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Billiards in ideal hyperbolic polygons

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  • 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.
    Mathematics Subject Classification: Primary: 37D40, 53A35; Secondary: 37C27.


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  • [1]

    H. Ahmedov, Casimir effect in hyperbolic polygons, J. Phys. A, 40 (2007), 10611-10623.doi: 10.1088/1751-8113/40/34/016.


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


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


    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.


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


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


    T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Classical Quantum Gravity, 20 (2003), R145-R22.doi: 10.1088/0264-9381/20/9/201.


    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.


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


    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.


    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.


    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


    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.


    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.


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


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


    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.


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


    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.


    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.


    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.


    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.


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


    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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