Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-13T08:04:32.151Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

Published online by Cambridge University Press:  19 September 2008

Ya. B. Pesin
Ya. B. Pesin
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 1 , March 1992 , pp. 123 - 151
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

[1]Alekseev, V. M.. Quasirandom dynamical systems. Mat. USSR-Sbornik 7 (1969), 1–43.CrossRefGoogle Scholar
[2]Anosov, D. V.. On one class of invariant sets of smooth dynamical systems. Trudy V Int. Conf. on Nonlinear Oscillations. Vol. 2. Inst. Mat. Akad. Nauk USSR: Kiev, 1970, pp. 39–45.Google Scholar
[3]Anosov, D. V. & Sinai, Ya. G.. Some smooth ergodic systems. Russ. Math. Surv. 22 (1967), 103–196.CrossRefGoogle Scholar
[4]Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P.. On appearance and structure of Lorenz attractor. Dokl. Akad. Nauk USSR 234 (1977), 336–339.Google Scholar
[5]Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P.. On attracting nonstructurally stable limit sets of type of Lorenz attractor. Trudy Moskov. Mat. Obshch. 44 (1983), 150–212.Google Scholar
[6]Afraimovich, V. S. & Pesin, Ya. B.. The Dimension of Lorenz Type Attractors. Gordon and Breach: Harwood Academic, Sov. Math. Phys. Rev. 6 (1987).Google Scholar
[7]Belykh, V. M.. Qualitative methods of the theory of nonlinear oscillations in point systems. Gorki University Press (1980).Google Scholar
[8]Bunimovich, L. A.. Systems of hyperbolic type with singularities. In: Modern Problems of Mathematics. Vol. 2 Sinai, Ya. G., ed., Springer-Verlag: Berlin, 1989.Google Scholar
[9]Bunimovich, L. A., & Sinai, Ya. G.. Stochasticity of the attractor in the Lorenz model. Nonlinear Waves (Proc. Winter School, Moscow). Nauka: Moscow, 1980, pp 212–226.Google Scholar
[10]Katok, A. & Strelcyn, J.-M.. Invariant manifolds, entropy and billiards: Smooth maps with singularities. Springer Lecture Notes in Mathematics 1222. Springer-Verlag: Berlin, 1986.CrossRefGoogle Scholar
[11]Ledrappier, F.. Propriétés ergodiques des mesures de Sinai. C. R. Acad. Sci., Paris, Ser. 1, 294, N17 (1982), pp 593–595.Google Scholar
[12]Levy, Y.. Ergodic properties of the Lozi map. Springer Lecture Notes in Mathematics, 1109. Springer-Verlag: Berlin, 1985, pp 103–116.Google Scholar
[13]Lozi, R.. Un attracteur étrange du type attracteur de Hénon. J. Phys., Paris 39, Coll. C5 (1978), 9–10.Google Scholar
[14]Misiurewicz, M.. Strange attractors for the Lozi mappings. In: Nonlinear Dynamics. Helleman, R. G., ed. Academic: New York, 1980, pp 398–358.Google Scholar
[15]Pesin, Ya. B.. Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv. 32 (1977), 55–114.CrossRefGoogle Scholar
[16]Pesin, Ya. B.. Ergodic theory of smooth dynamical systems. In: Modern Problems in Mathematics. Vol. 2. Sinai, Ya. G., ed. Springer-Verlag: Berlin, 1989.Google Scholar
[17]Pesin, Ya. B.. Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic. Proc. Int. Cong. Math. (1986).Google Scholar
[18]Pesin, Ya. B.. Families of invariant manifolds corresponding to nonzero characteristic exponents. Math USSR Izv. 10 (1976), 1281–1305.CrossRefGoogle Scholar
[19]Pesin, Ya. B. & Sinai, Ya. G.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 9 (1982), 417–438.CrossRefGoogle Scholar
[20]Ruelle, D.. Ergodic theory of differential dynamical systems. Publ Math IHES 50 (1979), 27–58.CrossRefGoogle Scholar
[21]Sinai, Ya. G.. Several rigorous results concerning the decay of time-correlation functions. Appendix in: Statistical Irreversibility in Nonlinear Systems. By Zaslovski, G. M.. Nauka: Moscow, 1979.Google Scholar
[22]Young, L. S.. Bowen-Ruelle-Sinai measures for certain piecewise hyperbolic maps. Trans. Am. Math. Soc. 287 (1985), 41–48.CrossRefGoogle Scholar
[23]Williams, R. F.. Structure of Lorenz attractors. Springer Lecture Notes in Mathematics 675. Springer-Verlag: Berlin, 1977, pp 94–112.Google Scholar
[24]Guckenheimer, J. & Williams, R.. Structural stability of the Lorenz attractors. Publ Math. IHES 50 (1980), 59–72.CrossRefGoogle Scholar
[25]Rychlik, M.. Invariant measures and the variation principle for Lozi mappings. PhD dissertation. University of California, Berkeley, 1983.Google Scholar