The Khinchin Theorem for interval-exchange transformations

Previous Article
Integrability and Lyapunov exponents
JMD Home
This Issue
Next Article
Tori with hyperbolic dynamics in 3-manifolds

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

The Khinchin Theorem for interval-exchange transformations

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

Received April 2010 Revised December 2010 Published April 2011

Abstract
Related Papers

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. 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. doi: 10.1007/s11511-007-0012-1. Google Scholar
[4]	P. Billingsley, Probability and measure,, Wiley Series in Probability and Mathematical Statistics, (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. doi: 10.1017/S0143385708080565. Google Scholar
[6]	M. Boshernitzan, Rank two interval-exchange transformations,, Ergodic Theory Dynam. Systems, 8 (1988), 379. 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], 307 (1988), 409. Google Scholar
[10]	G. H. Hardy and E. M. Wright, "An introduction to the Theory of Numbers,'', 3rd ed., (1954). Google Scholar
[11]	M. Keane, Interval-exchange transformations,, Math. Z., 141 (2002), 25. 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. doi: 10.1017/S0143385700002881. Google Scholar
[13]	Khinchin, "Continued Fractions,'', Translated by Peter Wynn. P. Noordhoff, (1963). 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. 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. 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. 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), 229. Google Scholar
[20]	G. Rauzy, Échanges d'intervalles et transformations induites,, (French) Acta Arith., 34 (1979), 315. Google Scholar
[21]	W. Veech, Interval exchange transformations,, J. Analyse Math., 33 (1978), 222. 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. doi: 10.2307/1971391. Google Scholar
[23]	J.-C. Yoccoz, "Echanges d'Intervalles,'', Cours Coll\`ege de France, (2005). Google Scholar
[24]	J.-C. Yoccoz, Interval-exchange maps and translation surfaces,, CMI summer school course, (2007). 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. Google Scholar
[26]	A. Zorich, Flat surfaces,, Frontiers in Number Theory, 1 (2006), 437. 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. 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. doi: 10.1007/s11511-007-0012-1. Google Scholar
[4]	P. Billingsley, Probability and measure,, Wiley Series in Probability and Mathematical Statistics, (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. doi: 10.1017/S0143385708080565. Google Scholar
[6]	M. Boshernitzan, Rank two interval-exchange transformations,, Ergodic Theory Dynam. Systems, 8 (1988), 379. 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], 307 (1988), 409. Google Scholar
[10]	G. H. Hardy and E. M. Wright, "An introduction to the Theory of Numbers,'', 3rd ed., (1954). Google Scholar
[11]	M. Keane, Interval-exchange transformations,, Math. Z., 141 (2002), 25. 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. doi: 10.1017/S0143385700002881. Google Scholar
[13]	Khinchin, "Continued Fractions,'', Translated by Peter Wynn. P. Noordhoff, (1963). 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. 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. 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. 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), 229. Google Scholar
[20]	G. Rauzy, Échanges d'intervalles et transformations induites,, (French) Acta Arith., 34 (1979), 315. Google Scholar
[21]	W. Veech, Interval exchange transformations,, J. Analyse Math., 33 (1978), 222. 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. doi: 10.2307/1971391. Google Scholar
[23]	J.-C. Yoccoz, "Echanges d'Intervalles,'', Cours Coll\`ege de France, (2005). Google Scholar
[24]	J.-C. Yoccoz, Interval-exchange maps and translation surfaces,, CMI summer school course, (2007). 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. Google Scholar
[26]	A. Zorich, Flat surfaces,, Frontiers in Number Theory, 1 (2006), 437. Google Scholar

[1]	Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139
[2]	Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[3]	Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253
[4]	Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[5]	Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[6]	Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
[7]	Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
[8]	Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[9]	Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
[10]	Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[11]	Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[12]	Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[13]	Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[14]	Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489
[15]	Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209
[16]	Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251
[17]	Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[18]	Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039
[19]	Matteo Costantini, André Kappes. The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve. Journal of Modern Dynamics, 2017, 11: 17-41. doi: 10.3934/jmd.2017002
[20]	David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences