@article{AIHPA_1998__68_4_421_0, author = {Bunimovich, Leonid A. and Rehacek, Jan}, title = {On the ergodicity of many-dimensional focusing billiards}, journal = {Annales de l'I.H.P. Physique th\'eorique}, pages = {421--448}, publisher = {Gauthier-Villars}, volume = {68}, number = {4}, year = {1998}, mrnumber = {1634311}, zbl = {0907.58037}, language = {en}, url = {http://www.numdam.org/item/AIHPA_1998__68_4_421_0/} }
TY - JOUR AU - Bunimovich, Leonid A. AU - Rehacek, Jan TI - On the ergodicity of many-dimensional focusing billiards JO - Annales de l'I.H.P. Physique théorique PY - 1998 SP - 421 EP - 448 VL - 68 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/item/AIHPA_1998__68_4_421_0/ LA - en ID - AIHPA_1998__68_4_421_0 ER -
Bunimovich, Leonid A.; Rehacek, Jan. On the ergodicity of many-dimensional focusing billiards. Annales de l'I.H.P. Physique théorique, Tome 68 (1998) no. 4, pp. 421-448. http://www.numdam.org/item/AIHPA_1998__68_4_421_0/
[A] Geodesic Flows on Riemannian Manifolds with Negative Curvature, Proc. Steklov Inst., 90, 1967. | MR | Zbl
,[AS] Certain Smooth Ergodic Systems, Russ. Math. Surveys 22, 1967, pp. 103-167. | MR | Zbl
, ,[B1] On the Ergodic Properties of Nowhere Dispersing Billards, Comm. Math. Phys., Vol. 65, 1979, pp. 295-312. | MR | Zbl
,[B2] Many-Dimensional Nowhere Dispersing Billiards with chaotic behavior, Physica D, 33, 1988, pp.58-64. | MR | Zbl
,[B3] On Absolutely Focusing Mirrors. In Ergodic Theory and Related Topics III. (ed. by U.Krengel), Lect. Notes in Math. 1514, New York: Springer-Verlag, 1992, pp. 62-82. | MR | Zbl
,[BG] Real Analytic Bernoulli Geodesic Flows on S2. Ergod. Th. Dyn. Sys., 9, 1989, pp. 27-45. | MR | Zbl
, ,[BR1] Nowhere Dispersing 3D Billiards with Non-vanishing Lyapunov Exponents, Comm. Math. Phys., 189, 1997, pp. 729-758. | MR | Zbl
, ,[BR2] How Many Dimensional Stadia Look Like, Preprint, 1997. | Zbl
, ,[BS] On the Fundamental Theorem of the Theory of Dispersing Billiards, Math. Sb., 90, 1973, pp. 415-431. | MR | Zbl
, ,[C] Treatise on Reflection and Retraction of Light, Simpkin & Marshall, London 1829.
,[Ch] Local Ergodicity of Hyperbolic Systems with Singularities, Funct. Anal. Appl., 27, 1993, pp. 51-54. | MR | Zbl
,[ChH] Nonuniformly Hyperbolic K-Systems are Bernoulli, Ergodic Theory and Dynamical Systems, 16, 1996, pp. 19-44. | MR | Zbl
, ,[ChT] Ergodicity of Billiards in Polygons with Pockets, preprint 1998. | MR | Zbl
, ,[D1] Geodesic flow on the two-sphere. I: Positive measure entropy. Ergod. Th. Dyn. Sys., Vol. 8, 1988, pp. 531-553. | MR | Zbl
,[D2] Geodesic flow on the two-sphere. II: Ergodicity. In Dynamical Systems (ed. by J. C. Alexander), Lect. Notes 1342, New York: Springer-Verlag 1988, pp. 112-153. | MR | Zbl
,[GO] Billiards and Bernoulli Schemes, Comm. Math. Phys., 38, 1974, pp. 83-101. | MR | Zbl
, ,[H] Statistik der Geodätischen Linien in Mannigfaltigkeiten Negativer Krummung, Ber. Verh. Sächs. Akad. Wiss., 91, 1939, pp. 261-304. | JFM | MR | Zbl
,[KS] Smooth Maps with Singularities. Invariant Manifolds, Entropy and Billiards, Lect. Notes in Math. 1222, Berlin, Springer 1986. | Zbl
, ,[KSS] A Transversal Fundamental Theorem for Semi-Dispersing Billiards, Comm. Math. Phys., 129, 1990, pp. 535-560. | MR | Zbl
, and ,[L] On the Existence of Caustics for the Billiard Ball Problem in a Convex Domain, Math. USSR Izv., Vol. 37, 1973, pp. 183-216. | Zbl
,[LW] Ergodicity in Hamiltonian Systems, Dynamics Reported, 4, 1993, pp. 130-302. | MR | Zbl
, ,[M] New Ergodic Billiards: Exact Results, Nonlinearity, Vol. 6, 1993, pp. 819-841. | MR | Zbl
,[S1] Dynamical Systems with Elastic Reflections, Russ. Math. Surveys, 25, 1970, pp. 137-189.
,[S2] Dynamical Systems with Countably Multiple Lebesgue Spectrum II., Izv. Akad. Nauk. USSR. Ser. Math., 30, 1966, pp. 15-68. | MR | Zbl
,[SCh] Ergodic Properties of Certain Systems of Two-dimensional Discs and Three-dimensional Balls, Russ. Math. Surveys, 42, 1987, pp. 181-207. | MR | Zbl
, ,[Sm] Differentiable Dynamical Systems, Bull. of AMS. 73, 1967, pp. 747-817. | MR | Zbl
,[W1] Linearly Stable Orbits in 3 Dimensional Billiards. Comm. Math. Phys., 129, 1990, pp. 319-327. | MR | Zbl
,[W2] Invariant Families of Cones and Lyapunov Exponents. Ergod. Th. & Dyn. Sys. 5, 1985, pp. 145-161. | Zbl
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