word is a Lyndon word, but the converse is false.
The object of the article under review is to study the map \(f\mapsto\rho(f)\) which associates the slope \(\rho(f)\) and its continued fraction expansion to the Christoffel word \(f\). The author discusses the operation \(\oplus\) which is defined by \(\rho(f\oplus g)=\rho(f)+\rho(g)\). The operation is made consistent, and given new insight on the rather intricate algorithms which add continued fractions. The author also investigates the product of a continued fraction by an integer.
The article under review, even though fascinating, is somewhat difficult to read since the notations and concept are not always defined: one is often asked to refer to the author’s thesis [”Addition et multiplication par un entier des mots de Christoffel”, Thèse, Limoges, 1995; per bibl.].


11B85 Automata sequences
68R15 Combinatorics on words
Full Text: DOI Numdam EuDML EMIS


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