Hard ball systems are completely hyperbolic

Annals of Mathematics

Volumn 149, Issue 1, 1999, Pages 35-96

  Simányi, Nándor ^a   Szász, Domokos ^b  

^a UNIVERSITY OF SZEGED   (Hungary)

^b ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS   (Hungary)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033241652     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/121019     Document Type: Article

References (33)

1. L. A. BUNIMOVICH, On the ergodic properties of nowhere dispersing billiards, Comm. Math. Phys. 65 (1979), 295-312.
2. D. BURAGO, S. FERLEGER, and A. KONONENKO, Uniform estimates on the number of collisions in semi-dispersing billiards, Annals of Math. 147 (1998), 695-708.
3. L. A. BUNIMOVICH and YA. G. SINAI, The fundamental theorem of the theory of scattering billiards Math. USSR-Sb. 19 (1973), 407-423.
4. I. P. CORNFELD, S. V. FOMIN, and YA. G. SINAI, Ergodic Theory, Springer-Verlag, New York, 1982.
5. N. I. CHERNOV and C. HASKELL, Nonuniformly hyperbolic K-systems are Bernoulli Ergodic Theory & Dynam. Systems 16 (1996), 19-44.
6. G. A. GALPERIN, On systems of locally interacting and repelling particles moving in space, Trudy MMO 43 142-196 (1981).
7. G. GALLAVOTTI and D. ORNSTEIN, Billiards and Bernoulli schemes, Comm. Math. Phys. 38 (1974), 83-101.
8. G. A. HEDLUND, The dynamics of geodesic flows, Bull. A.M.S. 45 (1939), 241-260.
9. E. HOPF, Statistik der geodetischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261-304.
10. N. S. KRYLOV, The processes of relaxation of statistical systems and the criterion of mechanical instability, Thesis, Moscow, (1942); Republished in English by Princeton University Press, Princeton N. J. (1979).
11. A. KATOK and J.-M. STRELCYN, Invariant manifolds, entropy and billiards; smooth maps with singularities, LNM 1222 (1986), Springer-Verlag, New York.
12. A. KRÁMLI, N. SIMÁNYI, and D. SZÁSZ, Ergodic properties of semi-dispersing billiards I. Two cylindric scatterers in the 3D torus, Nonlinearity 2 (1989), 311-326.
13. _, A "transversal" fundamental theorem for semi-dispersing billiards, Comm. Math. Phys. 129 (1990), 535-560.
14. _, The K-property of three billiard balls, Annals of Math. 133 (1991), 37-72.
15. _, The K-property of four billiard balls, Comm. Math. Phys. 144 (1992), 107-148.
16. D. ORNSTEIN and B. WEISS, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory & Dynam. Systems 18 (1998), 441-456.
17. YA. PESIN, Characteristic exponents and smooth ergodic theory Russian Math. Surveys 32 (1977), 55-114 .
18. N. SIMÁNYI, The K-property of N billiard balls I, Invent. Math. 108 (1992), 521-548; II. ibidem 110 (1992), 151-172.
19. N. SIMÁNYI, The K-property of N billiard balls I, Invent. Math. 108 (1992), 521-548; II. ibidem 110 (1992), 151-172.
20. YA. G. SINAI, On the foundation of the ergodic hypothesis for a dynamical system of statistical mechanics, Soviet Math. Dokl. 4 (1963), 1818-1822.
21. _, Dynamical systems with elastic reflections, Russian Math. Surveys 25 (1970), 137-189.
22. _, Development of Krylov's ideas, Afterword to N. S. Krylov's "Works on the foundations of statistical physics", see reference [K(1979)] Princeton University Press (1979).
23. I. STEWART, Galois Theory, Chapman and Hall, London (1973).
24. YA. G. SINAI and N. I. CHERNOV, Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls, Russian Math. Surveys 42 (1987), 181-207.
25. N. SIMÁNYI and D. SZÁSZ, The K-property of 4D billiards with nonorthogonal cylindric scatterers, J. Statist. Phys. 76 (1994), 587-604.
26. _, The K-property of Hamiltonian systems with restricted hard ball interactions, Math. Res. Lett. 2 (1995), 751-770.
27. _, Non-integrability of cylindric billiards and transitive Lie group actions, Ergod. Theory & Dynam. Systems, to appear.
28. N. SIMÁNYI and M. WOJTKOWSKI, Two-particle billiard system with arbitrary mass ratio, Ergod. Theory. & Dynam. Systems. 9 (1989), 165-171.
29. D. SZÁSZ, Ergodicity of classical billiard balls, Physica A 194 (1993), 86-92.
30. _, The K-property of 'orthogonal' cylindric billiards, Comm. Math. Phys. 160 (1994), 581-597.
31. _, Boltzmann's ergodic hypothesis, a conjecture for centuries?, Studia Sci. Math. Hungar. 31 (1996), 299-322.
32. L. N. VASERSTEIN, On systems of particles with finite range and/or repulsive interactions, Comm. Math. Phys. 69 (1979), 31-56.
33. M. WOJTKOWSKI, Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys. 105 (1986), 391-414.