Structure of three interval exchange transformations I: an arithmetic study

Sébastien Ferenczi[1]; Charles Holton[2]; Luca Q. Zamboni[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 (2001)

  • Volume: 51, Issue: 4, page 861-901
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Ferenczi, Sébastien, Holton, Charles, and Zamboni, Luca Q.. "Structure of three interval exchange transformations I: an arithmetic study." Annales de l’institut Fourier 51.4 (2001): 861-901. <http://eudml.org/doc/115939>.

@article{Ferenczi2001,
abstract = {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 $\{\mathbb \{Q\}\}.$},
affiliation = {Laboratoire de Mathématiques et Physique Théorique, CNRS, UPRES-A 6083, Parc de Grandmont, 37200 Tours (France); University of California, Department of Mathematics, Berkeley CA 94720-3840 (USA); University of North Texas, Department of Mathematics, Denton TX 76203-5116 (USA)},
author = {Ferenczi, Sébastien, Holton, Charles, Zamboni, Luca Q.},
journal = {Annales de l’institut Fourier},
keywords = {Generalized continued fraction; interval exchange transformations; generalized Gauss maps; generalized continued fraction; negative slope algorithm},
language = {eng},
number = {4},
pages = {861-901},
publisher = {Association des Annales de l'Institut Fourier},
title = {Structure of three interval exchange transformations I: an arithmetic study},
url = {http://eudml.org/doc/115939},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Ferenczi, Sébastien
AU - Holton, Charles
AU - Zamboni, Luca Q.
TI - Structure of three interval exchange transformations I: an arithmetic study
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 4
SP - 861
EP - 901
AB - 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 ${\mathbb {Q}}.$
LA - eng
KW - Generalized continued fraction; interval exchange transformations; generalized Gauss maps; generalized continued fraction; negative slope algorithm
UR - http://eudml.org/doc/115939
ER -

References

top
  1. J. Aaronson, M. Keane, The visits to zero of some deterministic random walks, Proc. London Math. Soc. 44 (1982), 535-553 Zbl0489.60006MR656248
  2. V.I. Arnold, A-graded algebras and continued fractions, Comm. Pure Applied Math. XLII (1989), 993-1000 Zbl0692.16012MR1008799
  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 116 (1988), 489-500 Zbl0703.58045MR1005392
  4. P. Arnoux, V. Berthe, S. Ito, Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions, (1999) Zbl1017.11006
  5. P. Arnoux, G. Rauzy, Représentation géométrique de suites de complexité 2 n + 1 , Bull. Soc. Math. France 119 (1991), 199-215 Zbl0789.28011MR1116845
  6. L. Bernstein, The Jacobi-Perron algorithm; its theory and applications, no 207 (1971), Springer-Verlag Zbl0213.05201MR285478
  7. V. Berthe, L. Vuillon, Tilings and rotations: a two-dimensional generalization of Sturmian sequences, Discrete Math. 223 (2000), 27-53 Zbl0970.68124MR1782038
  8. M. Boshernitzan, C. Carroll, An extension of Lagrange's theorem to interval exchange transformations over quadratic fields, J. Anal. Math. 72 (1997), 21-44 Zbl0931.28013MR1482988
  9. A.J. Brentjes, Multi-dimensional continued fraction algorithms, Math. Centre Tracts, Amsterdam 145 (1981) Zbl0471.10024MR638474
  10. E. Burger, On simultaneous diophantine approximation in the vector space + α , J. Number Theory 82 (2000), 12-24 Zbl0985.11031MR1755151
  11. E. Burger, On real quadratic number fields and simultaneous diophantine approximation, Monats. Math. 128 (1999), 201-209 Zbl0971.11039MR1719356
  12. J. Cassaigne, S. Ferenczi, L.Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Ann. Inst. Fourier 50 (2000), 1265-1276 Zbl1004.37008MR1799745
  13. N. Chekhova, P. Hubert, A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de tribonacci, J. Théorie des Nombres de Bordeaux (2001) Zbl1038.37010MR1879664
  14. E.M. Coven, G.A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1972), 138-153 Zbl0256.54028MR322838
  15. A. del Junco, A family of counterexamples in ergodic theory, Israël J. Math. 44 (1983), 160-188 Zbl0522.28012MR693358
  16. S. Ferenczi, C. Holton, L.Q. Zamboni, Structure of three-interval exchange transformations II: a combinatorial description of the trajectories, (2001) Zbl1130.37324MR1981920
  17. S. Ferenczi, C. Holton, L.Q. Zamboni, Structure of three-interval exchange transformations III: ergodic and spectral properties, (2001) Zbl1094.37005MR2110326
  18. T. Garrity, On periodic sequences for algebraic numbers, (1999) Zbl1015.11031
  19. Y. Hara-Mimachi, S. Ito, A characterization of real quadratic numbers by diophantine algorithms, Tokyo J. Math. 14 (1991), 251-267 Zbl0751.11034MR1138165
  20. G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Oxford University Press Zbl0058.03301MR67125
  21. C. Hermite, Letter to C.D.J. Jacobi, J. reine. angew Math. 40 (1839) 
  22. A. Hurwitz, Über eine besondere Art der Kettenbruchentwicklung reeller Grössen, Acta Math. 12 (1889), 367-405 
  23. A.B. Katok, A.M. Stepin, Approximations in ergodic theory, Usp. Math. Nauk. 22 (1967), 81-106 Zbl0172.07202MR219697
  24. E. Korkina, La périodicité des fractions continues multidimensionnelles, C.R. Acad. Sci. Paris, Série I 319 (1994), 777-780 Zbl0836.11023MR1300940
  25. C. Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 (1991), 1-39 Zbl0721.11029MR1093246
  26. J.L. Lagrange, Sur la solution des problèmes indéterminés du second degré, Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin 23 (1769) 
  27. H. Minkowski, Ein Kriterium für algebraishcen Zahlen, Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse, 293-315 
  28. H. Minkowski, Über periodische Approximationen algebraischer Zahlen, Acta Math. 26, 333-351 
  29. M. Morse, G.A. Hedlund, Symbolic dynamics, Amer. J. Math. 60 (1938), 815-866 Zbl0019.33502MR1507944
  30. M. Morse, G.A. Hedlund, Symbolic dynamics II: Sturmian sequences, Amer. J. Math. 62 (1940), 1-42 Zbl0022.34003MR745
  31. O. Perron, Die Lehre von den Kettenbrüchen (in German), (1929), Teubner Verlag Zbl55.0262.09
  32. G. Rauzy, Une généralization du développement en fraction continue, Séminaire de Théorie des Nombres, Paris (1975-1977) Zbl0369.28015
  33. G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith. 34 (1979), 315-328 Zbl0414.28018MR543205
  34. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178 Zbl0522.10032MR667748
  35. R. RISLEY, L.Q. ZAMBONI, A generalization of Sturmian sequences; combinatorial properties and transcendence, Acta Arith. 95 (2000), 167-184 Zbl0953.11007MR1785413
  36. F. Schweiger, The metrical theory of Jacobi-Perron algorithm, 334 (1973), Springer-Verlag, Berlin Zbl0287.10041MR345925
  37. F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, (1995), Oxford Science Publications Zbl0819.11027
  38. C. Szekeres, Multidimensional continued fractions, Ann. Univ. Sci. Budapest Sect. Math. 13 (1970), 113-140 Zbl0214.30101MR313198
  39. W. Veech, Interval exchange transformations, J. Anal. Math. 33 (1978), 222-278 Zbl0455.28006MR516048
  40. W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. 115 (1982), 201-242 Zbl0486.28014MR644019
  41. W. Veech, The metric theory of interval exchange transformations I, II, III, Amer. J. Math. 106 (1984), 1331-1421 Zbl0631.28006MR765582
  42. N. Wozny, L.Q. Zamboni, Frequencies of factors in Arnoux-Rauzy sequences, Acta Arith. 96 (2001), 261-278 Zbl0973.11030MR1814281
  43. L.Q. Zamboni, Une généralisation du théorème de Lagrange sur le développement en fraction continue, C.R. Acad. Sci. Paris, Série I 327 (1998), 527-530 Zbl1039.11500MR1647296
  44. F. Klein, Sur une représentation géométrique du développement en fraction continue ordinaire, Nouv. Ann. Math. 15, 321-331 
  45. A.B. Katok, A.M. Stepin, Approximations in ergodic theory, Russian Math. Surveys 22 (1967), 76-102 Zbl0172.07202MR219697

Citations in EuDML Documents

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  1. Louis-Sébastien Guimond, Zuzana Masáková, Edita Pelantová, Combinatorial properties of infinite words associated with cut-and-project sequences
  2. Sébastien Ferenczi, Luca Q. Zamboni, Eigenvalues and simplicity of interval exchange transformations
  3. Sébastien Ferenczi, A generalization of the self-dual induction to every interval exchange transformation
  4. Petr Ambrož, Zuzana Masáková, Edita Pelantová, Morphisms fixing words associated with exchange of three intervals
  5. Amy Glen, Jacques Justin, Episturmian words: a survey

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