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

Billiards in ideal hyperbolic polygons

Simon Castle 1, , Norbert Peyerimhoff 1, and  Karl Friedrich Siburg 2,

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.
Keywords: Billiards, hyperbolic polygons, periodic trajectories..
Mathematics Subject Classification: Primary: 37D40, 53A35; Secondary: 37C2.
Citation: Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete & 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.  Google Scholar

[2]

J. W. Anderson, "Hyperbolic Geometry," Second edition, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd. London, 2005.  Google Scholar

[3]

E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamburger Mathematische Abhandlungen, 3 (1924), 170-175. doi: 10.1007/BF02954622.  Google Scholar

[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.  Google Scholar

[5]

A. F. Beardon, "The Geometry of Discrete Groups," Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.  Google Scholar

[6]

P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progress in Mathematics, 106, Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Foltin, Billiards with positive topological entropy, Nonlinearity, 15 (2002), 2053-2076. doi: 10.1088/0951-7715/15/6/314.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Gutzwiller, "Chaos in Classical and Quantum Mechanics," Interdisciplinary Applied Mathematics, 1, Springer-Verlag, New York, 1990.  Google Scholar

[16]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesicques, J. Math. Pures App. (5), 4 (1898), 27-73. Google Scholar

[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.  Google Scholar

[18]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Tabachnikov and F. Dogru, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. doi: 10.1007/BF02985854.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. W. Anderson, "Hyperbolic Geometry," Second edition, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd. London, 2005.  Google Scholar

[3]

E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Hamburger Mathematische Abhandlungen, 3 (1924), 170-175. doi: 10.1007/BF02954622.  Google Scholar

[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.  Google Scholar

[5]

A. F. Beardon, "The Geometry of Discrete Groups," Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.  Google Scholar

[6]

P. Buser, "Geometry and Spectra of Compact Riemann Surfaces," Progress in Mathematics, 106, Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Foltin, Billiards with positive topological entropy, Nonlinearity, 15 (2002), 2053-2076. doi: 10.1088/0951-7715/15/6/314.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Gutzwiller, "Chaos in Classical and Quantum Mechanics," Interdisciplinary Applied Mathematics, 1, Springer-Verlag, New York, 1990.  Google Scholar

[16]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodesicques, J. Math. Pures App. (5), 4 (1898), 27-73. Google Scholar

[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.  Google Scholar

[18]

S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 87-132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Tabachnikov and F. Dogru, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. doi: 10.1007/BF02985854.  Google Scholar

[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.  Google Scholar

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