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Lyapunov spectrum for geodesic flows of rank 1 surfaces
A note on integrable mechanical systems on surfaces
1. | Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48859, United States |
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein. Corrected reprint of the second (1989) edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 199?. |
[2] |
M. Bialy, Integrable geodesic flows on surfaces, Geom. Funct. Anal., 20 (2010), 357-367.
doi: 10.1007/s00039-010-0069-4. |
[3] |
A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Z., 246 (2004), 213-236.
doi: 10.1007/s00209-003-0596-x. |
[4] |
L. T. Butler, Invariant fibrations of geodesic flows, Topology, 44 (2005), 769-789.
doi: 10.1016/j.top.2005.01.004. |
[5] |
_______, An optical Hamiltonian and obstructions to integrability, Nonlinearity, 19 (2006), 2123-2135.
doi: 10.1088/0951-7715/19/9/008. |
[6] |
_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces, Preprint arXiv:1208.1460v1 (2012), 1-7. |
[7] |
P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251. |
[8] |
E. Glasmachers and G. Knieper, Characterization of geodesic flows on $T^2$ with and without positive topological entropy, Geom. Funct. Anal., 20 (2010), 1259-1277.
doi: 10.1007/s00039-010-0087-2. |
[9] |
_______, Minimal geodesic foliation on $T^2$ in case of vanishing topological entropy, J. Topol. Anal., 3 (2011), 511-520.
doi: 10.1142/S1793525311000623. |
[10] |
V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302. |
[11] |
______, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 31, Springer-Verlag, Berlin, 1996, Translated from the Russian manuscript by S. V. Bolotin, D. Treshchev and Yuri Fedorov. |
[12] |
Y. Long, Collection of problems proposed at International Conference on Variational Methods, Front. Math. China, 3 (2008), 259-273.
doi: 10.1007/s11464-008-0017-x. |
[13] |
N. N. Nehorošev, Action-angle variables, and their generalizations, Trudy Moskov. Mat. Obšč., 26 (1972), 181-198. |
[14] |
I. A. Taĭmanov, Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 429-435, 448. |
show all references
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein. Corrected reprint of the second (1989) edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 199?. |
[2] |
M. Bialy, Integrable geodesic flows on surfaces, Geom. Funct. Anal., 20 (2010), 357-367.
doi: 10.1007/s00039-010-0069-4. |
[3] |
A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Z., 246 (2004), 213-236.
doi: 10.1007/s00209-003-0596-x. |
[4] |
L. T. Butler, Invariant fibrations of geodesic flows, Topology, 44 (2005), 769-789.
doi: 10.1016/j.top.2005.01.004. |
[5] |
_______, An optical Hamiltonian and obstructions to integrability, Nonlinearity, 19 (2006), 2123-2135.
doi: 10.1088/0951-7715/19/9/008. |
[6] |
_______, A generalization of Kozlov's theorem on integrable mechanical systems on surfaces, Preprint arXiv:1208.1460v1 (2012), 1-7. |
[7] |
P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251. |
[8] |
E. Glasmachers and G. Knieper, Characterization of geodesic flows on $T^2$ with and without positive topological entropy, Geom. Funct. Anal., 20 (2010), 1259-1277.
doi: 10.1007/s00039-010-0087-2. |
[9] |
_______, Minimal geodesic foliation on $T^2$ in case of vanishing topological entropy, J. Topol. Anal., 3 (2011), 511-520.
doi: 10.1142/S1793525311000623. |
[10] |
V. V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302. |
[11] |
______, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 31, Springer-Verlag, Berlin, 1996, Translated from the Russian manuscript by S. V. Bolotin, D. Treshchev and Yuri Fedorov. |
[12] |
Y. Long, Collection of problems proposed at International Conference on Variational Methods, Front. Math. China, 3 (2008), 259-273.
doi: 10.1007/s11464-008-0017-x. |
[13] |
N. N. Nehorošev, Action-angle variables, and their generalizations, Trudy Moskov. Mat. Obšč., 26 (1972), 181-198. |
[14] |
I. A. Taĭmanov, Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 429-435, 448. |
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