Ergodic Property of Lyapunov Exponents

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),

$$\mathop {{\rm lim}}\limits_{n \to+ \infty } \frac{1}{n}\mathop\sum \limits_{k = 0}^{n - 1} \delta _{^f } k_x= \mu+,$$

((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

$${\rm \{ (}\lambda _{\rm 1} {\rm (}\mu _{\rm+ } {\rm , }f{\rm),}m_{\rm 1} {\rm (}\mu _{\rm+ } {\rm , }f{\rm)), \cdot \cdot\cdot , (}\lambda _{\rm r} {\rm (}\mu _{\rm+ } {\rm , }f{\rm ),}m_{\rm r} {\rm (}\mu _{\rm+ } {\rm , }f{\rm))\} }{\rm .}$$

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.