Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T17:51:08.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities

Published online by Cambridge University Press:  19 September 2008

Tyll Krüger  and
Serge Troubetzkoy
Tyll Krüger
Affiliation:
BiBoS, Fakultät für Physik, Universität Bielefeld, D-4800 Bielefeld 1, Germany
Serge Troubetzkoy
Affiliation:
BiBoS, Fakultät für Physik, Universität Bielefeld, D-4800 Bielefeld 1, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 487 - 508
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adler, R. & Weiss, B.. Entropy a complete metric invariant for automorphisms of the torus. Proc. Nat Acad. Sci. USA 57 no. 6 (1967), 1573–76.CrossRefGoogle ScholarPubMed
Bowen, R.. Markov partitions for axiom A diffeomorphisms. Amer. J. Math. 92 (1970), 725–47.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Mathematics 470. Springer, Berlin, 1975.CrossRefGoogle Scholar
Bunimovich, L. A.. A theorem on ergodicity of two-dimensional hyperbolic billiards. Commun. Math. Phys. 130 (1990), 599–621.CrossRefGoogle Scholar
Bunimovich, L. A. & Krüger, T.. Ergodicity of 3-dimensional piece-wise convex billiards. In preparation.Google Scholar
Bunimovich, L. A. & Sinai, Ya. G.. Markov partitions for dispersed billiards. Commun. Math. Phys. 78 (1980), 247–80.CrossRefGoogle Scholar
Bunimovich, L. A. & Sinai, Ya. G.. Erratum-Markov partition for dispersed billiards. Commun. Math. Phys. 107 (1986), 357–58.CrossRefGoogle Scholar
Bunimovich, L. A., Sinai, Ya. G. & Chernov, N. I.. Markov partitions for two dimensional hyperbolic billiards (in Russian). Usp. Math. Nauk 45–3–273 (1990), 97–134.Google Scholar
Burns, K. & Gerber, M.. Real analytic Bernoulli geodesic flows on S 2. Ergod. Th. & Dynam. Sys. 9 (1989), 27–45.CrossRefGoogle Scholar
Donnay, V.. Geodesic flow on the two-sphere, Part I: Positive measure entropy. Ergod. Th. & Dynam. Sys. 8 (1988a), 531–53.CrossRefGoogle Scholar
Donnay, V.. Geodesic Flow on the Two-Sphere, Part II: Ergodicity; Dynamical Systems Proc; Univ. of Maryland 1986–87. Alexander, J. C., ed. Springer Lecture Notes 1342. Springer, Berlin, 1988b. pp 112–53.Google Scholar
Donnay, V. & Liverani, C.. Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys. 135 (1991), 267–302.CrossRefGoogle Scholar
Gallavotti, G.. Lectures on the Billiards. Springer Lecture Notes in Physics 38. Springer, Berlin, 1975. pp 236–295.Google Scholar
Hadamard, J.. Les surfaces à courbures opposées et leur Lignes Geodesiques. J. Math. Pure Appl. 4 (1898), 27–73.Google Scholar
Katok, A.. Dynamical system with hyperbolic structure. Amer. Math. Soc. Transl. 116 (2) (1981), 43–94.Google Scholar
Katok, A. & Strelcyn, J. -M.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Springer Lecture Notes in Mathematics 1222. Springer, Berlin, 1986.CrossRefGoogle Scholar
Kubo, I.. Perturbed billiard systems I. The ergodicity of the motion of a particle in a compound central field. Nagoya Math. J. 61 (1976), 1–57.CrossRefGoogle Scholar
Levy, Y.. A note on Sinai and Bunimovich's Markov partition for billiards. J. Stat. Phys. 45 no. 1/2 (1986), 63–68.CrossRefGoogle Scholar
Misiurewicz, M.. Strange attractors for the Lozi mappings. Int. Conf. New York 1979. Ann. NY Acad. Sci. 357 (1980), 348–358.Google Scholar
Morse, M.. A one-to-one representation of geodesies on a surface of negative curvature. Amer. J. Math. 43 (1921), 33–51.CrossRefGoogle Scholar
Sinai, Ya. G.. Construction of Markov partitions. Fund. Anal Appl. 2 (1968), 70–80.CrossRefGoogle Scholar
Smale, S.. Diffeomorphisms with Many Periodic Points; Differential and Combinatorial Topology. Princeton University Press, Princeton, 1965. pp 63–80.Google Scholar