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Compactly supported Hamiltonian loops with a non-zero Calabi invariant
Pseudo-Anosov eigenfoliations on Panov planes
1. | Clemson University, E-1b Martin Hall, Clemson, SC 29634, United States |
2. | Clemson University, O-229 Martin Hall, Clemson, SC 29634, United States |
Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.
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K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses,, to appear in \emph{Nonlinearity}., (). Google Scholar |
[16] |
K. Frączek and M. Schmoll, Dynamics on quadratic differentials in the determinant locus,, in preparation., (). Google Scholar |
[17] |
K. Frączek and C. Ulcigrai, Non-ergodic $\Z$-periodic billiards and infinite translation surfaces,, to appear in \emph{Inventiones Math.}, (). Google Scholar |
[18] |
K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles,, to appear in \emph{Communications in Mathematical Physics}, (). Google Scholar |
[19] |
J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum,, \arXiv{1307.3481v1}., (). Google Scholar |
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W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, \arXiv{0905.3736v1}., (). Google Scholar |
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, P. Hubert,, Oral communication., (). Google Scholar |
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Topology and its Applications, 161 (2014), 73-94.
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C. Johnson and M. Schmoll, Dynamics on Panov planes,, in final preparation., (). Google Scholar |
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[32] |
Duke Math. J., 133 (2006), 569-590.
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[33] |
J. Mod. Dyn., 3 (2009), 589-594.
doi: 10.3934/jmd.2009.3.589. |
[34] |
Topology Appl., 154 (2007), 2365-2375.
doi: 10.1016/j.topol.2007.01.021. |
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Ph.D. Thesis, The University of Chicago,, 2005. |
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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] |
preprint, 2011/2012. Google Scholar |
[2] |
Ergodic Theory and Dynamical Systems, 29 (2009), 1417-1449.
doi: 10.1017/S0143385708000783. |
[3] |
J. Amer. Math. Soc., 17 (2004), 871-908.
doi: 10.1090/S0894-0347-04-00461-8. |
[4] |
arXiv:1305.1104, (2013). Google Scholar |
[5] |
Ergodic Theory Dyn. Systems, 32 (2012), 491-515.
doi: 10.1017/S0143385711001003. |
[6] |
Journal of Modern Dynamics, 7 (2013), 1-29.
doi: 10.3934/jmd.2013.7.1. |
[7] |
preprint, arXiv:1107.1810v3, Annales Scientifiques de l'Ecole Normale Supérieure, 47 (2014), 28 pp. Google Scholar |
[8] |
Translated from the German by Michael J. Moravcsik, Reprint of the 1959 English edition, Dover Publications, Inc., New York, 1990. |
[9] |
A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\slr$ action on moduli space,, \arXiv{1302.3320}., (). Google Scholar |
[10] |
Publications Mathématiques de l'IHÉS, (2013), 1-127, arXiv:1112.5872. Google Scholar |
[11] |
Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012. |
[12] |
Translation from the 1979 French original by Djun M. Kim and Dan Margalit, Mathematical Notes, 48, Princeton University Press, Princeton, NJ, 2012. |
[13] |
J. Anal. Math., 112 (2010), 289-328.
doi: 10.1007/s11854-010-0031-2. |
[14] |
Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392. |
[15] |
K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses,, to appear in \emph{Nonlinearity}., (). Google Scholar |
[16] |
K. Frączek and M. Schmoll, Dynamics on quadratic differentials in the determinant locus,, in preparation., (). Google Scholar |
[17] |
K. Frączek and C. Ulcigrai, Non-ergodic $\Z$-periodic billiards and infinite translation surfaces,, to appear in \emph{Inventiones Math.}, (). Google Scholar |
[18] |
K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles,, to appear in \emph{Communications in Mathematical Physics}, (). Google Scholar |
[19] |
J. Grivaux and P. Hubert, Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum,, \arXiv{1307.3481v1}., (). Google Scholar |
[20] |
Duke Math. J., 103 (2000), 191-213.
doi: 10.1215/S0012-7094-00-10321-3. |
[21] |
J. Math. Phys., 21 (1980), 1802-1808.
doi: 10.1063/1.524633. |
[22] |
W. P. Hooper, The invariant measures of some infinite interval exchange maps,, \arXiv{1005.1902}., (). Google Scholar |
[23] |
W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, \arXiv{0905.3736v1}., (). Google Scholar |
[24] |
, P. Hubert,, Oral communication., (). Google Scholar |
[25] |
J. Reine Angew. Math., 656 (2011), 223-244.
doi: 10.1515/CRELLE.2011.052. |
[26] |
Compos. Math., 149 (2013), 1364-1380.
doi: 10.1112/S0010437X12000887. |
[27] |
Topology and its Applications, 161 (2014), 73-94.
doi: 10.1016/j.topol.2013.09.010. |
[28] |
C. Johnson and M. Schmoll, Dynamics on Panov planes,, in final preparation., (). Google Scholar |
[29] |
Duke Math. J., 66 (1992), 387-442.
doi: 10.1215/S0012-7094-92-06613-0. |
[30] |
in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089.
doi: 10.1016/S1874-575X(02)80015-7. |
[31] |
J. Amer. Math. Soc., 16 (2003), 857-885 (electronic).
doi: 10.1090/S0894-0347-03-00432-6. |
[32] |
Duke Math. J., 133 (2006), 569-590.
doi: 10.1215/S0012-7094-06-13335-5. |
[33] |
J. Mod. Dyn., 3 (2009), 589-594.
doi: 10.3934/jmd.2009.3.589. |
[34] |
Topology Appl., 154 (2007), 2365-2375.
doi: 10.1016/j.topol.2007.01.021. |
[35] |
Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431.
doi: 10.1090/S0273-0979-1988-15685-6. |
[36] |
Ph.D. Thesis, The University of Chicago,, 2005. |
[37] |
Invent. Math., 97 (1989), 533-683.
doi: 10.1007/BF01388890. |
[38] |
in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583.
doi: 10.1007/978-3-540-31347-2_13. |
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