# American Institute of Mathematical Sciences

October  2012, 6(4): 477-497. doi: 10.3934/jmd.2012.6.477

## Ergodic infinite group extensions of geodesic flows on translation surfaces

 1 SUNY College at Old Westbury, Mathematics/CIS Department, P.O. Box 210, Old Westbury, NY 11568 2 Aix-Marseille University, CNRS, CPT, IML, Frumam, 13288 Marseille Cedex 09

Received  May 2012 Published  January 2013

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. K. Frączek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases.
Citation: David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
##### References:
 [1] J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs, 50 (1997). Google Scholar [2] M. Boshernitzan, A condition for weak mixing of induced IETs,, in, 567 (2012), 53. doi: 10.1090/conm/567/11251. Google Scholar [3] M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons,, Trans. Amer. Math. Soc., 350 (1998), 3523. doi: 10.1090/S0002-9947-98-02089-3. Google Scholar [4] J. Chaika and P. Hubert, Ergodicity of skew products over interval exchange transformations,, in preparation., (). Google Scholar [5] J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. doi: 10.1090/conm/485/09492. Google Scholar [6] J.-P. Conze and K. Frączek, Cocycles over interval exchange transformations and multivalued Hamiltonian flows,, Adv. Math., 226 (2011), 4373. doi: 10.1016/j.aim.2010.11.014. Google Scholar [7] J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces,, Ergodic Theory Dynam. Systems, 32 (2012), 491. doi: 10.1017/S0143385711001003. Google Scholar [8] V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, preprint., (). Google Scholar [9] K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows,, Fund. Math., 185 (2005), 117. doi: 10.4064/fm185-2-2. Google Scholar [10] K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, \arXiv{1109.4584}, (2011). Google Scholar [11] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar [12] W. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Disc. Cont. Dyn. Sys., (). Google Scholar [13] W. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, Annales de l'Institut Fourier, 62 (2012), 1581. Google Scholar [14] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Isr. J. Math., 151 (2006), 281. doi: 10.1007/BF02777365. Google Scholar [15] P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, Journal fuer die Reine und Angewandte Mathematik (Crelle's Journal), 656 (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar [16] P. Hubert and T. A. Schmidt, An introduction to Veech surfaces,, in, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar [17] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, preprint., (). Google Scholar [18] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials,, The Annals of Mathematics (2), 124 (1986), 293. doi: 10.2307/1971280. Google Scholar [19] 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 [20] E. Lanneau and D.-M. Nguyen, Teichmüller curves generatedby Weierstrass Prym eigenforms in genus three and genus four,, preprint., (). Google Scholar [21] H. Masur, Ergodic theory of translation surfaces,, in, (2006), 527. doi: 10.1016/S1874-575X(06)80032-9. Google Scholar [22] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar [23] C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin,, Mathematische Annalen, 333 (2005), 87. doi: 10.1007/s00208-005-0666-y. Google Scholar [24] S. J. Patterson, Diophantine approximation in Fuchsian groups,, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527. Google Scholar [25] C. Pugh and M. Shub, Ergodic elements of ergodic actions,, Compositio Math., 23 (1971), 115. Google Scholar [26] D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, preprint, (2012). Google Scholar [27] K. Schmidt, "Cocycles of Ergodic Transformation Groups,", Lecture Notes in Mathematics, (1977). Google Scholar [28] _____, A cylinder flow arising from irregularity of distribution,, Compositio Mathematica, 36 (1978), 225. Google Scholar [29] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215. doi: 10.1007/BF02392354. Google Scholar [30] S. Troubetzkoy, Recurrence in generic staircases,, Discrete Contin. Dyn. Syst., 32 (2012), 1047. doi: 10.3934/dcds.2012.32.1047. Google Scholar [31] W. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation,, Erg. Thry. Dyn. Sys., 7 (1987), 149. doi: 10.1017/S0143385700003862. Google Scholar [32] A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

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##### References:
 [1] J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs, 50 (1997). Google Scholar [2] M. Boshernitzan, A condition for weak mixing of induced IETs,, in, 567 (2012), 53. doi: 10.1090/conm/567/11251. Google Scholar [3] M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons,, Trans. Amer. Math. Soc., 350 (1998), 3523. doi: 10.1090/S0002-9947-98-02089-3. Google Scholar [4] J. Chaika and P. Hubert, Ergodicity of skew products over interval exchange transformations,, in preparation., (). Google Scholar [5] J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. doi: 10.1090/conm/485/09492. Google Scholar [6] J.-P. Conze and K. Frączek, Cocycles over interval exchange transformations and multivalued Hamiltonian flows,, Adv. Math., 226 (2011), 4373. doi: 10.1016/j.aim.2010.11.014. Google Scholar [7] J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces,, Ergodic Theory Dynam. Systems, 32 (2012), 491. doi: 10.1017/S0143385711001003. Google Scholar [8] V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, preprint., (). Google Scholar [9] K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows,, Fund. Math., 185 (2005), 117. doi: 10.4064/fm185-2-2. Google Scholar [10] K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, \arXiv{1109.4584}, (2011). Google Scholar [11] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar [12] W. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Disc. Cont. Dyn. Sys., (). Google Scholar [13] W. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, Annales de l'Institut Fourier, 62 (2012), 1581. Google Scholar [14] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Isr. J. Math., 151 (2006), 281. doi: 10.1007/BF02777365. Google Scholar [15] P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, Journal fuer die Reine und Angewandte Mathematik (Crelle's Journal), 656 (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar [16] P. Hubert and T. A. Schmidt, An introduction to Veech surfaces,, in, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar [17] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, preprint., (). Google Scholar [18] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials,, The Annals of Mathematics (2), 124 (1986), 293. doi: 10.2307/1971280. Google Scholar [19] 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 [20] E. Lanneau and D.-M. Nguyen, Teichmüller curves generatedby Weierstrass Prym eigenforms in genus three and genus four,, preprint., (). Google Scholar [21] H. Masur, Ergodic theory of translation surfaces,, in, (2006), 527. doi: 10.1016/S1874-575X(06)80032-9. Google Scholar [22] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar [23] C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin,, Mathematische Annalen, 333 (2005), 87. doi: 10.1007/s00208-005-0666-y. Google Scholar [24] S. J. Patterson, Diophantine approximation in Fuchsian groups,, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527. Google Scholar [25] C. Pugh and M. Shub, Ergodic elements of ergodic actions,, Compositio Math., 23 (1971), 115. Google Scholar [26] D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, preprint, (2012). Google Scholar [27] K. Schmidt, "Cocycles of Ergodic Transformation Groups,", Lecture Notes in Mathematics, (1977). Google Scholar [28] _____, A cylinder flow arising from irregularity of distribution,, Compositio Mathematica, 36 (1978), 225. Google Scholar [29] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215. doi: 10.1007/BF02392354. Google Scholar [30] S. Troubetzkoy, Recurrence in generic staircases,, Discrete Contin. Dyn. Syst., 32 (2012), 1047. doi: 10.3934/dcds.2012.32.1047. Google Scholar [31] W. Veech, Boshernitzan's criterion for unique ergodicity of an interval exchange transformation,, Erg. Thry. Dyn. Sys., 7 (1987), 149. doi: 10.1017/S0143385700003862. Google Scholar [32] A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13. Google Scholar
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