July  2011, 29(3): 893-908. doi: 10.3934/dcds.2011.29.893

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

Received  November 2009 Revised  June 2010 Published  November 2010

We consider billiard trajectories in ideal hyperbolic polygons and present a conjecture about the minimality of the average length of cyclically related billiard trajectories in regular hyperbolic polygons. We prove this conjecture in particular cases, using geometric and algebraic methods from hyperbolic geometry.
Citation: Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893
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.

[1]

W. Patrick Hooper. Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons. Journal of Modern Dynamics, 2007, 1 (4) : 649-663. doi: 10.3934/jmd.2007.1.649

[2]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155

[3]

Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132

[4]

Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903

[5]

Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1

[6]

R. Bartolo, Anna Maria Candela, J.L. Flores, Addolorata Salvatore. Periodic trajectories in plane wave type spacetimes. Conference Publications, 2005, 2005 (Special) : 77-83. doi: 10.3934/proc.2005.2005.77

[7]

Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719

[8]

Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31

[9]

Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino. Hartman-Grobman theorems along hyperbolic stationary trajectories. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 281-292. doi: 10.3934/dcds.2007.17.281

[10]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505

[11]

Gunter M. Ziegler. Projected products of polygons. Electronic Research Announcements, 2004, 10: 122-134.

[12]

Paweł Pilarczyk. Topological-numerical approach to the existence of periodic trajectories in ODE's. Conference Publications, 2003, 2003 (Special) : 701-708. doi: 10.3934/proc.2003.2003.701

[13]

Alexander M. Krasnosel'skii, Edward O'Grady, Alexei Pokrovskii, Dmitrii I. Rachinskii. Periodic canard trajectories with multiple segments following the unstable part of critical manifold. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 467-482. doi: 10.3934/dcdsb.2013.18.467

[14]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911

[15]

Anton Izosimov. Pentagrams, inscribed polygons, and Prym varieties. Electronic Research Announcements, 2016, 23: 25-40. doi: 10.3934/era.2016.23.004

[16]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[17]

Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615

[18]

Morimichi Kawasaki, Ryuma Orita. Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories. Journal of Modern Dynamics, 2017, 11: 313-339. doi: 10.3934/jmd.2017013

[19]

Zhanying Yang. Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method. Communications on Pure and Applied Analysis, 2014, 13 (1) : 249-272. doi: 10.3934/cpaa.2014.13.249

[20]

Xiaolong Li, Katsutoshi Shinohara. On super-exponential divergence of periodic points for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1707-1729. doi: 10.3934/dcds.2021169

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (195)
  • HTML views (0)
  • Cited by (0)

[Back to Top]