×

zbMATH — the first resource for mathematics

Local instability of orbits in polygonal and polyhedral billiards. (English) Zbl 0924.58043
Summary: We classify when local instability of orbits of closely points can occur for billiards in two-dimensional polygons, for billiards inside three-dimensional polyhedra and for geodesic flows on surfaces of three-dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.

MSC:
37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [BKM] Boldrighini, C., Keane, M., Marchetti, F.: Billiards in polygons. Ann. Prob.6, 532–540 (1978) · Zbl 0377.28014 · doi:10.1214/aop/1176995475
[2] [B] Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc.153, 401–414 (1971) · Zbl 0212.29201 · doi:10.1090/S0002-9947-1971-0274707-X
[3] [CGa] Chernov, N.I., Galperin, G.A.: Billiards and chaos. Moscow: Znania 1991 (in Russian)
[4] [C] Coxeter, H.M.S.: Introduction to geometry. New York: Wiley 1961 · Zbl 0095.34502
[5] [F] Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton NJ: Princeton University Press 1981 · Zbl 0459.28023
[6] [Ga] Galperin, G.A.: Nonperiodic and noteverywhere dense billiard trajectories in convex polygons and polyhedrons. Commun. Math. Phys.91, 187–211 (1983). · Zbl 0529.70001 · doi:10.1007/BF01211158
[7] [GaSV] Galperin, G.A., Stepin, A.M., Vorobetz, Ya.B.: Periodic billiard trajectories in polygons: the mechanism of their appearance. Russian Math. Surveys47, 5–80 (1992) · Zbl 0777.58031 · doi:10.1070/RM1992v047n03ABEH000893
[8] [GaZ] Galperin, G.A., Zemlyakov, A.N.: Mathematical billiards. Moscow: Nauka 1990 (in Russian)
[9] [Gu1] Gutkin, E.: Billiards on almost integrable polyhedral surfaces. Erg. Th. Dyn. Sys.4, 569–584 (1984) · Zbl 0569.58028
[10] [Gu2] Gutkin, E.: Billiards in polygons. Physica D19, 311–333 (1986) · Zbl 0593.58016 · doi:10.1016/0167-2789(86)90062-X
[11] [K] Katok, A.: The Growth rate for the number of singular and periodic orbits for a polygonal billiard. Commun. Math. Phys.111, 151–160 (1987) · Zbl 0631.58020 · doi:10.1007/BF01239021
[12] [PP] Pesin, Ya.B., Pitskel, B.S.: Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl.18, 307–318 (1984) · Zbl 0567.54027 · doi:10.1007/BF01083692
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.