Finiteness in polygonal billiards on hyperbolic plane
DOI:
https://doi.org/10.12775/TMNA.2021.003Keywords
Hyperbolic plane, polygonal billiards, pointed geodesics, subshifts of finite type, Hausdorff metric, space of all subshiftsAbstract
J. Hadamard studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating ``Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in ``rational polygons'' on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just `Subshifts of Finite Type' or their dense subsets. We further show that `Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.References
A. Abrams and S. Katok, Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature, Studia Math. 246 (2019), 167–202.
R. Adler and L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull. Amer. Math. Soc. 25 (1991), 229–334.
E. Akin, The general topology of dynamical systems, Graduate Studies in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1993.
E. Akin, J. Auslander and A. Nagar, Variations on the concept of topological transitivity, Studia Math. 235 (2016), 225–249.
E. Akin, J. Auslander and A. Nagar, Dynamics of induced systems, Ergodic Theory Dynam. Systems 37 (2017), no. 7, 2034–2059.
E. Akin and J. Wiseman, Varieties of mixing, Trans. Amer. Math. Soc. 372 (2019), no. 6, 4359–4390.
J.W. Anderson, Hyperbolic Geometry, Springer–Verlag London Limited, 2005.
A. Barwell, Ch. Good, R. Knight and B.E. Raines, A characterization of ω-limit sets in shift spaces, Ergodic Theory Dynam. Systems 30 (2010), 21–31.
A.F. Beardon, The Geometry of Discrete Groups, Springer–Verlag New York Inc., 1983.
T. Bedford, M. Keane, C. Series, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, 1991.
L. Bunimovich, The ergodic properties of certain billiards, Funkc. Anal. Prilozen 8 (1974), 73–74. (in Russian)
D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
S. Castle, N. Peyerimhoff and K.F. Siburg, Billiards in ideal hyperbolic polygons, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 893–908.
N. Chernov and R. Markarian, Chaotic Billiards, American Mathematical Society, 2006.
M.-J. Giannoni and D. Ullmo, Coding chaotic billiards I, Physica D. 41 (1990), 371–390.
M.-J. Giannoni and D. Ullmo, Coding Chaotic billiards II, Phys. D 84 (1995), 329–356.
W.H. Gottschalk and G.A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloquium Publications, vol. 36, 1955.
M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, based on the 1981 French original, transl. from the French by Sean Michael Bates, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999.
E. Gutkin, Billiards in polygons, Physica D 19 (1986), 311–333.
E. Gutkin, Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces, Regul. Chaotic Dyn. 15 (2010), no. 4–5, 482–503.
E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems, Chaos 22 (2012), no. 2, 116–128.
E. Gutkin and V. Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces, Geom. Dedicata 118 (2006), 185–208.
M. Gutzwiller, Classical quantization of a Hamiltonian with ergodic behavior, Phys. Rev. Lett. 45 (1980), no. 3, 150–153.
M. Gutzwiller, The geometry of quantum chaos, Physica Scripta T9 (1985), 184–192.
M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, vol. 1, Springer–Verlag, New York, 1990.
G.A. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc., 45 (1939), 241–260.
J.E. Hofer, Topological entropy for non compact spaces, Michijan Math. J. 21 (1974), 235–242.
A. Illanes and S.B. Nadler Jr., Hyperspaces: Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Mathematics, vol. 216, Marcel Dekker, New York, 1999.
A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes 18 (1975), 760–764.
S. Katok, Fuchsian Groups, The University of Chicago Press, 1992.
S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull Amer. Math. Soc. 44 (2007), 87–132.
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995.
E. Micheal, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951).
M. Morse and G. Hedlund, Symbolic dynamics, Amer. J. Math. 60 (1938), 815–866.
D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl. 156 (2008), no. 2, 251–261.
R.E. Schwartz, Obtuse triangular billiards I. Near the (2, 3, 6) triangle, Exp. Math. 15 (2006), no. 2, 161–182.
R.E. Schwartz, Obtuse triangular billiards II. Near the (2, 3, 6) triangle, Exp. Math. 18 (2009), no. 2, 137–171.
C. Series, Symbolic dynamics for geodesic flows, Acta Math. 146, (1981), 103–128.
C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynam. Systems 6 (1986), 601–625.
S. Tabachnikov, Geometry and Billiards, American Mathematical Society, 2005.
L.W. Tu, An Introduction to Manifolds, Springer, 2008.
J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, vol. 257, Kluwer, Dordrecht, 1993.
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Anima Nagar, Pradeep Singh
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0