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.  Google Scholar

[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.  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), 3523-3535. 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 "Ergodic Theory," Contemporary Mathematics, 485, Amer. Math. Soc., Providence, RI, (2009), 45-70. 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), 4373-4428. 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), 491-515. 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), 117-142. 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), 191-213. 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), 1581-1600. Google Scholar

[14]

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

[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.  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), 293-311. 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), 631-678. 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 "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 527-547. doi: 10.1016/S1874-575X(06)80032-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. J. Patterson, Diophantine approximation in Fuchsian groups, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527-563.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  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), 3523-3535. 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 "Ergodic Theory," Contemporary Mathematics, 485, Amer. Math. Soc., Providence, RI, (2009), 45-70. 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), 4373-4428. 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), 491-515. 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), 117-142. 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), 191-213. 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), 1581-1600. Google Scholar

[14]

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

[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.  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), 293-311. 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), 631-678. 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 "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 527-547. doi: 10.1016/S1874-575X(06)80032-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. J. Patterson, Diophantine approximation in Fuchsian groups, Philos. Trans. Roy. Soc. London Ser. A, 282 (1976), 527-563.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[13]

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

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[15]

Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
[16]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437
[17]

Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
[18]

Wooyeon Kim, Seonhee Lim. Notes on the values of volume entropy of graphs. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5117-5129. doi: 10.3934/dcds.2020221
[19]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
[20]

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

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences