On the ergodicity of many-dimensional focusing billiards
Annales de l'I.H.P. Physique théorique, Tome 68 (1998) no. 4, pp. 421-448.
@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  - 
%0 Journal Article
%A Bunimovich, Leonid A.
%A Rehacek, Jan
%T On the ergodicity of many-dimensional focusing billiards
%J Annales de l'I.H.P. Physique théorique
%D 1998
%P 421-448
%V 68
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/item/AIHPA_1998__68_4_421_0/
%G en
%F AIHPA_1998__68_4_421_0
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] D.V. Anosov, Geodesic Flows on Riemannian Manifolds with Negative Curvature, Proc. Steklov Inst., 90, 1967. | MR | Zbl

[AS] D.V. Anosov, Ya G. Sinai, Certain Smooth Ergodic Systems, Russ. Math. Surveys 22, 1967, pp. 103-167. | MR | Zbl

[B1] L.A. Bunimovich, On the Ergodic Properties of Nowhere Dispersing Billards, Comm. Math. Phys., Vol. 65, 1979, pp. 295-312. | MR | Zbl

[B2] L.A. Bunimovich, Many-Dimensional Nowhere Dispersing Billiards with chaotic behavior, Physica D, 33, 1988, pp.58-64. | MR | Zbl

[B3] L.A. Bunimovich, 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] K. Burns, M. Gerber, Real Analytic Bernoulli Geodesic Flows on S2. Ergod. Th. Dyn. Sys., 9, 1989, pp. 27-45. | MR | Zbl

[BR1] L.A. Bunimovich, J. Rehacek, Nowhere Dispersing 3D Billiards with Non-vanishing Lyapunov Exponents, Comm. Math. Phys., 189, 1997, pp. 729-758. | MR | Zbl

[BR2] L.A. Bunimovich, J. Rehacek, How Many Dimensional Stadia Look Like, Preprint, 1997. | Zbl

[BS] L.A. Bunimovich, Ya G. Sinai, On the Fundamental Theorem of the Theory of Dispersing Billiards, Math. Sb., 90, 1973, pp. 415-431. | MR | Zbl

[C] H. Coddington, Treatise on Reflection and Retraction of Light, Simpkin & Marshall, London 1829.

[Ch] N.I. Chernov, Local Ergodicity of Hyperbolic Systems with Singularities, Funct. Anal. Appl., 27, 1993, pp. 51-54. | MR | Zbl

[ChH] N.I. Chernov, C. Haskell, Nonuniformly Hyperbolic K-Systems are Bernoulli, Ergodic Theory and Dynamical Systems, 16, 1996, pp. 19-44. | MR | Zbl

[ChT] N.I. Chernov, S. Troubetzkoy, Ergodicity of Billiards in Polygons with Pockets, preprint 1998. | MR | Zbl

[D1] V.J. Donnay, Geodesic flow on the two-sphere. I: Positive measure entropy. Ergod. Th. Dyn. Sys., Vol. 8, 1988, pp. 531-553. | MR | Zbl

[D2] V.J. Donnay, 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] G. Gallavotti, D.S. Ornstein, Billiards and Bernoulli Schemes, Comm. Math. Phys., 38, 1974, pp. 83-101. | MR | Zbl

[H] E. Hopf, Statistik der Geodätischen Linien in Mannigfaltigkeiten Negativer Krummung, Ber. Verh. Sächs. Akad. Wiss., 91, 1939, pp. 261-304. | JFM | MR | Zbl

[KS] A. Katok, J.M. Strelcyn, Smooth Maps with Singularities. Invariant Manifolds, Entropy and Billiards, Lect. Notes in Math. 1222, Berlin, Springer 1986. | Zbl

[KSS] A. Krámli, N. Simányi and D. Szász, A Transversal Fundamental Theorem for Semi-Dispersing Billiards, Comm. Math. Phys., 129, 1990, pp. 535-560. | MR | Zbl

[L] V.F. Lazutkin, 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] C. Liverani, M. Wojtkowski, Ergodicity in Hamiltonian Systems, Dynamics Reported, 4, 1993, pp. 130-302. | MR | Zbl

[M] R. Markarian, New Ergodic Billiards: Exact Results, Nonlinearity, Vol. 6, 1993, pp. 819-841. | MR | Zbl

[S1] Ya G. Sinai, Dynamical Systems with Elastic Reflections, Russ. Math. Surveys, 25, 1970, pp. 137-189.

[S2] Ya G. Sinai, Dynamical Systems with Countably Multiple Lebesgue Spectrum II., Izv. Akad. Nauk. USSR. Ser. Math., 30, 1966, pp. 15-68. | MR | Zbl

[SCh] Ya G. Sinai, N.I. Chernov, Ergodic Properties of Certain Systems of Two-dimensional Discs and Three-dimensional Balls, Russ. Math. Surveys, 42, 1987, pp. 181-207. | MR | Zbl

[Sm] S. Smale, Differentiable Dynamical Systems, Bull. of AMS. 73, 1967, pp. 747-817. | MR | Zbl

[W1] M.P. Wojtkowski, Linearly Stable Orbits in 3 Dimensional Billiards. Comm. Math. Phys., 129, 1990, pp. 319-327. | MR | Zbl

[W2] M.P. Wojtkowski, Invariant Families of Cones and Lyapunov Exponents. Ergod. Th. & Dyn. Sys. 5, 1985, pp. 145-161. | Zbl