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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 |
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. |
[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). |
[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. |
[4] |
A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499.
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, 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. |
[7] |
A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.
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), 93-141.
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, arXiv:1302.3320, (2014). |
[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). |
[12] |
S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, arXiv:1307.7314, (2014). |
[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. |
[14] |
G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940, (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), 241-298.
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), 797-815.
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), 11-71.
doi: 10.1070/RM1989v044n05ABEH002214. |
[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. |
[20] |
D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 30 (2004), 1551-1560.
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), 823-872.
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), 169-200.
doi: 10.2307/1971341. |
[24] |
W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.
doi: 10.2307/1971391. |
[25] |
R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9488-4. |
[26] |
A. Zorich, Deviation for interval exchange transformations, Erogodic Theory Dynam. Systems, 17 (1997), 1477-1499.
doi: 10.1017/S0143385797086215. |
[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. |
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. |
[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). |
[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. |
[4] |
A. Bufetov, Limit theorems for translation flows, Ann. of Math. (2), 179 (2014), 431-499.
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, 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. |
[7] |
A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.
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), 93-141.
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, arXiv:1302.3320, (2014). |
[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). |
[12] |
S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, arXiv:1307.7314, (2014). |
[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. |
[14] |
G. Forni, Sobolev regularity of solutions of the cohomological equation, arXiv:0707.0940, (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), 241-298.
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), 797-815.
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), 11-71.
doi: 10.1070/RM1989v044n05ABEH002214. |
[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. |
[20] |
D. Kleinbock and B. Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not., 30 (2004), 1551-1560.
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), 823-872.
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), 169-200.
doi: 10.2307/1971341. |
[24] |
W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.
doi: 10.2307/1971391. |
[25] |
R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9488-4. |
[26] |
A. Zorich, Deviation for interval exchange transformations, Erogodic Theory Dynam. Systems, 17 (1997), 1477-1499.
doi: 10.1017/S0143385797086215. |
[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. |
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