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January  2011, 5(1): 123-183. doi: 10.3934/jmd.2011.5.123

The Khinchin Theorem for interval-exchange transformations

Luca Marchese 1,

1. 

Section de mathématiques, case postale 64, 2–4 Rue du Lièvre, 1211 Genève, Switzerland

Received  April 2010 Revised  December 2010 Published  April 2011

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.
Keywords: Rauzy– Veech induction, Interval-exchange transformations, Teichmüller dynamics., Diophantine conditions.
Mathematics Subject Classification: Primary: 37A20, 37E05; Secondary: 32G15, 11K5.
Citation: Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123
References:
[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.  Google Scholar

[2]

A. Avila and M. J.Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, preprint \arXiv{0908.1102}., ().   Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

M. Boshernitzan and J. Chaika, Diophantine properties of IET and general systems: Quantitative proximality and connectivity,, preprint \arXiv{0910.5422}., ().   Google Scholar

[8]

J. Chaika, Shrinking targets for IETs: Extending a theorem of Kurzweil,, preprint \arXiv{0910.2694}., ().   Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

L. Marchese, Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow,, to appear in Bull. Soc. Math. France., ().   Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

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

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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). Google Scholar

[25]

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

[26]

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

show all references

References:
[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.  Google Scholar

[2]

A. Avila and M. J.Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials,, preprint \arXiv{0908.1102}., ().   Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

M. Boshernitzan and J. Chaika, Diophantine properties of IET and general systems: Quantitative proximality and connectivity,, preprint \arXiv{0910.5422}., ().   Google Scholar

[8]

J. Chaika, Shrinking targets for IETs: Extending a theorem of Kurzweil,, preprint \arXiv{0910.2694}., ().   Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

L. Marchese, Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow,, to appear in Bull. Soc. Math. France., ().   Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

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

[21]

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

[22]

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.  Google Scholar

[23]

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

[24]

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). Google Scholar

[25]

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

[26]

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

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