Ergodic Hyperbolic Attractors of Endomorphisms

Let μ be an SRB-measure on an Axiom A attractor Δ of a C ²-endomorphism (M, f). As is known, μ-almost every x ϵ Δ is positively regular and the Lyapunov exponents of (f, T f) at x are constants λ⁽ⁱ⁾(f, μ), 1 ≤ i ≤ s. In this paper, we prove that Lebesgue-almost every x in a small neighborhood of Δ is positively regular and the Lyapunov exponents of (f, T f) at x are the constants λ⁽ⁱ⁾(f, μ), 1 ≤ i ≤ s. This result is then generalized to nonuniformly completely hyperbolic attractors of endomorphisms. The generic property of SRB-measures is also proved. 2000 Mathematics Subject Classification. 37D20, 37D25, 37C40. This work was supported by the 973 Funds of China for Nonlinear Science, the NSFC 10271008, and the Doctoral Program Foundation of the Ministry of Education.