-
Previous Article
Connecting orbits for families of Tonelli Hamiltonians
- JMD Home
- This Issue
-
Next Article
An algebraic characterization of expanding Thurston maps
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 |
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., (). 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. |
[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., (). 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. |
[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. |
[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. |
[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., (). 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. |
[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., (). Google Scholar |
[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). Google Scholar |
[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., (). 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. |
[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., (). 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. |
[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. |
[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. |
[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., (). 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. |
[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., (). Google Scholar |
[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). Google Scholar |
[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. |
[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 - A, 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 - A, 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 - A, 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 - A, 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 - A, 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 - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401 |
[13] |
Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119 |
[14] |
Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022 |
[15] |
Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004 |
[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] |
Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149 |
[19] |
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 |
[20] |
Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073 |
2018 Impact Factor: 0.295
Tools
Metrics
Other articles
by authors
[Back to Top]