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Mat. Sb. (N.S.), 1973, Volume 90(132), Number 3, Pages 415–431 (Mi msb3057)  

This article is cited in 31 scientific papers (total in 33 papers)

On a fundamental theorem in the theory of dispersing billiards

L. A. Bunimovich, Ya. G. Sinai


Abstract: Billiards are considered within domains in the plane or on the two-dimensional torus with the euclidian metric, where the boundaries of these domains are everywhere convex inward. It is shown that the flow $\{S_t\}$ generated by such a billiard is a $K$-system. A fundamental place is here assigned to the proof of the theorem showing that transversal fibers for the flow $\{S_t\}$ consist “on the whole” of sufficiently long regular segments. From this theorem follow assertions on the absolute continuity of transversal fibers for the billiards in question.
Figures: 8.
Bibliography: 6 titles.

Full text: PDF file (1636 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1973, 19:3, 407–423

Bibliographic databases:

UDC: 519.25
MSC: 28A65
Received: 02.02.1972

Citation: L. A. Bunimovich, Ya. G. Sinai, “On a fundamental theorem in the theory of dispersing billiards”, Mat. Sb. (N.S.), 90(132):3 (1973), 415–431; Math. USSR-Sb., 19:3 (1973), 407–423

Citation in format AMSBIB
\Bibitem{BunSin73}
\by L.~A.~Bunimovich, Ya.~G.~Sinai
\paper On~a~fundamental theorem in the theory of dispersing billiards
\jour Mat. Sb. (N.S.)
\yr 1973
\vol 90(132)
\issue 3
\pages 415--431
\mathnet{http://mi.mathnet.ru/msb3057}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=367153}
\zmath{https://zbmath.org/?q=an:0252.58006}
\transl
\jour Math. USSR-Sb.
\yr 1973
\vol 19
\issue 3
\pages 407--423
\crossref{https://doi.org/10.1070/SM1973v019n03ABEH001786}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Shnirelman, “Ergodicheskie svoistva sobstvennykh funktsii”, UMN, 29:6(180) (1974), 181–182  mathnet  mathscinet  zmath
    2. L. A. Bunimovich, “On billiards close to dispersing”, Math. USSR-Sb., 23:1 (1974), 45–67  mathnet  crossref  mathscinet  zmath
    3. Ya. B. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory”, Russian Math. Surveys, 32:4 (1977), 55–114  mathnet  crossref  mathscinet  zmath
    4. O Penrose, Rep Prog Phys, 42:12 (1979), 1937  crossref  adsnasa  isi
    5. Ya. G. Sinai, “Ergodic properties of the Lorentz gas”, Funct. Anal. Appl., 13:3 (1979), 192–202  mathnet  crossref  mathscinet  zmath
    6. George M. Zaslavsky, “Stochasticity in quantum systems”, Physics Reports, 80:3 (1981), 157  crossref  elib
    7. N. I. Chernov, “Structure of transversal leaves in multidimensional semidispersing billiards”, Funct. Anal. Appl., 16:4 (1982), 270–280  mathnet  crossref  mathscinet  zmath  isi
    8. Ya. G. Sinai, N. I. Chernov, “Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls”, Russian Math. Surveys, 42:3 (1987), 181–207  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. M. Paramio, J. Sesma, “Breakdown of KAM tori in the standard map”, Physics Letters A, 124:6-7 (1987), 345  crossref
    10. J. P Bouchaud, A Georges, P. Le Doussal, “Towards a Scaling Theory of Finite-Time Properties in Dynamical Systems?”, Europhys Lett, 5:2 (1988), 119  crossref  adsnasa
    11. Maciej P. Wojtkowski, “Measure theoretic entropy of the system of hard spheres”, Ergod Th Dynam Sys, 8:1 (1988)  crossref  mathscinet
    12. A. Krámli, N. Simányi, D. Szász, “Dispersing billiards without focal points on surfaces are ergodic”, Comm Math Phys, 125:3 (1989), 439  crossref  mathscinet
    13. Nándor Simányi, Maciej P. Wojtkowski, “Two-particle billiard system with arbitrary mass ratio”, Ergod Th Dynam Sys, 9:1 (1989)  crossref  mathscinet
    14. Per Dahlqvist, Gunnar Russberg, “Existence of stable orbits in the x^{2}y^{2} potential”, Phys Rev Letters, 65:23 (1990), 2837  crossref  mathscinet  zmath
    15. L. A. Bunimovich, Ya. G. Sinai, N. I. Chernov, “Markov partitions for two-dimensional hyperbolic billiards”, Russian Math. Surveys, 45:3 (1990), 105–152  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. Kari Eloranta, “Alpha-congruence for dispersive billiards”, Ergod Th Dynam Sys, 11:2 (1991)  crossref  mathscinet
    17. I. K. Babenko, “On the behavior of trajectories of scattering billiards on the flat torus”, Math. USSR-Sb., 72:1 (1992), 207–220  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. N. I. Chernov, “Topological entropy and periodic points of two-dimensional hyperbolic billiards”, Funct. Anal. Appl., 25:1 (1991), 39–45  mathnet  crossref  mathscinet  zmath  isi
    19. L. A. Bunimovich, Ya. G. Sinai, N. I. Chernov, “Statistical properties of two-dimensional hyperbolic billiards”, Russian Math. Surveys, 46:4 (1991), 47–106  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    20. N. I. Chernov, S. Troubetzkoy, “Measures with infinite Lyapunov exponents for the periodic Lorentz gas”, J Statist Phys, 83:1-2 (1996), 193  crossref  mathscinet  zmath  adsnasa
    21. Tamás Tasnádi, “The behavior of nearby trajectories in magnetic billiards”, J Math Phys (N Y ), 37:11 (1996), 5577  crossref  mathscinet  zmath  adsnasa
    22. Leonid A. Bunimovich, Herbert Spohn, “Viscosity for a periodic two disk fluid: An existence proof”, Comm Math Phys, 176:3 (1996), 661  crossref  mathscinet  zmath
    23. S. P. Novikov, L. A. Bunimovich, A. M. Vershik, B. M. Gurevich, E. I. Dinaburg, G. A. Margulis, V. I. Oseledets, S. A. Pirogov, K. M. Khanin, N. N. Chentsova, “Yakov Grigor'evich Sinai (on his sixtieth birthday)”, Russian Math. Surveys, 51:4 (1996), 765–778  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    24. G. M. Zaslavsky, M. Edelman, “Maxwell’s demon as a dynamical model”, Phys Rev E, 56:5 (1997), 5310  crossref  mathscinet  adsnasa  isi
    25. G. M. Zaslavsky, M. Edelman, “Hierarchical structures in the phase space and fractional kinetics: I. Classical systems”, Chaos, 10:1 (2000), 135  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. L. A. Bunimovich, “Rasseyanie, defokusirovka i astigmatizm”, Matem. prosv., ser. 3, 5, MTsNMO, M., 2001, 106–124  mathnet
    27. Marco Lenci, “Semidispersing billiards with an infinite cusp. II”, Chaos, 13:1 (2003), 105  crossref  mathscinet  zmath  isi
    28. Nándor Simányi, “Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems”, Invent math, 154:1 (2003), 123  crossref  isi  elib
    29. G. M. Zaslavsky, “Polynomial dispersion of trajectories in sticky dynamics”, Phys Rev E, 72:3 (2005), 036204  crossref  mathscinet  adsnasa  isi
    30. N. Chernov, “Advanced Statistical Properties of Dispersing Billiards”, J Statist Phys, 122:6 (2006), 1061  crossref  mathscinet  zmath  adsnasa  isi
    31. L. A. Bunimovich, “Criterion of absolute focusing for focusing component of billiards”, Reg Chaot Dyn, 14:1 (2009), 42  crossref  mathscinet  isi
    32. V. Rom-Kedar, D. Turaev, “Billiards: A singular perturbation limit of smooth Hamiltonian flows”, Chaos, 22:2 (2012), 026102  crossref
    33. Luis Barreira, Davor Dragičević, Claudia Valls, “Lyapunov Functions and Cone Families”, J Stat Phys, 2012  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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