Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T04:30:15.986Z Has data issue: false hasContentIssue false
  • English
  • Français

A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities

Published online by Cambridge University Press:  09 July 2012

GIANLUIGI DEL MAGNO  and
ROBERTO MARKARIAN
GIANLUIGI DEL MAGNO
Affiliation:
CEMAPRE, ISEG, Universidade Tecnica de Lisboa, 1200 Lisbon, Portugal (email: [email protected])
ROBERTO MARKARIAN
Affiliation:
Instituto de Matemática y Estadística ‘Prof. Ing. Rafael Laguardia’ (IMERL), Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 983 - 1007
Copyright
Copyright © 2012 Cambridge University Press 

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

[An]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235.Google Scholar
[Ar]Arnold, V. I.. Mathematical Methods of Classical Mechanics. Springer, New York, 1997.Google Scholar
[Bu]Bunimovich, L. A.. On absolutely focusing mirrors. Ergodic Theory and Related Topics, III (Güstrow, Germany 1990). Springer, Berlin, 1992, pp. 62–82.Google Scholar
[BS]Bunimovich, L. A. and Sinai, Ya. G.. The fundamental theorem of the theory of scattering billiards. Mat. Sb. (N.S.) 90(132) (1973), 415–431.Google Scholar
[BG]Burns, K. and Gerber, M.. Continuous invariant cone families and ergodicity of flows in dimension three. Ergod. Th. & Dynam. Sys. 9 (1989), 19–25.CrossRefGoogle Scholar
[C]Chernov, N. I.. Local ergodicity of hyperbolic systems with singularities. Funct. Anal. Appl. 27 (1993), 51–54.Google Scholar
[CH]Chernov, N. I. and Haskell, C.. Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. & Dynam. Sys. 16 (1996), 19–44.CrossRefGoogle Scholar
[CM]Chernov, N. I. and Markarian, R.. Chaotic Billiards (Mathematical Surveys and Monographs, 127). American Mathematical Society, Providence, RI, 2006.CrossRefGoogle Scholar
[Do]Donnay, V.. Using integrability to produce chaos: billiards with positive entropy. Comm. Math. Phys. 141 (1991), 225–257.CrossRefGoogle Scholar
[Ho]Hopf, E.. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
[KB]Katok, A. and Burns, K.. Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems. Ergod. Th. & Dynam. Sys. 14 (1994), 757–785.Google Scholar
[KS]Katok, A. and Strelcyn, J.-M.. Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, New York, 1986.Google Scholar
[KSS]Krámli, A., Simányi, N. and Szász, D.. A ‘transversal’ fundamental theorem for semi-dispersing billiards. Comm. Math. Phys. 129 (1990), 535–560; Erratum, Comm. Math. Phys. 138 (1991), 207–208.Google Scholar
[LW]Liverani, C. and Wojtkowski, M.. Ergodicity in Hamiltonian systems. Dynamics Reported: Expositions in Dynamical Systems. New Series Vol. 4. Springer, Berlin, 1995, pp. 130–202.CrossRefGoogle Scholar
[M1]Markarian, R.. Billiards with Pesin region of measure one. Comm. Math. Phys. 118 (1988), 87–97.Google Scholar
[M2]Markarian, R.. The fundamental theorem of Sinai-Chernov for dynamical systems with singularities. Dynamical Systems (Santiago, 1990) (Pitman Research Notes in Mathematics, 285). Longman, Harlow, 1993, pp. 131–158.Google Scholar
[OW]Ornstein, D. S. and Weiss, B.. On the Bernoulli nature of systems with hyperbolic structure. Ergod. Th. & Dynam. Sys. 18 (1998), 441–456.Google Scholar
[P]Pesin, Ya. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32 (1977), 55–114.Google Scholar
[S]Sinai, Ya. G.. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math. Surveys 25 (1970), 137–189.Google Scholar
[SC]Sinai, Ya. G. and Chernov, N. I.. Ergodic properties of some systems of two-dimensional disks and three-dimensional balls. Russian Math. Surveys 42 (1987), 181–207.Google Scholar
[W1]Wojtkowski, M.. Invariant families of cones and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 145–161.CrossRefGoogle Scholar
[W2]Wojtkowski, M.. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. 105 (1986), 391–414.Google Scholar