Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci

Chekhova, Nataliya; Hubert, Pascal; Messaoudi, Ali

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 371-394.

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.

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.

[1] P. Alessandri, V. Berthé, Three distances theorems and combinatorics on words. L'Enseignement Mathématique 44 (1998), 103-132. | MR | Zbl

[2] P. Arnoux, Un exemple de semi-conjugaison entre un échange d'intervalles et une rotation sur le tore. Bull. Soc. Math. France 116 (1988), 489-500. | Numdam | MR | Zbl

[3] P. Arnoux, A. Fisher, The scenery flow for geometric structures on the torus: the linear setting. Chinese Annals of Math., à paraître. | Zbl

[4] P. Arnoux, G. Rauzy, Représentation géométrique de suites de complexité 2n +1. Bull. Soc. Math. France 119 (1991), 101-117. | Numdam | MR | Zbl

[5] J. Berstel, Recent results in sturmian words. Developments in language theory (Magedburg 1995), World Scientific (96), 13-24. | MR | Zbl

[6] J. Berstel, M. Pocciola, A geometric proof of the enumeration formula for sturmian words. Internat. J. Algebra Comput. 3 (1993), 349-355. | MR | Zbl

[7] V. Berthé, Fréquences des facteurs des suites sturmiennes. Theoret. Comp. Sci. 165 (1996), 295-309. | MR | Zbl

[8] A. Bertrand-Mathis, Développement en base θ, répartition modulo un de la suite (xθn)n≽0, langages codés et θ-shift. Bull. Soc. Math. France 114 (1986), 271-323. | Numdam | Zbl

[9] J. Cassaigne, Limit values of the reccurence quotient of sturmian sequencies. Theoret. Comp. Sci. 218 (1999), 3-12. | MR | Zbl

[10] J. Cassaigne, Communication privée.

[11] J.W.S. Cassels, An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics 45, Hafner Publishing Co., New York, 1972. An introduction to diophantine approximation. Cambridge Tracs in Mathematics and Mathematical physics 45 (1972), Cambridge Univ. Press. | MR | Zbl

[12] N. Chekhova, Covering numbers of rotations. Theoret. Comput. Sci. 230 (2000), 97-116. | MR | Zbl

[13] N. Chekhova, Nombres de recouvrement. Thèse de l'Université de la Mediterranée, Aix-Marseille II, 1997.

[14] N. Chevallier, Meilleures approximations d'un élément du tore T2 et géométrie de la suite des multiples de cet élément. Acta Arithmetica 78 (1996), 19-35. | EuDML | MR | Zbl

[15] E.M. Coven, G.A. Hedlund, Sequences with minimal block growth. Math. Systems Theory 7 (1973), 138-153. | MR | Zbl

[16] F.M. Dekking, Recurrent Sets. Adv. Math. 44 (1982), 78-104. | MR | Zbl

[17] F.M. Dekking, On the Thue-Morse measure Acta Univ. Carolin. Math. Phys. 33 (1992), 35-40. | EuDML | MR | Zbl

[18] A. Del Junco, Transformations with discrete spectrum are stacking transformations. Can. J. Math. 24 (1976), 836-839. | MR | Zbl

[19] A. Del Junco, A family of counterexamples in ergodic theory. Israel J. Math. 44 (1983), 160-188. | MR | Zbl

[20] S. Dulucq, D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm. Theoret. Comp. Sci. 71 (1990), 381-400. | MR | Zbl

[21] S. Ferenczi, Systems of finite rank. Colloq. Math. 73 (1997), 35-65. | EuDML | MR | Zbl

[22] S. Ferenczi, C. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory 47 (1997), 146-161. | MR | Zbl

[23] A.S. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985), 105-114. | MR | Zbl

[24] C. Frougny, B. Solomyak, Finite Beta-expansions. Ergodic Theory Dynam. Systems 12 (1992), 713-723. | MR | Zbl

[25] W.J. Gilbert, Complex numbers with three radix expansions. Can. J. Math. 34 (1982), 1335-1348. | MR | Zbl

[26] P. Halmos, Lectures on ergodic theory. Publications of the Mathematical Society of Japan 3, The Mathematical Society of Japan, 1956. | MR | Zbl

[27] G.A. Hedlund, M. Morse, Symbolic dynamics. Amer. J. Math. 60 (1938), 815-866. | JFM | MR | Zbl

[28] G.A. Hedlund, M. Morse, Symbolic dynamics, part II: Sturmian trajectories. Amer. J. Math. 62 (1940) 1-42. | JFM | MR | Zbl

[29] S. Ito, M. Mizutani, Potato Exchange Transformations with Fractal Domains. Preprint.

[30] S. Ito, M. Kimura, On the Rauzy Fractal. Japan J. Indust. Appl. Math. 8 (1991), 461-486. | MR | Zbl

[31] D.E. Knuth, E. Donald, The art of computer Programming, Vol 2. Seminumerical algorithms, Second edition. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass., 1981. | MR | Zbl

[32] T. Komatsu, On the characteristic word of the inhomogeneous Beatty sequence. Bull. Austr. Math. Soc. 51 (1995), 337-351. | MR | Zbl

[33] T. Komatsu, A certain power series associated with a Beatty sequence. Acta Arith. 76 (1996), 109-129. | EuDML | MR | Zbl

[34] T. Komatsu, A certain power series and the inhomogeneous continued fraction expansions. J. Number Th. 59 (1996), 291-312. | MR | Zbl

[35] T. Komatsu, The fractional part of nθ + ϕ and Beatty sequences. J. Théor. Nombres Bordeaux 7 (1995), 387-406. | EuDML | Numdam | Zbl

[36] A. De Luca, F. Mignosi, Some combinatorial properties of sturmian words. Theoret. Comp. Sci. 136 (1994), 361-385. | MR | Zbl

[37] A. Messaoudi, Autour du fractal de Rauzy. Thèse de l'Université de la Mediterranée, Aix-Marseille II, 1996.

[38] A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy. J. Théor. Nombres Bordeaux 10 (1998), 135-162. | EuDML | Numdam | MR | Zbl

[39] A. Messaoudi, Frontière du fractal de Rauzy et système de numération complexe. Acta Arith. 95 (2000), 195-224. | MR | Zbl

[40] F. Mignosi On the number of factors of Sturmian words. Theoret. Comp. Sci. 82 (1991), 71-84. | MR | Zbl

[41] B. Mossé, Notions de reconnaissabilité pour les substitutions et complexité des suites automatiques. Bull. Soc. Math. France 124 (1996), 101-108. | Numdam | MR | Zbl

[42] K. Nishioka, J. Tamura, I. Shiokawa, Arithmetical properties of a certain power series. J. Number Th. 42 (1992), 61-87. | MR | Zbl

[43] B. Praggastis, Markov partitions for hyperbolic toral automorphisms. Ph. D. Thesis, Univ. of Washington, 1992.

[44] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis. Lecture Notes in Mathematics 1294, Springer-Verlag, Berlin, 1987. | MR | Zbl

[45] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147-178. | EuDML | Numdam | MR | Zbl

[46] G. Rauzy, Suites à termes dans un alphabet fini. Sém. Théor. Nombres Bordeaux (1982-1983), 1-16. | EuDML | MR | Zbl

[47] G. Rauzy, Mots infinis en arithmétique. Automata on infinite words, 165-171, Lecture Notes in Comput. Sci. 192, Springer, Berlin, 1985. | MR | Zbl

[48] P. Séébold, Fibonacci morphisms and sturmian words. Theoret. Comp. Sci. 88 (1991), 367-384. | MR | Zbl

[49] V. Sirvent, Properties of geometrical realisations of substitutions associated to a family of Pisot numbers. Ph. D. Thesis, Warwick, 1993.

[50] V. Sirvent, Relationships between the dynamical systems associated to the Rauzy substitutions. Theoret. Comp. Sci. 164 (1996), 41-57. | MR | Zbl

[51] V. Sirvent, On some dynamical subsets of the Rauzy fractal. Theoret. Comp. Sci. 180 (1997), 363-370. | MR | Zbl

[52] V.T. Sós, On the distribution mod 1 of the sequence nα. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 1 (1958), 127-134. | Zbl

[53] E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liège 41 (1972), 179-182. | MR | Zbl