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Strong stochastic stability for non-uniformly expanding maps

Published online by Cambridge University Press:  06 August 2012

JOSÉ F. ALVES  and
HELDER VILARINHO
JOSÉ F. ALVES
Affiliation:
CMUP, Rua do Campo Alegre 687, 4169-007 Porto, Portugal (email: [email protected])
HELDER VILARINHO
Affiliation:
Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6200-001 Covilhã, Portugal (email: [email protected])
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Abstract

We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the $L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398] and the second one is the class of Viana maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013 , pp. 647 - 692
Copyright
Copyright © 2012 Cambridge University Press

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