Stochastic stability of diffeomorphisms with dominated splitting

Details Stochastic stability of diffeomorphisms with dominated splitting Stochastic stability of diffeomorphisms with dominated splitting

2004-04-07

arxiv.org/abs/math/0404160v3

Stochastics and Dynamics, 7 (3), 299--333, 2007.

Mathematics

Dynamical Systems

32 pages; introduction revised and proofs detailed

Scientific paper

We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain $C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our setting encompasses that of partially hyperbolic diffeomorphisms as well. Moreover, the techniques used enable us to obtain SRB measures in this setting through zero-noise limit measures.

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