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On uniform convergence in ergodic theorems for a class of skew product transformations
Billiards in ideal hyperbolic polygons
1. | Department of Math. Sciences, Durham University, Durham DH1 3LE, United Kingdom |
2. | Technische Universität Dortmund, Fakultät für Mathematik, 44221 Dortmund, Germany |
References:
[1] |
H. Ahmedov, Casimir effect in hyperbolic polygons, J. Phys. A, 40 (2007), 10611-10623.
doi: 10.1088/1751-8113/40/34/016. |
[2] |
J. W. Anderson, "Hyperbolic Geometry," Second edition, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd. London, 2005. |
[3] |
E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamburger Mathematische Abhandlungen, 3 (1924), 170-175.
doi: 10.1007/BF02954622. |
[4] |
M. Bauer and A. Lopes, A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$, Discrete Contin. Dynam. Systems, 3 (1997), 107-116. |
[5] |
A. F. Beardon, "The Geometry of Discrete Groups," Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983. |
[6] |
P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progress in Mathematics, 106, Birkhäuser Boston, Inc., Boston, MA, 1992. |
[7] |
T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Classical Quantum Gravity, 20 (2003), R145-R22.
doi: 10.1088/0264-9381/20/9/201. |
[8] |
V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234.
doi: 10.1016/S0393-0440(02)00219-X. |
[9] |
C. Foltin, Billiards with positive topological entropy, Nonlinearity, 15 (2002), 2053-2076.
doi: 10.1088/0951-7715/15/6/314. |
[10] |
M.-J. Giannoni and D. Ullmo, Coding chaotic billiards. I. Noncompact billiards on a negative curvature manifold, Phys. D, 41 (1990), 371-390.
doi: 10.1016/0167-2789(90)90005-A. |
[11] |
M.-J. Giannoni and D. Ullmo, Coding chaotic billiards. II. Compact billiards defined on the pseudosphere, Phys. D., 84 (1995), 329-356.
doi: 10.1016/0167-2789(95)00064-B. |
[12] |
Ch. Grosche, Energy fluctuation analysis in integrable billiards in hyperbolic geometry, in "Emerging Applications of Number Theory" (Minneapolis, MN, 1996), IMA Vol. Math. Appl., 109, Springer, New York, (1999), 269-289 |
[13] |
B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Commun. Math. Phys., 208 (1999), 65-90.
doi: 10.1007/s002200050748. |
[14] |
E. Gutkin and S. Tabachnikov, Complexity of piecewise convex transfomrationen to polygonal billiards on surfaces of constant curvature, Mosc. Math. J., 6 (2006), 673-701, 772. |
[15] |
M. Gutzwiller, "Chaos in Classical and Quantum Mechanics," Interdisciplinary Applied Mathematics, 1, Springer-Verlag, New York, 1990. |
[16] |
J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesicques, J. Math. Pures App. (5), 4 (1898), 27-73. |
[17] |
L. Karp and N. Peyerimhoff, Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane, Arch. Math. (Basel), 79 (2002), 223-231. |
[18] |
S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132. |
[19] |
C. W. Misner, The mixmaster cosmological metrics, in "Deterministic Chaos in General Relativity" (Kananaskis, AB, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 332, Plenum, New York, (1994), 317-328. |
[20] |
Ch. Schmit, "Quantum and Classical Properties of Some Billiards on the Hyperbolic Plane," Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, (1991), 331-370. |
[21] |
C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80.
doi: 10.1112/jlms/s2-31.1.69. |
[22] |
S. Tabachnikov, A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards, Geom. Dedicata, 129 (2007), 83-87.
doi: 10.1007/s10711-007-9196-y. |
[23] |
S. Tabachnikov and F. Dogru, Dual billiards, Math. Intelligencer, 27 (2005), 18-25.
doi: 10.1007/BF02985854. |
[24] |
A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.
doi: 10.1016/0393-0440(90)90021-T. |
show all references
References:
[1] |
H. Ahmedov, Casimir effect in hyperbolic polygons, J. Phys. A, 40 (2007), 10611-10623.
doi: 10.1088/1751-8113/40/34/016. |
[2] |
J. W. Anderson, "Hyperbolic Geometry," Second edition, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd. London, 2005. |
[3] |
E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamburger Mathematische Abhandlungen, 3 (1924), 170-175.
doi: 10.1007/BF02954622. |
[4] |
M. Bauer and A. Lopes, A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$, Discrete Contin. Dynam. Systems, 3 (1997), 107-116. |
[5] |
A. F. Beardon, "The Geometry of Discrete Groups," Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983. |
[6] |
P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progress in Mathematics, 106, Birkhäuser Boston, Inc., Boston, MA, 1992. |
[7] |
T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Classical Quantum Gravity, 20 (2003), R145-R22.
doi: 10.1088/0264-9381/20/9/201. |
[8] |
V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234.
doi: 10.1016/S0393-0440(02)00219-X. |
[9] |
C. Foltin, Billiards with positive topological entropy, Nonlinearity, 15 (2002), 2053-2076.
doi: 10.1088/0951-7715/15/6/314. |
[10] |
M.-J. Giannoni and D. Ullmo, Coding chaotic billiards. I. Noncompact billiards on a negative curvature manifold, Phys. D, 41 (1990), 371-390.
doi: 10.1016/0167-2789(90)90005-A. |
[11] |
M.-J. Giannoni and D. Ullmo, Coding chaotic billiards. II. Compact billiards defined on the pseudosphere, Phys. D., 84 (1995), 329-356.
doi: 10.1016/0167-2789(95)00064-B. |
[12] |
Ch. Grosche, Energy fluctuation analysis in integrable billiards in hyperbolic geometry, in "Emerging Applications of Number Theory" (Minneapolis, MN, 1996), IMA Vol. Math. Appl., 109, Springer, New York, (1999), 269-289 |
[13] |
B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Commun. Math. Phys., 208 (1999), 65-90.
doi: 10.1007/s002200050748. |
[14] |
E. Gutkin and S. Tabachnikov, Complexity of piecewise convex transfomrationen to polygonal billiards on surfaces of constant curvature, Mosc. Math. J., 6 (2006), 673-701, 772. |
[15] |
M. Gutzwiller, "Chaos in Classical and Quantum Mechanics," Interdisciplinary Applied Mathematics, 1, Springer-Verlag, New York, 1990. |
[16] |
J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesicques, J. Math. Pures App. (5), 4 (1898), 27-73. |
[17] |
L. Karp and N. Peyerimhoff, Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane, Arch. Math. (Basel), 79 (2002), 223-231. |
[18] |
S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132. |
[19] |
C. W. Misner, The mixmaster cosmological metrics, in "Deterministic Chaos in General Relativity" (Kananaskis, AB, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 332, Plenum, New York, (1994), 317-328. |
[20] |
Ch. Schmit, "Quantum and Classical Properties of Some Billiards on the Hyperbolic Plane," Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, (1991), 331-370. |
[21] |
C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80.
doi: 10.1112/jlms/s2-31.1.69. |
[22] |
S. Tabachnikov, A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards, Geom. Dedicata, 129 (2007), 83-87.
doi: 10.1007/s10711-007-9196-y. |
[23] |
S. Tabachnikov and F. Dogru, Dual billiards, Math. Intelligencer, 27 (2005), 18-25.
doi: 10.1007/BF02985854. |
[24] |
A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.
doi: 10.1016/0393-0440(90)90021-T. |
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