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Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties. (English) Zbl 0774.58029

This is a fundamental and very important paper introducing and studying a new class of maps having generalized hyperbolic attractors. They are rather strongly unstable. In the linear approximation their unstability is as strong as it is in classical hyperbolic attractors. However, the considered maps are discontinuous on some closed subset. The hyperbolicity of the described maps is as weak as one encounters in systems with nonzero Lyapunov exponents. Thus the systems with generalized hyperbolic attractors describe a rather widespread way for the appearance of stochasticity.
The aim of this work is to describe the ergodic properties of the dynamical systems having the generalized hyperbolic attractors. One proves the existence of Gibbs \(u\)-measures analogous to Bowen-Ruelle- Sinai measures for classical hyperbolic attractors. The description of ergodic properties of the systems is given with respect to Gibbs \(u\)- measure.
One studies some topological properties of maps on generalized hyperbolic attractors. One proves an analog of the theorem of spectrum decomposition on basic sets for axiom \({\mathcal A}\) diffeomorphisms. For the considered case a number of components of topological transitivity is in general taken into account. For a typical point (with respect to the Riemannian volume) in a basin of a generalized hyperbolic attractor (with positive Riemannian volume) its trajectory can be considered to be “quite stochastic”. But this basin is not in general a neighborhood of the attractor. It can happen that there exists a subset of a positive Riemannian volume in a small neighborhood of the attractor consisting of points whose trajectories go to the attractor but are not “stochastic”.
The famous examples such as the Lorenz type attractor, the Lozi attractor, Belykh attractors are examples of generalized hyperbolic attractors. The results can be considered as a dissipative version of the theory of systems with singularities preserving the smooth measure.

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
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[1] DOI: 10.2307/2000396 · Zbl 0552.58022 · doi:10.2307/2000396
[2] Williams, Structure of Lorenz attractors pp 94– (1977)
[3] Ruelle, Publ Math IHES 50 pp 27– (1979) · Zbl 0426.58014 · doi:10.1007/BF02684768
[4] Bunimovich, Nonlinear Waves pp 212– (1980)
[5] Bunimovich, Modern Problems of Mathematics 2 (1989)
[6] Belykh, Qualitative methods of the theory of nonlinear oscillations in point systems (1980) · Zbl 0529.34001
[7] Afraimovich, The Dimension of Lorenz Type Attractors 6 (1987)
[8] Afraimovich, Trudy Moskov. Mat. Obshch. 44 pp 150– (1983)
[9] Afraimovich, Dokl. Akad. Nauk USSR 234 pp 336– (1977)
[10] DOI: 10.1070/RM1967v022n05ABEH001228 · Zbl 0177.42002 · doi:10.1070/RM1967v022n05ABEH001228
[11] Anosov, Trudy V Int. Conf. on Nonlinear Oscillations 2 pp 39– (1970)
[12] DOI: 10.1070/SM1969v007n01ABEH001076 · Zbl 0198.57002 · doi:10.1070/SM1969v007n01ABEH001076
[13] Pesin, Ergod. Th. & Dynam. Sys. 9 pp 417– (1982)
[14] DOI: 10.1070/IM1976v010n06ABEH001835 · Zbl 0383.58012 · doi:10.1070/IM1976v010n06ABEH001835
[15] Pesin, Proc. Int. Cong. Math. none pp none– (1986)
[16] Pesin, Modern Problems in Mathematics 2 (1989)
[17] DOI: 10.1070/RM1977v032n04ABEH001639 · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[18] Misiurewicz, Nonlinear Dynamics pp 398– (1980)
[19] Lozi, J. Phys. 39 pp 9– (1978)
[20] Levy, Ergodic properties of the Lozi map pp 103– (1985) · Zbl 0614.58031
[21] Ledrappier, C. R. Acad. Sci. 294 pp 593– (1982)
[22] Katok, Invariant manifolds, entropy and billiards: Smooth maps with singularities (1986) · Zbl 0658.58001 · doi:10.1007/BFb0099031
[23] Guckenheimer, Publ Math. IHES 50 pp 59– (1980) · Zbl 0436.58018 · doi:10.1007/BF02684769
[24] Sinai, Statistical Irreversibility in Nonlinear Systems (1979)
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