Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, p. 371-394

We study combinatoric, ergodic and arithmetic properties of the fixed point of Tribonacci substitution (first introduced by G. Rauzy) and of the related rotation of the two dimentional torus. We give a geometric generalization of the three distances theorem and an explicit formula for the recurrence function of the fixed point of the substitution. We state Diophantine approximation’s properties of the vector of the rotation of 𝕋 2 : we prove that, for a suitable norm, the sequence of best approximation of this vector is the sequence of Tribonacci numbers. We compute the ergodic invariants F and F C of the symbolic system related to the substitution.

Nous étudions certaines propriétés combinatoires, ergodiques et arithmétiques du point fixe de la substitution de Tribonacci (introduite par G. Rauzy) et de la rotation du tore 𝕋 2 qui lui est associée. Nous établissons une généralisation géométrique du théorème des trois distances et donnons une formule explicite pour la fonction de récurrence du point fixe. Nous donnons des propriétés d’approximation diophantienne du vecteur de la rotation de 𝕋 2 : nous montrons, que pour une norme adaptée, la suite de meilleure approximation de ce vecteur est la suite des nombres de Tribonacci. Nous calculons enfin les invariants ergodiques F et F C du système dynamique associé à la substitution.

@article{JTNB_2001__13_2_371_0,
     author = {Chekhova, Nataliya and Hubert, Pascal and Messaoudi, Ali},
     title = {Propri\'et\'es combinatoires, ergodiques et arithm\'etiques de la substitution de Tribonacci},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     pages = {371-394},
     zbl = {1038.37010},
     mrnumber = {1879664},
     language = {fr},
     url = {http://www.numdam.org/item/JTNB_2001__13_2_371_0}
}
Chekhova, Nataliya; Hubert, Pascal; Messaoudi, Ali. Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 371-394. http://www.numdam.org/item/JTNB_2001__13_2_371_0/

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