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Billiards in polygons. (English) Zbl 0593.58016
This paper surveys the results on billiards in polygons, as a typical example for the parabolic case. (Elliptic and hyperbolic billiards will be discussed in a forthcoming survey paper by A. Katok and M. Wojtkowski.) The main questions of interest are ergodicity, topological transitivity and existence of periodic orbits.
The results on rational and almost integrable billiards are contained in section 4, those for irrational billiards in section 5.
Reviewer: M.Denker

MSC:
37A99 Ergodic theory
28D10 One-parameter continuous families of measure-preserving transformations
54H20 Topological dynamics (MSC2010)
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[1] Arnold, V.I., Mathematical methods of classical mechanics, (1978), Springer Berlin · Zbl 0386.70001
[2] Boldrighini, C.; Keane, M.; Marchetti, F., Billiards in polygons, Ann. prob., 6, 532-540, (1978) · Zbl 0377.28014
[3] Boshernitzan, M., A condition for minimal interval exchange maps to be uniquely ergodic, Duke math. J., (1985), to appear in · Zbl 0602.28009
[4] Bunimovich, L.A., On the ergodic properties of nowhere dispersing billiards, Comm. math. phys., 65, 295-312, (1976) · Zbl 0421.58017
[5] De Verdiere, Y.Colin, Sur LES longueurs des trajectories periodiques d’un billiard, (), 122-139
[6] Fox, R.H.; Kershner, R.B., Concerning the transitive properties of geodesics on a rational polyhedron, Duke math. J., 2, 147-150, (1936) · JFM 62.0817.01
[7] Galperin, G.A., Nonperiodic and not everywhere dense billiards trajectories in convex polygons and polyhedrons, Comm. math. phys., 91, 187-211, (1983) · Zbl 0529.70001
[8] Galperin, G.A., Elastic collisions of particles on a line, Russ. math. surveys, 33, 211-212, (1978)
[9] E. Gutkin, Geodesic flows on rational polyhedra, preprint. · Zbl 1203.37037
[10] Gutkin, E., Billiards on almost integrable polyhedral surfaces, Ergod. th. and dynam. sys., 4, 569-584, (1984) · Zbl 0569.58028
[11] E. Gutkin, private communication.
[12] Henyey, F.S.; Pomphrey, N., The autocorrelation function of a pseudointegrable system, Physica, 6D, 78-94, (1982) · Zbl 1194.37127
[13] A. Katok, The growth rate for the number of saddle connections for a polygonal billiard, preprint. · Zbl 0631.58020
[14] A. Katok and M. Wojtkowski, Billiards, in preparation.
[15] Katok, A., Ergodicity of generic irrational billiards, (), 13
[16] Keane, M., Coding problems in ergodic theory, ()
[17] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, preprint. · Zbl 0637.58010
[18] Kornfeld, I.P.; Fomin, S.V.; Sinai, Y.G., Ergodic theory, (1982), Springer Berlin
[19] Lasutkin, V.F., The existence of caustics for a billiard problem in a convex domain, Math. of USSR-izv., 7, 185-214, (1973)
[20] H. Masur, Rational billiards have periodic orbits, preprint.
[21] Mayer, A., Trajectories on the orientable surfaces, Mat. sbornik, 12, 71-84, (1943), (in Russian) · Zbl 0063.03856
[22] Richens, P.T.; Berry, M.V., Pseudointegrable systems in classical and quantum mechanics, Physica, 2D, 495-512, (1981) · Zbl 1194.37150
[23] Sataev, E.A., On the number of invariant measures for flows on orientable surfaces, Math. of USSR-izv., 9, 813-830, (1976) · Zbl 0336.28007
[24] Sinai, Y.G., Dynamical systems with elastic reflections. ergodic properties of dispersing billiards, Russ. math. surveys, 25, 137-189, (1970) · Zbl 0263.58011
[25] Sinai, Y.G., Billiard trajectories in a polyhedral angle, Russ. math. surveys, 33, 229-230, (1978) · Zbl 0415.28021
[26] Veech, W., Finite group extensions of irrational rotations, Isr. J. math., 21, 240-259, (1975) · Zbl 0334.28014
[27] M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents, preprint. · Zbl 0602.58029
[28] A. Katok and J.M. Strelcyn, Smooth maps with singularities: invariant manifolds, entropy and billiards, Universit√© Paris-Nord, Prepublications math., n 56. · Zbl 0658.58001
[29] Zemlyakov, A.N.; Katok, A.B., Topological transitivity of billiards in polygon,, Math. notes, 18, 760-764, (1975) · Zbl 0323.58012
[30] Katok, A., Interval exchange transformations and some special flows are not mixing, Isr. J. math., 35, 301-310, (1980) · Zbl 0437.28009
[31] Kerkhoff, S.; Masur, H.; Smillie, J., A rational billiard flow is uniquely ergodic in almost every direction, Bull. AMS, 13, 141-142, (1985) · Zbl 0574.58020
[32] M. Bishernitzan, Rank two interval exchange maps, in preparation.
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