## Construction de mots de Christoffel. (Construction of Christoffel words).(French)Zbl 0742.11013

The authors first define a (finite) Christoffel word: this is a Lyndon word lexicographically maximal with a given slope. They generalize this notion to infinite Christoffel words, and prove that the action of $$PGL_ 2(\mathbb{Z})$$ on $$\mathbb{P}^ 1$$ can be identified with the action of certain substitutions on a subset of the set of infinite Christoffel words. This is related to Beatty sequences and Sturmian words, and some applications are announced for a future note.

### MSC:

 11B85 Automata sequences 68R15 Combinatorics on words 68Q45 Formal languages and automata 11J70 Continued fractions and generalizations 03D05 Automata and formal grammars in connection with logical questions