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Structure of three-interval exchange transformations III: Ergodic and spectral properties

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Abstract

In this paper, we present a detailed study of the spectral/ergodic properties of three-interval exchange transformations. Our approach is mostly combinatorial and relies on the diophantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally, we give affirmative answers to two questions posed by W. A. Veech on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum.

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References

  1. P. Arnoux,Un exemple de semi-conjugaison entre un échange d'intervalles et une translation sur le tore, Bull. Soc. Math. France116 (1988), 489–500.

    MATH  MathSciNet  Google Scholar 

  2. P. Arnoux and A. M. Fischer,The scenery flow for geometric structures on the torus: the linear setting, Chinese Ann. Math. Ser. B22 (2001), 427–470.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Arnoux and G. Rauzy,Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France119 (1991), 199–215.

    MATH  MathSciNet  Google Scholar 

  4. V. Berthé, N. Chekhova and S. Ferenczi,Covering numbers: arithmetics and dynamics for rotations and interval exchanges, J. Analyse Math.79 (1999), 1–31.

    MATH  Google Scholar 

  5. M. Boshernitzan,A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J.52 (1985), 723–752.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Boshernitzan,Rank two interval exchange maps, Ergodic Theory Dynam. Systems8 (1988), 379–394.

    MATH  MathSciNet  Google Scholar 

  7. M. Boshernitzan and C. R. Carroll,An extension of Lagrange's Theorem to interval exchange transformations over quadratic fields, J. Analyse Math.72 (1997), 21–44.

    MATH  MathSciNet  Google Scholar 

  8. M. Boshernitzan and A. Nogueira,Mixing properties of interval exchange transformations, in preparation.

  9. R. V. Chacon,A geometric construction of measure-preserving transformations, Proc. of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, 1965, pp. 335–360.

  10. J. Chaves and A. Nogueira,Spectral properties of interval exchange maps, Monatsh. Math.134 (2001), 89–102.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. R. Choksi and M. G. Nadkarni,The group of eigenvalues of a rank-one transformation, Canad. Math. Bull.38 (1995), 42–54.

    MATH  MathSciNet  Google Scholar 

  12. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai,Ergodic Theory, Springer-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  13. E. M. Coven and G. A. Hedlund,Sequences with minimal block growth, Math. Systems Theory7 (1972), 138–153.

    Article  MathSciNet  Google Scholar 

  14. A. del Junco,A family of counterexamples in ergodic theory, Israel J. Math.44 (1983), 160–188.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. del Junco and D. J. Rudolph,A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems7 (1987), 229–247.

    MATH  MathSciNet  Google Scholar 

  16. S. Ferenczi,Systems of finite rank, Colloq. Math.73 (1997), 35–65.

    MATH  MathSciNet  Google Scholar 

  17. S. Ferenczi and C. Mauduit,Transcendence of numbers with a low complexity expansion, J. Number Theory67 (1997), 146–161.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Ferenczi, C. Holton and L. Zamboni,The structure of three-interval exchange transformations I: an arithmetic study, Ann. Inst. Fourier (Grenoble)51 (2001), 861–901.

    MATH  MathSciNet  Google Scholar 

  19. S. Ferenczi, C. Holton and L. Zamboni,The structure of three-interval exchange transformations II: a combinatorial description of the trajectories, J. Analyse Math.89 (2003), 239–276.

    MATH  MathSciNet  Google Scholar 

  20. A. B. Katok and A. M. Stepin,Approximations in ergodic theory, Usp. Mat. Nauk22 (1967), 81–106 (in Russian), translated in Russian Math. Surveys22 (1967), 76–102.

    MATH  MathSciNet  Google Scholar 

  21. M. S. Keane,Interval exchange transformations, Math. Z.141 (1975), 25–31.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. S. Keane,Non-ergodic interval exchange transformations, Israel J. Math.26 (1977), 188–196.

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Keynes and D. Newton,A “minimal” non-uniquely ergodic interval exchange transformation, Math. Z.148 (1976), 101–106.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Masur,Interval exchange transformations and measured foliations, Ann. of Math.115 (1982), 169–200.

    Article  MathSciNet  Google Scholar 

  25. K. D. Merrill,Cohomology of step functions under irrational rotations, Israel J. Math.52 (1985), 320–340.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Morse and G. A. Hedlund,Symbolic dynamics II. Sturmian trajectories, Amer. J. Math.62 (1940), 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Nogueira and D. Rudolph,Topological weak-mixing of interval exchange maps, Ergodic Theory Dynam. Systems17 (1997), 1183–1209.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Rauzy,Échanges d'intervalles et transformations induites, Acta Arith.34 (1979), 315–328.

    MATH  MathSciNet  Google Scholar 

  29. M. Stewart,Irregularities of uniform distribution, Acta Math. Acad. Sci. Hungar.37 (1981), 185–221.

    Article  MATH  MathSciNet  Google Scholar 

  30. W. A. Veech,Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Trans. Amer. Math. Soc.140 (1969), 1–33.

    Article  MATH  MathSciNet  Google Scholar 

  31. W. A. Veech,Interval exchange transformations, J. Analyse Math.33 (1978), 222–272.

    MATH  MathSciNet  Google Scholar 

  32. W. A. Veech,Projective Swiss cheeses and uniquely ergodic interval exchange transformation, Progr. Math.10 (1981), 113–193.

    MathSciNet  Google Scholar 

  33. W. A. Veech,Gauss measures for transformations on the space of interval exchange maps, Ann. of Math.115 (1982), 201–242.

    Article  MathSciNet  Google Scholar 

  34. W. A. Veech,The metric theory of interval exchange transformations I: Generic spectral properties, Amer. J. Math.106 (1984), 1331–1359.

    Article  MATH  MathSciNet  Google Scholar 

  35. W. A. Veech,The metric theory of interval exchange transformations II: Approximation by primitive interval exchanges, Amer. J. Math.106 (1984), 1361–1387.

    Article  MATH  MathSciNet  Google Scholar 

  36. W. A. Veech,The metric theory of interval exchange transformations III: The Sah-Arnoux-Fathi invariant, Amer. J. Math.106 (1984), 1389–1422.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Luminy, CNRS, UPR 9016, 163 av. de Luminy, Case 907, F13288, Marseille Cedex 9, France

    Sébastien Ferenczi

  2. CNRS-FR 2291, Fédération de Recherche des Unités de Mathématiques de Marseille, 163 av. de Luminy, Case 909, F13288, Marseille Cedex 9, France

    Sébastien Ferenczi

  3. Department of Mathematics RLM 8.100, University of Texas at Austin, 26th and Speedway, 78712-1082, Austin, TX, USA

    Charles Holton

  4. Department of Mathematics, University of North Texas, 76203-5116, Denton, TX, USA

    Luca Q. Zamboni

Authors
  1. Sébastien Ferenczi
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  2. Charles Holton
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  3. Luca Q. Zamboni
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Correspondence to Sébastien Ferenczi.

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Partially supported by NSF grant INT-9726708.

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Ferenczi, S., Holton, C. & Zamboni, L.Q. Structure of three-interval exchange transformations III: Ergodic and spectral properties. J. Anal. Math. 93, 103–138 (2004). https://doi.org/10.1007/BF02789305

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  • DOI: https://doi.org/10.1007/BF02789305

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