The complete hyperbolicity of cylindric billiards

Ergodic Theory and Dynamical Systems

Volumn 22, Issue 1, 2002, Pages 281-302

  Simányi, Nándor ^a  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035981913     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0143385702000135     Document Type: Article

References (29)

1. [Ba] P. Bálint. Chaotic and ergodic properties of cylindric billiards. Ergod. Th. & Dynam. Sys. 19 (1999), 1127-1156.
2. [Bu] L.A. Bunimovich. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65 (1979), 295-312.
3. [BFK] D. Barago, S. Ferleger and A. Kononenko. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. Math. 147 (1998), 695-708.
4. [CH] N.I. Chernov and C. Haskell. Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. & Dynam. Sys. 16 (1996), 19-44.
5. [Ga] G. Galperin. On systems of locally interacting and repelling particles moving in space. Trudy MMO 43 (1981), 142-196.
6. [Hed] G.A. Hedlund. The dynamics of geodesic flows. Bull. Amer. Math. Soc. 45 (1939), 241-260.
7. [Hel] S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic, 1978.
8. [Ho] E. Hopf. Statistik der geodetischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261-304.
9. [KS] A. Katok and J.-M. Strelcyn. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, 1986.
10. [KSSz89] A. Krámli, N. Simányi and D. Szász. Ergodic properties of semi-dispersing billiards I. Two cylindric scatterers ia the 3-D Torus. Nonlinearity 2 (1989), 311-326.
11. [KSSz90] A. Krámli, N. Simányi and D. Szász. A 'Transversal' fundamental theorem for semi-dispersing billiards. Commun. Math. Phys. 129 (1990), 535-560.
12. [KSSz91] A. Krámli, N. Simányi and D. Szász. The K-property of three billiard balls. Ann. Math. 133 (1991 ), 37-72.
13. [KSSz92] A. Krámli, N. Simányi and D. Szász. The K-property of four billiard balls. Commun. Math. Phys. 144 (1992), 107-148.
14. [OW] D. Ornstein and B. Weiss. On the Bernoulli nature of systems with some hyperbolic structure. Ergod. Th. & Dynam. Sys. 18 (1998), 441-456.
15. [Pe] Ya. Pesin. Characteristic exponents and smooth ergodic theory. Russian Math. Surveys 32 (1977), 55-114.
16. [Sim92] N. Simányi. The K-property of N billiard balls I. Invent. Math. 108 (1992), 521-548; The K-property of N billiard balls II. Invent. Math. 110 (1992), 151-172.
17. [Sim92] N. Simányi. The K-property of N billiard balls I. Invent. Math. 108 (1992), 521-548; The K-property of N billiard balls II. Invent. Math. 110 (1992), 151-172.
18. [Sim99] N. Simányi. Ergodicity of hard spheres in a box. Ergod. Th. & Dynam. Sys. 19 (1999), 741-766.
19. [SSz94] N. Simányi and D. Szász. The K-property of 4-D billiards with non-orthogonal cylindric scatterers. J. Stat. Phys. 76(1/2) (1994), 587-604.
20. [SSz99] N. Simányi and D. Szász. Hard ball systems are completely hyperbolic. Ann. Math. 149 (1999), 35-96.
21. [SSz00] N. Simányi and D. Szász. Non-integrability of cylindric billiards and transitive lie group actions, Ergod. Th. & Dynam. Sys. 20 (2000), 593-610.
22. [Sin63] Ya.G. Sinai. On the foundation of the ergodic hypothesis for a dynamical system of statistical mechanics. Sov. Math. Dokl. 4 (1963), 1818-1822.
23. [Sin70] Ya.G. Sinai. Dynamical systems with elastic reflections. Russian Math. Surveys 25(2) (1970), 137-189.
24. [SCh] Ya.G. Sinai and N.I. Chernov. Ergodic properties of certain systems of 2-D discs and 3-D balls. Russian Math. Surveys 42(3) (1987), 181-207.
25. [Sz93] D. Szász. Ergodicity of classical billiard balls. Physica A 194 (1993), 86-92.
26. [Sz94] D. Szász. The K-property of 'orthogonal' cylindric billiards. Commun. Math. Phys. 160 (1994), 581-597.
27. [Va] L.N. Vaserstein. On systems of particles with finite range and/or repulsive interactions. Commun. Math. Phys. 69 (1979), 31-56.
28. [Wo85] M. Wojtkowski. Invariant families of cones and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 145-161.
29. [Wo86] M. Wojtkowski. Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105 (1986), 391-414.