Ergodic infinite group extensions of geodesic flows on translation surfaces

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

Ergodic infinite group extensions of geodesic flows on translation surfaces

David Ralston ^1, and Serge Troubetzkoy ^2,

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

Abstract
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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.

Keywords: Translation surface, essential values., ergodicity, diophantine approximation.

Mathematics Subject Classification: Primary: 37A25, 37A40; Secondary: 37A20, 37C4.

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).
[2]	M. Boshernitzan, A condition for weak mixing of induced IETs,, in, 567 (2012), 53. doi: 10.1090/conm/567/11251.
[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.
[4]	J. Chaika and P. Hubert, Ergodicity of skew products over interval exchange transformations,, in preparation., ().
[5]	J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. doi: 10.1090/conm/485/09492.
[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.
[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.
[8]	V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, preprint., ().
[9]	K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows,, Fund. Math., 185 (2005), 117. doi: 10.4064/fm185-2-2.
[10]	K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, \arXiv{1109.4584}, (2011).
[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.
[12]	W. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Disc. Cont. Dyn. Sys., ().
[13]	W. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, Annales de l'Institut Fourier, 62 (2012), 1581.
[14]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Isr. J. Math., 151 (2006), 281. doi: 10.1007/BF02777365.
[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.
[16]	P. Hubert and T. A. Schmidt, An introduction to Veech surfaces,, in, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7.
[17]	P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, preprint., ().
[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.
[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.
[20]	E. Lanneau and D.-M. Nguyen, Teichmüller curves generatedby Weierstrass Prym eigenforms in genus three and genus four,, preprint., ().
[21]	H. Masur, Ergodic theory of translation surfaces,, in, (2006), 527. doi: 10.1016/S1874-575X(06)80032-9.
[22]	H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.
[23]	C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin,, Mathematische Annalen, 333 (2005), 87. doi: 10.1007/s00208-005-0666-y.
[24]	S. J. Patterson, Diophantine approximation in Fuchsian groups,, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527.
[25]	C. Pugh and M. Shub, Ergodic elements of ergodic actions,, Compositio Math., 23 (1971), 115.
[26]	D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, preprint, (2012).
[27]	K. Schmidt, "Cocycles of Ergodic Transformation Groups,", Lecture Notes in Mathematics, (1977).
[28]	_____, A cylinder flow arising from irregularity of distribution,, Compositio Mathematica, 36 (1978), 225.
[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.
[30]	S. Troubetzkoy, Recurrence in generic staircases,, Discrete Contin. Dyn. Syst., 32 (2012), 1047. doi: 10.3934/dcds.2012.32.1047.
[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.
[32]	A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

show all references

References:

[1]	J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs, 50 (1997).
[2]	M. Boshernitzan, A condition for weak mixing of induced IETs,, in, 567 (2012), 53. doi: 10.1090/conm/567/11251.
[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.
[4]	J. Chaika and P. Hubert, Ergodicity of skew products over interval exchange transformations,, in preparation., ().
[5]	J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. doi: 10.1090/conm/485/09492.
[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.
[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.
[8]	V. Delecroix, P. Hubert and S. Leliévre, Diffusion for the periodic wind-tree model,, preprint., ().
[9]	K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows,, Fund. Math., 185 (2005), 117. doi: 10.4064/fm185-2-2.
[10]	K. Frączek and C. Ulcigrai, Non-ergodic Z-periodic billiards and infinite translation surfaces,, \arXiv{1109.4584}, (2011).
[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.
[12]	W. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Disc. Cont. Dyn. Sys., ().
[13]	W. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, Annales de l'Institut Fourier, 62 (2012), 1581.
[14]	P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Isr. J. Math., 151 (2006), 281. doi: 10.1007/BF02777365.
[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.
[16]	P. Hubert and T. A. Schmidt, An introduction to Veech surfaces,, in, (2006), 501. doi: 10.1016/S1874-575X(06)80031-7.
[17]	P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, preprint., ().
[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.
[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.
[20]	E. Lanneau and D.-M. Nguyen, Teichmüller curves generatedby Weierstrass Prym eigenforms in genus three and genus four,, preprint., ().
[21]	H. Masur, Ergodic theory of translation surfaces,, in, (2006), 527. doi: 10.1016/S1874-575X(06)80032-9.
[22]	H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.
[23]	C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin,, Mathematische Annalen, 333 (2005), 87. doi: 10.1007/s00208-005-0666-y.
[24]	S. J. Patterson, Diophantine approximation in Fuchsian groups,, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527.
[25]	C. Pugh and M. Shub, Ergodic elements of ergodic actions,, Compositio Math., 23 (1971), 115.
[26]	D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, preprint, (2012).
[27]	K. Schmidt, "Cocycles of Ergodic Transformation Groups,", Lecture Notes in Mathematics, (1977).
[28]	_____, A cylinder flow arising from irregularity of distribution,, Compositio Mathematica, 36 (1978), 225.
[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.
[30]	S. Troubetzkoy, Recurrence in generic staircases,, Discrete Contin. Dyn. Syst., 32 (2012), 1047. doi: 10.3934/dcds.2012.32.1047.
[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.
[32]	A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

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