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The Khinchin Theorem for interval-exchange transformations

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  • We define a Diophantine condition for interval-exchange transformations. When the number of intervals is two, that is, for rotations on the circle, our condition coincides with the classical Khinchin condition. We prove for interval-exchange transformations the same dichotomy as in the Khinchin Theorem. We also develop several results relating the Rauzy-Veech algorithm with homogeneous approximations for interval-exchange transformations.
    Mathematics Subject Classification: Primary: 37A20, 37E05; Secondary: 32G15, 11K55.


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  • [1]

    A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211.


    A. Avila and M. J.ResendeExponential mixing for the Teichmüller flow in the space of quadratic differentials, preprint arXiv:0908.1102.


    A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Mathematica, 198 (2007), 1-56.doi: 10.1007/s11511-007-0012-1.


    P. Billingsley, Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1979.


    C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, Ergodic Theory Dynam. Systems, 29 (2009), 767-816.doi: 10.1017/S0143385708080565.


    M. Boshernitzan, Rank two interval-exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394.


    M. Boshernitzan and J. ChaikaDiophantine properties of IET and general systems: Quantitative proximality and connectivity, preprint arXiv:0910.5422.


    J. ChaikaShrinking targets for IETs: Extending a theorem of Kurzweil, preprint arXiv:0910.2694.


    C. Danthony and A. Nogueira, Involutions linéaires et feuilletages mesurés, (French) [Linear involutions and measured foliations], C. R. Acad. Sci. Paris Sér I Math., 307(1988), 409-412.


    G. H. Hardy and E. M. Wright, "An introduction to the Theory of Numbers,'' 3rd ed., Oxford, at the Clarendon Press, 1954.


    M. Keane, Interval-exchange transformations, Math. Z., 141 (2002), 25-31.doi: 10.1007/BF01236981.


    S. P. Kerckhoff, Symplicial systems for interval-exchange maps and measured foliations, Ergodic Theory Dynam Systems, 5 (1985), 257-271.doi: 10.1017/S0143385700002881.


    Khinchin, "Continued Fractions,'' Translated by Peter Wynn. P. Noordhoff, Ltd., Groningen 1963 iii+101 pp.


    M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153(2003), 631-678.doi: 10.1007/s00222-003-0303-x.


    L. Marchese, "The Khinchin Theorem for Intervals Exchange Transformations and its Consequences for the Teichmüller Flow,'' PhD thesis.


    L. MarcheseKhinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow, to appear in Bull. Soc. Math. France.


    S. Marmi, P. Moussa and J.-C.Yoccoz, The cohomological equation for Roth type interval-exchange maps, J. American Math. Soc., 18 (2005), 823-872.doi: 10.1090/S0894-0347-05-00490-X.


    H. Masur, Interval exchange transformation and measured foliations, Ann. of Math. (2) , 115 (1982), 169-200,.doi: 10.2307/1971341.


    H. Masur, Logarithmic law for geodesic in moduli space, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), 229-245, Contemp. Math., 150, Amer. Math. Soc., Providence, RI, 1993.


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


    W. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272.doi: 10.1007/BF02790174.


    W. Veech, Gauss measures for transformations on the space of interval-exchange maps, Annals of Mathematics (2), 115 (1982), 201-242.doi: 10.2307/1971391.


    J.-C. Yoccoz, "Echanges d'Intervalles,'' Cours Coll\`ege de France, Janvier-Mars, 2005.


    J.-C. Yoccoz, Interval-exchange maps and translation surfaces, CMI summer school course, Centro di ricerca matematica Ennio de Giorgi, Pisa, June-July 2007 (in preparation).


    A. Zorich, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Annales de l'Institut Fourier (Grenoble), 46 (1996), 325-370.


    A. Zorich, Flat surfaces, Frontiers in Number Theory, Physics and Geometry, Vol. 1, 437-583, Springer, Berlin, (2006).

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