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Random perturbations of nonuniformly expanding maps. (English) Zbl 1043.37016
de Melo, Welington (ed.) et al., Geometric methods in dynamics (I). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-138-4/pbk). Astérisque 286, 25-62 (2003).
The authors study properties of random perturbations of smooth, nonuniformly expanding maps on a compact manifold $$M$$. An SRB measure for a nonuniformly expanding map $$f$$ is an $$f$$-invariant Borel probability measure $$\mu$$ for which the time average along the $$f$$-orbit of $$x\in M$$ converges to $$\int \phi\;d\mu$$ for all continuous $$\phi$$ and all $$x$$ in a set of positive Lebesgue measure. A Borel probability measure is called physical for a family of random perturbations of $$f$$ if it has similar properties for almost every perturbation in the family. The map $$f$$ is stochastically stable if the weak-* limit points of the physical measures as the size of the perturbation goes to 0 are linear combinations of the SRB measures.
Necessary conditions and sufficient conditions for stochastic stability in this context are developed. It is shown also that the number of physical measures is finite and independent of the size of the random perturbations, provided that these are sufficiently small, and that this number is at most the (finite) number of SRB measures.
Examples are given where the two numbers are different. These results are extended to the case where $$f$$ has a critical set of Lebesgue measure 0, along with certain technical assumptions about the dynamics near the critical set. The results are applied to classes of maps studied previously in [M. Viana, Publ. Math., Inst. Hautes Étud. Sci. 85, 63–96 (1997; Zbl 1037.37016)] and [J. F. Alves, C. Bonatti and M. Viana, Invent. Math 140, 351–398 (2000; Zbl 0962.37012)].
For the entire collection see [Zbl 1029.00027].

MSC:
 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37A05 Dynamical aspects of measure-preserving transformations 37H99 Random dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)