Strong stochastic stability for non-uniformly expanding maps
March 30, 2010
1:00 pm 646 PGH
Abstract
We address the strong stochastic stability of a broad class of
discrete-time dynamical systems - non-uniformly expanding maps - when some
random noise is introduced in the deterministic dynamics. A weaker form of
stochastic stability for this systems was established by J. F. Alves and V.
Araújo (2003) in the sense of convergence of the physical stationary
probability measure to the SRB probability measure in the weak* topology.
We present a strategy to improve this result in order to obtain the strong
stochastic stability, i.e., the convergence of the density of the physical
measure to the density of the SRB measure in the L1-norm,
and in a more general framework of random perturbations. We illustrate our
main result for two examples of non-uniformly expanding maps: the first is
related to an open class of local diffeomorphisms introduced by J. F.
Alves, C. Bonatti and M. Viana (2000) and the second to Viana maps - an
example with critical set introduced by M. Viana (1997). This is a joint
work with J. F. Alves.
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