On the ergodic properties of nowhere dispersing billiards. (English) Zbl 0421.58017


37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37N99 Applications of dynamical systems
37A99 Ergodic theory
Full Text: DOI


[1] Sinai, Ya. G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Usp. Mat. Nauk25, 141-192 (1970); Russ. Math. Surv.25, 137-189 (1970) · Zbl 0252.58005
[2] Bunimovich, L. A., Sinai, Ya. G.: On a fundamental theorem in the theory of dispersing billiards. Mat. Sb.90, 415-431 (1973); Math. USSR Sb.19, 407-424 (1973)
[3] Arnold, V.J.: Small denominators and problems of stability of motion in classical and celectial mechanics. Usp. Mat. Nauk18, 91-192 (1963); Russ. Math. Surv.18, 85-191 (1963)
[4] Hopf, E.: Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II. Math. Ann.117, 590-608 (1940) · Zbl 0023.26801
[5] Bunimovich, L. A.: On billiards close to dispersing. Mat. Sb.94, 49-73 (1974); Math. USSR Sb.23, 45-67 (1974)
[6] Birkhoff, G.D.: Dynamical systems. Am. Math. Soc. Colloq. Dubl., Vol. 9. Providence, R.I.: AMS 1927; rev. ed. 1966 · JFM 53.0732.01
[7] Lazutkin, V.F.: The existence of coustics for the billiard problem in a convex domain. Izv. A.N. SSSR, ser matem37, 186-216 (1973); Math. USSR Izv.37, 186-216 (1973) · Zbl 0256.52001
[8] Bunimovich, L.A.: On the ergodic properties of some billiards. Funkt. Anal. Jego Prilog.8, 73-74 (1974); Funct. Anal. Appl.8, 73-74 (1974) · Zbl 0309.42013
[9] Rohlin, V.A.: Selected topics from the metric theory of dynamical systems. Usp. Mat. Nauk4, 57-128 (1949); English transl.: Am. Math. Soc. Transl. (2)49, 171-240 (1965)
[10] Zemlyakov, A. N., Katok, A. B.: The topological transitivity of billiards in polygons. Mat. zametki18, 291-301 (1975) · Zbl 0315.58014
[11] Boldrighini, C., Keane, M., Marchetti, F.: Billiards in polygons. Preprint, Univ. di Camerino, Italy (1977) · Zbl 0377.28014
[12] Ornstein, D.S.: An example of a Kolmogorov auromorphism that is not a Bernoulli shift. Adv. Math.10, 49-62 (1973) · Zbl 0245.28011
[13] Hovanskij, A. N.: Application of continued fractions and their generalizations to problems in approximate theory. Moscow: Gittl 1956; English transl.: Groningen: Noordhoff 1963
[14] Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov90, 3-209 (1967); Proc. Steklov Inst. Math.90, 1-227 (1969) · Zbl 0176.19101
[15] Brin, M.J., Pesin, Ya. B.: Partially hyperbolic dynamical systems. Izv. A.N.SSSR, ser. matem.38, 170-212 (1974); Math. USSR Izv.7, 185-227 (1973) · Zbl 0285.58010
[16] Gallavotti, G.: Lectures on billiards. In: Dynamical systems, theory and applications. Lecture notes in physics, Vol. 38. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0324.28013
[17] Gallavotti, G., Ornstein, D.S.: Billiards and Bernoulli schemes. Commun. math. Phys.38, 83-101 (1974) · Zbl 0313.58017
[18] Rüssmann, H.: Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisrings. Nachr. Acad. Wiss., Göttingen, Math. Phys. Kl. II, 67-105 (1970)
[19] Bennettin, G., Strelcyn, J.M.: Numerical experiments on a billiard stochastic transition and entropy. Preprint, Univ. Paris-Nord, France (1977)
[20] Sinai, Ya. G.: Introduction to ergodic theory. Math. notes, Vol. 18. Princeton: Princeton University Press 1976 · Zbl 0375.28011
[21] Dvorin, M.M., Lazutkin, F.F.: Existence of infinitely many elliptic and hyperbolic periodic trajectories for convex billiard. Funkt. Anal. Jego Prilog.7, 20-27 (1973); Funct. Anal. Appl.7, 103-112 (1974) · Zbl 0298.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.