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Strong statistical stability of non-uniformly expanding maps. (English) Zbl 1061.37020
Let $$M$$ be a compact Riemannian manifold and let $$m$$ be a volume form of $$M$$. Let $$F$$ be a family of $$C^k$$-maps, $$k\geq 2$$, of $$M$$ into itself endowed with the $$C^k$$-topology. We assume that each $$f\in F$$ admits a unique absolutely continuous $$f$$-invariant probability measure $$\mu_f$$ in $$M$$. We say that $$f_0\in F$$ is statistically stable if $$F\ni f\to d\mu_f/dm$$ is continuous at $$f_0$$ with respect to the $$L$$-norm in the space of densities. After giving the notions of nonuniformly expanding maps, the author gives sufficient conditions for certain classes of nonuniformly expanding maps to be statistically stable. For the proof, the author uses, among other things, the results of the author and M. Viana [Ergodic Theory Dyn. Syst. 22, 1–32 (2002; Zbl 1067.37034)] on Markovian maps.
Reviewer’s note: Theorem A (p. 1196) should be read Theorem 1.5.

##### MSC:
 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37C75 Stability theory for smooth dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
##### Keywords:
nonuniformly expanding maps; statistically stable maps
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