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2015, 9: 1-23. doi: 10.3934/jmd.2015.9.1

Every flat surface is Birkhoff and Oseledets generic in almost every direction

Jon Chaika 1, and  Alex Eskin 2,

1. 

Department of Mathematics, University of Utah, 155 S. 1400 E., Room 233, Salt Lake City, UT 84112, United States
2. 

Department of Mathematics, University of Chicago, Chicago, IL 60637

Received  November 2013 Revised  October 2014 Published  May 2015

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of $SL(2,\mathbb R)$ on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.
Keywords: flat surfaces., Ergodic theorems.
Mathematics Subject Classification: Primary: 37A10; Secondary: 37A2.
Citation: Jon Chaika, Alex Eskin. Every flat surface is Birkhoff and Oseledets generic in almost every direction. Journal of Modern Dynamics, 2015, 9: 1-23. doi: 10.3934/jmd.2015.9.1
References:
[1]

J. Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z.  Google Scholar

[2]

A. Avila, A. Eskin and M. Moeller, Symplectic and isometric $SL(2,\mathbb R)$ invariant subbundles of the Hodge bundle, arXiv:1209.2854, (2012). Google Scholar

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Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III), Ann. of Math.(2), 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5.  Google Scholar

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A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499. doi: 10.4007/annals.2014.179.2.2.  Google Scholar

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V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, , ().   Google Scholar

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P. Ehrenfest and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, (in German) Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S, 1912; English translation: The Conceptual Foundations of the Statistical Approach in Mechanics, Translated by M. J. Moravcsik, Cornell University Press, Itacha, NY, 1959.  Google Scholar

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A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.  Google Scholar

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A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984.  Google Scholar

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A. Eskin and C. Matheus, Semisimplicity of the Lyapunov spectrum for irreducible cocycles,, , ().   Google Scholar

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A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space, arXiv:1302.3320, (2014). Google Scholar

[11]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $SL(2,\mathbbR)$ action on moduli space, arXiv:1305.3015, (2014). Google Scholar

[12]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, arXiv:1307.7314, (2014). Google Scholar

[13]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150.  Google Scholar

[14]

G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940, (2007). Google Scholar

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K. Frączek and P. Hubert, Recurrence and non-ergodicity,, preprint., ().   Google Scholar

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K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$ periodic billiards and infinite translation surfaces, Inventiones mathematicae, 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z.  Google Scholar

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A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815. doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar

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I. Ya. Gol'dsheĭd and G. A. Margulis, Lyapunov indices of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71. doi: 10.1070/RM1989v044n05ABEH002214.  Google Scholar

[19]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.  Google Scholar

[20]

D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 30 (2004), 1551-1560. doi: 10.1155/S1073792804133412.  Google Scholar

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

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S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps,, , ().   Google Scholar

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

[24]

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

[25]

R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4.  Google Scholar

[26]

A. Zorich, Deviation for interval exchange transformations, Erogodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.  Google Scholar

[27]

A. Zorich, Flat Surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

show all references

References:
[1]

J. Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata, 119 (2006), 121-140. doi: 10.1007/s10711-006-9058-z.  Google Scholar

[2]

A. Avila, A. Eskin and M. Moeller, Symplectic and isometric $SL(2,\mathbb R)$ invariant subbundles of the Hodge bundle, arXiv:1209.2854, (2012). Google Scholar

[3]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III), Ann. of Math.(2), 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5.  Google Scholar

[4]

A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499. doi: 10.4007/annals.2014.179.2.2.  Google Scholar

[5]

V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, , ().   Google Scholar

[6]

P. Ehrenfest and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, (in German) Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S, 1912; English translation: The Conceptual Foundations of the Statistical Approach in Mechanics, Translated by M. J. Moravcsik, Cornell University Press, Itacha, NY, 1959.  Google Scholar

[7]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.  Google Scholar

[8]

A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984.  Google Scholar

[9]

A. Eskin and C. Matheus, Semisimplicity of the Lyapunov spectrum for irreducible cocycles,, , ().   Google Scholar

[10]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space, arXiv:1302.3320, (2014). Google Scholar

[11]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $SL(2,\mathbbR)$ action on moduli space, arXiv:1305.3015, (2014). Google Scholar

[12]

S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, arXiv:1307.7314, (2014). Google Scholar

[13]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150.  Google Scholar

[14]

G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940, (2007). Google Scholar

[15]

K. Frączek and P. Hubert, Recurrence and non-ergodicity,, preprint., ().   Google Scholar

[16]

K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$ periodic billiards and infinite translation surfaces, Inventiones mathematicae, 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z.  Google Scholar

[17]

A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797-815. doi: 10.1016/S0246-0203(97)80113-6.  Google Scholar

[18]

I. Ya. Gol'dsheĭd and G. A. Margulis, Lyapunov indices of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71. doi: 10.1070/RM1989v044n05ABEH002214.  Google Scholar

[19]

J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.  Google Scholar

[20]

D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 30 (2004), 1551-1560. doi: 10.1155/S1073792804133412.  Google Scholar

[21]

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

[22]

S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps,, , ().   Google Scholar

[23]

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

[24]

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

[25]

R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4.  Google Scholar

[26]

A. Zorich, Deviation for interval exchange transformations, Erogodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.  Google Scholar

[27]

A. Zorich, Flat Surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

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