American Institute of Mathematical Sciences

Advanced
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

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, American Mathematical Society, Providence, RI, 1997.

[2]

M. Boshernitzan, A condition for weak mixing of induced IETs, in "Dynamical Systems and Group Actions," Contemp. Math., 567, Amer. Math. Soc., Providence, RI, (2012), 53-65. 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), 3523-3535. 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 "Ergodic Theory," Contemporary Mathematics, 485, Amer. Math. Soc., Providence, RI, (2009), 45-70. 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), 4373-4428. 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), 491-515. 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), 117-142. 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), 191-213. 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), 1581-1600.

[14]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$, Isr. J. Math., 151 (2006), 281-321. 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), 223-244. doi: 10.1515/CRELLE.2011.052.

[16]

P. Hubert and T. A. Schmidt, An introduction to Veech surfaces, in "Handbook of dynamical systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 501-526. 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), 293-311. 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), 631-678. 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 "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 527-547. doi: 10.1016/S1874-575X(06)80032-9.

[22]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[23]

C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Mathematische Annalen, 333 (2005), 87-130. 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), 527-563.

[25]

C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122.

[26]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, preprint, arXiv:1201.3738, (2012).

[27]

K. Schmidt, "Cocycles of Ergodic Transformation Groups," Lecture Notes in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, India, 1977.

[28]

_____, A cylinder flow arising from irregularity of distribution, Compositio Mathematica, 36 (1978), 225-232.

[29]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. doi: 10.1007/BF02392354.

[30]

S. Troubetzkoy, Recurrence in generic staircases, Discrete Contin. Dyn. Syst., 32 (2012), 1047-1053. 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), 149-153. doi: 10.1017/S0143385700003862.

[32]

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. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997.

[2]

M. Boshernitzan, A condition for weak mixing of induced IETs, in "Dynamical Systems and Group Actions," Contemp. Math., 567, Amer. Math. Soc., Providence, RI, (2012), 53-65. 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), 3523-3535. 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 "Ergodic Theory," Contemporary Mathematics, 485, Amer. Math. Soc., Providence, RI, (2009), 45-70. 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), 4373-4428. 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), 491-515. 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), 117-142. 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), 191-213. 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), 1581-1600.

[14]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$, Isr. J. Math., 151 (2006), 281-321. 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), 223-244. doi: 10.1515/CRELLE.2011.052.

[16]

P. Hubert and T. A. Schmidt, An introduction to Veech surfaces, in "Handbook of dynamical systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 501-526. 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), 293-311. 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), 631-678. 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 "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 527-547. doi: 10.1016/S1874-575X(06)80032-9.

[22]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.

[23]

C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Mathematische Annalen, 333 (2005), 87-130. 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), 527-563.

[25]

C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122.

[26]

D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, preprint, arXiv:1201.3738, (2012).

[27]

K. Schmidt, "Cocycles of Ergodic Transformation Groups," Lecture Notes in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, India, 1977.

[28]

_____, A cylinder flow arising from irregularity of distribution, Compositio Mathematica, 36 (1978), 225-232.

[29]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237. doi: 10.1007/BF02392354.

[30]

S. Troubetzkoy, Recurrence in generic staircases, Discrete Contin. Dyn. Syst., 32 (2012), 1047-1053. 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), 149-153. doi: 10.1017/S0143385700003862.

[32]

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.

[1]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019
[2]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[3]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[4]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
[5]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008
[6]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[7]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[8]

José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[9]

Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[10]

David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[11]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[12]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[13]

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[14]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[15]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[16]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
[17]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[18]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073
[19]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks and Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437
[20]

Wooyeon Kim, Seonhee Lim. Notes on the values of volume entropy of graphs. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5117-5129. doi: 10.3934/dcds.2020221

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy