Stochastic stability of non-uniformly hyperbolic diffeomorphisms

Carlos Vasquez | Alves, Jose | Araujo, Vitor

Keywords: dominated splitting; non-uniform hyperbolicity; SRB measure; random perturbation; stochastic stability

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction 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 non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [J. F. Alves; C. Bonatti and M. Viana, \emph{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 as uniformly hyperbolic diffeomorphisms.

Anexos:

• alarva51.pdf