Abstract
Let α be in the two-dimensional torus T 2 = R 2/Z 2. Assume that the translation map T : x → x + α acts ergodically. We present a symbolic coding of the map T which shares several properties with the Sturmian coding of a one-dimensional translation. The symbolic dynamical system is metrically isomorphic to the geometric dynamical system (T 2, T). The coding is of quadratic growth complexity and 2-balanced. Moreover, there is a geometric underpinning, the coding is related to a fundamental domain for the action of Z 2 on R 2 and also to bounded remainder sets.
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Arnoux P., Ito S.: Pisot substitutions and Rauzy fractals. (Marne-la-Valle, 2000). Journes Montoises d’Informatique Thorique. Bull. Belg. Math. Soc. Simon Stevin 8(2), 181–207 (2001)
Arnoux P., Rauzy G.: Représentation géométrique des suites de complexité 2n + 1. Bull. Soc. Math. France 119, 199–215 (1991)
Berthé V., Ferenczi S., Zamboni Q.: Interactions between Dynamics, Arithmetics and Combinatorics: the Good. The Bad, The Ugly Contemp. Math. 385, 333–364 (2005)
Chechkhova N., Hubert P., Messaoudi A.: Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Théor. Nombres Bordeaux 13, 371–394 (2001)
Chevallier N.: Best simultaneous diophantine approximations of some cubic algebraic numbers. J. Théor. Nombres Bordeaux 14, 403–414 (2002)
Denker M., Grillenberger C., Sigmund K.: Ergodic Theory on Compacts Spaces. LNM 527. Springer, Berlin (1976)
Ferenczi S.: Bounded remainder sets. Acta Arith. 61(4), 319–326 (1992)
Hecke E.: Analytische Funktionen und die Verteilung von Zahlen mod. eins, Abh Math. Semin, pp. 54–76. Hamburg University, Hamburg (1922)
Kesten, H.: On a conjecture of Erdös and Szüsz related to uniform distribution mod1, Acta Arith. bf 12, 193–212 (1966/1967)
Lagarias, J.C.: Some new results in simultaneous diophantine approximation. In: Ribenboim, P. (ed.) Proc. of the Queen’s Number Theory Conference 1979. Queen’s Paper in Pure and Applied Math. No. 54, pp. 453–474 (1980)
Liardet P.: Regularities of distribution. Compositio Math. 61(3), 267–293 (1987)
Pytheas fog N.: Substitution in Dynamics, Arithmetics and Combinatorics, LMN 1794. Springer, Berlin (2002)
Rauzy G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110, 147–178 (1982)
Rauzy, G.: Ensembles à restes bornés, Seminar on number theory, 1983–1984 (Talence, 1983/1984), Exp. No. 24, 12 pp., University of Bordeaux I, Talence (1984)
Steineder C., Winkler R.: Complexity of Hartman sequences. J. Théor. Nombres Bordeaux 17(1), 347–357 (2005)
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Communicated by K. Schmidt.
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Chevallier, N. Coding of a translation of the two-dimensional torus. Monatsh Math 157, 101–130 (2009). https://doi.org/10.1007/s00605-008-0074-y
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DOI: https://doi.org/10.1007/s00605-008-0074-y