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- W54923406 abstract "Generic property of SRB measures was first investigated by Bowen [10]. Theorem 4.12 in [10] says that if Ω is a hyperbolic attractor of a C 2 Axiom A diffeomorphism (M, f) and m is the volume measure on the compact Riemannian manifold M induced by the Riemannian metric, then for m-almost all x in the basin of attraction W s(Ω), $$mathop {{rm lim}}limits_{n to+ infty } frac{1}{n}mathopsum limits_{k = 0}^{n - 1} delta _{^f } k_x= mu+,$$ ((VIII.1)) where µ + is the SRB measure for f on Ω. As µ + is an ergodic measure, the Lyapunov exponents of system f : M ← are µ +-almost everywhere constants. Recently, by exploiting a Ruelle’s perturbation theorem [79, Theorem 4.1] Jiang et al. [29] proved that m-almost all x ∈ W s(Ω) is positively regular and the Lyapunov spectrum of the system (i.e., the Lyapunov exponents associated with their multiplicities) at x are the constants $${rm { (}lambda _{rm 1} {rm (}mu _{rm+ } {rm , }f{rm),}m_{rm 1} {rm (}mu _{rm+ } {rm , }f{rm)), cdot cdotcdot , (}lambda _{rm r} {rm (}mu _{rm+ } {rm , }f{rm ),}m_{rm r} {rm (}mu _{rm+ } {rm , }f{rm))} }{rm .}$$ This is called the ergodic property of Lyapunov exponents. Similar results have also been obtained in [29] for nonuniformly completely hyperbolic attractors of C 2 diffeomorphisms.KeywordsLyapunov ExponentAbsolute ContinuityConditional MeasureLyapunov SpectrumMarkov PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
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- W54923406 date "2009-01-01" @default.
- W54923406 modified "2023-09-26" @default.
- W54923406 title "Ergodic Property of Lyapunov Exponents" @default.
- W54923406 doi "https://doi.org/10.1007/978-3-642-01954-8_8" @default.
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