Propriétés d'invariance des mots sturmiens
Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 2, p. 351-369

An infinite binary word is said to be Sturmian if it is balanced and not ultimately periodic. We compute the slope and the intercept of f(x) for any Sturmian word x and any Sturmian morphism f. Using continued fraction expansions of Raney, we characterize the slopes of the words which are left invariant under a non-trivial substitution. Then we prove that the converse also holds for a particular class of sturmian words the intercept of which is an homography of the slope.

Un mot sturmien est un mot infini, binaire, équilibré et non ultimement périodique. On détermine l’évolution de la pente et de l’intercept d’un mot sturmien, sous l’action du monoïde de Sturm. À l’aide des matrices de Raney, on énonce une condition que doivent satisfaire les pentes des mots laissés fixes par une substitution non triviale. Puis on prouve que cette condition est suffisante pour un ensemble particulier de mots dont l’intercept est une homographie de la pente.

@article{JTNB_1997__9_2_351_0,
     author = {Parvaix, Bruno},
     title = {Propri\'et\'es d'invariance des mots sturmiens},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     pages = {351-369},
     zbl = {0904.11008},
     mrnumber = {1617403},
     language = {fr},
     url = {http://www.numdam.org/item/JTNB_1997__9_2_351_0}
}
Parvaix, Bruno. Propriétés d'invariance des mots sturmiens. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 351-369. http://www.numdam.org/item/JTNB_1997__9_2_351_0/

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