# American Institute of Mathematical Sciences

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

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

 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.
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. doi: 10.1007/s10711-006-9058-z. [2] A. Avila, A. Eskin and M. Moeller, Symplectic and isometric $SL(2,\mathbb R)$ invariant subbundles of the Hodge bundle,, , (2012). [3] Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III),, Ann. of Math.(2), 178 (2013), 1017. doi: 10.4007/annals.2013.178.3.5. [4] A. Bufetov, Limit theorems for translation flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. [5] V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, , (). [6] P. Ehrenfest and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, (in German) Encykl. d. Math. Wissensch. IV 2 II, (1912). [7] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443. doi: 10.1017/S0143385701001225. [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. doi: 10.2307/120984. [9] A. Eskin and C. Matheus, Semisimplicity of the Lyapunov spectrum for irreducible cocycles,, , (). [10] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space,, , (2014). [11] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $SL(2,\mathbbR)$ action on moduli space,, , (2014). [12] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle,, , (2014). [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. [14] G. Forni, Sobolev regularity of solutions of the cohomological equation,, , (2007). [15] K. Frączek and P. Hubert, Recurrence and non-ergodicity,, preprint., (). [16] K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$ periodic billiards and infinite translation surfaces,, Inventiones mathematicae, 197 (2014), 241. doi: 10.1007/s00222-013-0482-z. [17] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems,, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797. doi: 10.1016/S0246-0203(97)80113-6. [18] I. Ya. Gol'dsheĭd and G. A. Margulis, Lyapunov indices of a product of random matrices,, Russian Math. Surveys, 44 (1989), 11. doi: 10.1070/RM1989v044n05ABEH002214. [19] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, J. Math. Phys., 21 (1980), 1802. doi: 10.1063/1.524633. [20] D. Kleinbock and B. Weiss, Bounded geodesics in moduli space,, Int. Math. Res. Not., 30 (2004), 1551. doi: 10.1155/S1073792804133412. [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. doi: 10.1090/S0894-0347-05-00490-X. [22] S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps,, , (). [23] H. Masur, Interval exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. doi: 10.2307/1971341. [24] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [25] R. Zimmer, Ergodic Theory and Semisimple Groups,, Monographs in Mathematics, (1984). doi: 10.1007/978-1-4684-9488-4. [26] A. Zorich, Deviation for interval exchange transformations,, Erogodic Theory Dynam. Systems, 17 (1997), 1477. doi: 10.1017/S0143385797086215. [27] A. Zorich, Flat Surfaces,, in Frontiers in Number Theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

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##### References:
 [1] J. Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow,, Geom. Dedicata, 119 (2006), 121. doi: 10.1007/s10711-006-9058-z. [2] A. Avila, A. Eskin and M. Moeller, Symplectic and isometric $SL(2,\mathbb R)$ invariant subbundles of the Hodge bundle,, , (2012). [3] Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III),, Ann. of Math.(2), 178 (2013), 1017. doi: 10.4007/annals.2013.178.3.5. [4] A. Bufetov, Limit theorems for translation flows,, Ann. of Math. (2), 179 (2014), 431. doi: 10.4007/annals.2014.179.2.2. [5] V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, , (). [6] P. Ehrenfest and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, (in German) Encykl. d. Math. Wissensch. IV 2 II, (1912). [7] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces,, Ergodic Theory Dynam. Systems, 21 (2001), 443. doi: 10.1017/S0143385701001225. [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. doi: 10.2307/120984. [9] A. Eskin and C. Matheus, Semisimplicity of the Lyapunov spectrum for irreducible cocycles,, , (). [10] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL(2,\mathbbR)$ action on moduli space,, , (2014). [11] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $SL(2,\mathbbR)$ action on moduli space,, , (2014). [12] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle,, , (2014). [13] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. [14] G. Forni, Sobolev regularity of solutions of the cohomological equation,, , (2007). [15] K. Frączek and P. Hubert, Recurrence and non-ergodicity,, preprint., (). [16] K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$ periodic billiards and infinite translation surfaces,, Inventiones mathematicae, 197 (2014), 241. doi: 10.1007/s00222-013-0482-z. [17] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems,, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 797. doi: 10.1016/S0246-0203(97)80113-6. [18] I. Ya. Gol'dsheĭd and G. A. Margulis, Lyapunov indices of a product of random matrices,, Russian Math. Surveys, 44 (1989), 11. doi: 10.1070/RM1989v044n05ABEH002214. [19] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, J. Math. Phys., 21 (1980), 1802. doi: 10.1063/1.524633. [20] D. Kleinbock and B. Weiss, Bounded geodesics in moduli space,, Int. Math. Res. Not., 30 (2004), 1551. doi: 10.1155/S1073792804133412. [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. doi: 10.1090/S0894-0347-05-00490-X. [22] S. Marmi and J.-C. Yoccoz, Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps,, , (). [23] H. Masur, Interval exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169. doi: 10.2307/1971341. [24] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [25] R. Zimmer, Ergodic Theory and Semisimple Groups,, Monographs in Mathematics, (1984). doi: 10.1007/978-1-4684-9488-4. [26] A. Zorich, Deviation for interval exchange transformations,, Erogodic Theory Dynam. Systems, 17 (1997), 1477. doi: 10.1017/S0143385797086215. [27] A. Zorich, Flat Surfaces,, in Frontiers in Number Theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.
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