Structure of three interval exchange transformations I: an arithmetic study

Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q.

doi:10.5802/aif.1839

Structure of three interval exchange transformations I: an arithmetic study
[Structure de trois transformations d'échanges d'intervalles I : une étude arithmétique]

Ferenczi, Sébastien ¹ ; Holton, Charles ² ; Zamboni, Luca Q.³

¹ Laboratoire de Mathématiques et Physique Théorique, CNRS, UPRES-A 6083, Parc de Grandmont, 37200 Tours (France)
² University of California, Department of Mathematics, Berkeley CA 94720-3840 (USA)
³ University of North Texas, Department of Mathematics, Denton TX 76203-5116 (USA)

Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 861-901.

Résumé
Abstract

Dans cet article nous décrivons une généralisation à la dimension $2$ de l’ algorithme d’Euclide, qui provient de la dynamique des échanges de $3$ intervalles. Nous examinons diverses propriétés diophantiennes de cet algorithme, en particulier la qualité de l’approximation simultanée. Nous montrons qu’il vérifie un théorème de type Lagrange : l’algorithme est finalement périodique si et seulement si les paramètres sont dans la même extension quadratique de $ℚ$ .

In this paper we describe a $2$ -dimensional generalization of the Euclidean algorithm which stems from the dynamics of $3$ -interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of $ℚ .$

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.1839

Classification : 11J70, 11J13, 37A05
Keywords: Generalized continued fraction, interval exchange transformations
Mot clés : fractions continues généralisées, échanges d'intervalles

Affiliations des auteurs :

Ferenczi, Sébastien ¹ ; Holton, Charles ² ; Zamboni, Luca Q. ³

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     title = {Structure of three interval exchange transformations {I:} an arithmetic study},
     journal = {Annales de l'Institut Fourier},
     pages = {861--901},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
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     mrnumber = {1849209},
     zbl = {1029.11036},
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Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three interval exchange transformations I: an arithmetic study. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 861-901. doi : 10.5802/aif.1839. http://www.numdam.org/articles/10.5802/aif.1839/

Bibliographie
Cité par

[1] J. Aaronson; M. Keane The visits to zero of some deterministic random walks, Proc. London Math. Soc., Volume 44 (1982) no. 3, pp. 535-553 | DOI | MR | Zbl

[2] V.I. Arnold A-graded algebras and continued fractions, Comm. Pure Applied Math., Volume XLII (1989), pp. 993-1000 | DOI | MR | Zbl

[3] P. Arnoux Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore (in French), Bull. Soc. Math. France, Volume 116 (1988) no. 4, pp. 489-500 | Numdam | MR | Zbl

[4] P. Arnoux; V. Berthe; S. Ito Discrete planes, $ℤ^{2}$ -actions, Jacobi-Perron algorithm and substitutions (1999) (Preprint)

[5] P. Arnoux; G. Rauzy Représentation géométrique de suites de complexité $2 n + 1$ , Bull. Soc. Math. France, Volume 119 (1991) no. 2, pp. 199-215 | Numdam | MR | Zbl

[6] L. Bernstein The Jacobi-Perron algorithm; its theory and applications, Lecture Notes in Mathematics, no 207, Springer-Verlag, 1971 | MR | Zbl

[7] V. Berthe; L. Vuillon Tilings and rotations: a two-dimensional generalization of Sturmian sequences, Discrete Math., Volume 223 (2000), pp. 27-53 | DOI | MR | Zbl

[8] M. Boshernitzan; C. Carroll An extension of Lagrange's theorem to interval exchange transformations over quadratic fields, J. Anal. Math., Volume 72 (1997), pp. 21-44 | DOI | MR | Zbl

[9] A.J. Brentjes Multi-dimensional continued fraction algorithms, Math. Centre Tracts, Amsterdam, Volume 145 (1981) | MR | Zbl

[10] E. Burger On simultaneous diophantine approximation in the vector space $ℚ + ℚ α$ , J. Number Theory, Volume 82 (2000), pp. 12-24 | DOI | MR | Zbl

[11] E. Burger On real quadratic number fields and simultaneous diophantine approximation, Monats. Math., Volume 128 (1999), pp. 201-209 | DOI | MR | Zbl

[12] J. Cassaigne; S. Ferenczi; L.Q. Zamboni Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier, Volume 50 (2000) no. 4, pp. 1265-1276 | DOI | Numdam | MR | Zbl

[13] N. Chekhova; P. Hubert; A. Messaoudi Propriétés combinatoires, ergodiques et arithmétiques de la substitution de tribonacci, J. Théorie des Nombres de Bordeaux (2001) | Numdam | MR | Zbl

[14] E.M. Coven; G.A. Hedlund Sequences with minimal block growth, Math. Systems Theory, Volume 7 (1972) no. 2, pp. 138-153 | DOI | MR | Zbl

[15] A. del Junco A family of counterexamples in ergodic theory, Israël J. Math., Volume 44 (1983) no. 2, pp. 160-188 | DOI | MR | Zbl

[16] S. Ferenczi; C. Holton; L.Q. Zamboni Structure of three-interval exchange transformations II: a combinatorial description of the trajectories (2001) (Preprint, 32pp.) | MR | Zbl

[17] S. Ferenczi; C. Holton; L.Q. Zamboni Structure of three-interval exchange transformations III: ergodic and spectral properties (2001) (Preprint, 29 pp.) | MR | Zbl

[18] T. Garrity On periodic sequences for algebraic numbers (1999) (, http://front.math.ucdavis.edu/math.NT/9906016) | Zbl

[19] Y. Hara-Mimachi; S. Ito A characterization of real quadratic numbers by diophantine algorithms, Tokyo J. Math., Volume 14 (1991) no. 2, pp. 251-267 | DOI | MR | Zbl

[20] G.H. Hardy; E.M. Wright An introduction to the theory of numbers, Oxford University Press | MR | Zbl

[21] C. Hermite Letter to C.D.J. Jacobi, J. reine. angew Math., Volume 40 (1839) no. 286

[22] A. Hurwitz Über eine besondere Art der Kettenbruchentwicklung reeller Grössen, Acta Math., Volume 12 (1889), pp. 367-405 | DOI | JFM

[23] A.B. Katok; A.M. Stepin Approximations in ergodic theory, Usp. Math. Nauk., Volume 22 (1967) no. 5, pp. 81-106 | MR | Zbl

[23] A.B. Katok; A.M. Stepin Approximations in ergodic theory, Russian Math. Surveys, Volume 22 (1967) no. 5, pp. 76-102 | MR | Zbl

[24] F. Klein Sur une représentation géométrique du développement en fraction continue ordinaire, Nouv. Ann. Math., Volume 15, pp. 321-331 | JFM | Numdam

[25] E. Korkina La périodicité des fractions continues multidimensionnelles, C.R. Acad. Sci. Paris, Série I, Volume 319 (1994), pp. 777-780 | MR | Zbl

[26] C. Kraaikamp A new class of continued fraction expansions, Acta Arith., Volume 57 (1991), pp. 1-39 | MR | Zbl

[27] J.L. Lagrange Sur la solution des problèmes indéterminés du second degré, Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, Volume 23 (1769)

[28] H. Minkowski Ein Kriterium für algebraishcen Zahlen, Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse, pp. 293-315

[29] H. Minkowski Über periodische Approximationen algebraischer Zahlen, Acta Math., Volume 26, pp. 333-351 | DOI | JFM

[30] M. Morse; G.A. Hedlund Symbolic dynamics, Amer. J. Math., Volume 60 (1938), pp. 815-866 | DOI | JFM | MR

[31] M. Morse; G.A. Hedlund Symbolic dynamics II: Sturmian sequences, Amer. J. Math., Volume 62 (1940), pp. 1-42 | DOI | MR | Zbl

[32] O. Perron Die Lehre von den Kettenbrüchen (in German), Teubner Verlag, 1929 | JFM

[33] G. Rauzy Une généralization du développement en fraction continue, Séminaire de Théorie des Nombres, Paris (1975-1977) | Numdam

[34] G. Rauzy Échanges d'intervalles et transformations induites, Acta Arith., Volume 34 (1979), pp. 315-328 | MR | Zbl

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

[36] R. RISLEY; L.Q. ZAMBONI A generalization of Sturmian sequences; combinatorial properties and transcendence, Acta Arith., Volume 95 (2000) no. 2, pp. 167-184 | MR | Zbl

[37] F. Schweiger The metrical theory of Jacobi-Perron algorithm, Lecture Notes in Mathematics, 334, Springer-Verlag, Berlin, 1973 | MR | Zbl

[38] F. Schweiger Ergodic Theory of Fibred Systems and Metric Number Theory (1995), pp. 289 pp. | Zbl

[39] C. Szekeres Multidimensional continued fractions, Ann. Univ. Sci. Budapest Sect. Math., Volume 13 (1970), pp. 113-140 | MR | Zbl

[40] W. Veech Interval exchange transformations, J. Anal. Math., Volume 33 (1978), pp. 222-278 | DOI | MR | Zbl

[41] W. Veech Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982) no. 1, pp. 201-242 | DOI | MR | Zbl

[42] W. Veech The metric theory of interval exchange transformations I, II, III, Amer. J. Math., Volume 106 (1984), pp. 1331-1421 | DOI | MR | Zbl

[43] N. Wozny; L.Q. Zamboni Frequencies of factors in Arnoux-Rauzy sequences, Acta Arith., Volume 96 (2001) no. 3, pp. 261-278 | DOI | MR | Zbl

[44] L.Q. Zamboni Une généralisation du théorème de Lagrange sur le développement en fraction continue, C.R. Acad. Sci. Paris, Série I, Volume 327 (1998), pp. 527-530 | MR | Zbl

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