Ergodic Theory and Dynamical Systems



Statistical stability for robust classes of maps with non-uniform expansion


JOSÉ F. ALVES a1 and MARCELO VIANA a2
a1 Centro de Matemática da Universidade do Porto, Praça Gomes Teixeira, 4099-002 Porto Portugal (e-mail: jfalves@fc.up.pt)
a2 IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil (e-mail: viana@impa.br)

Abstract

We consider open sets of transformations in a manifold M, exhibiting non-uniformly expanding behavior in some forward invariant domain U\subset M. Assuming that each transformation has a unique Sinai–Ruelle–Bowen (SRB) measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L1-norm. We also describe a concrete application.

(Received May 8 2000)
(Revised January 19 2001)