596 research outputs found
Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation
Over the last 10 years or so, advanced statistical properties, including
exponential decay of correlations, have been established for certain classes of
singular hyperbolic flows in three dimensions. The results apply in particular
to the classical Lorenz attractor. However, many of the proofs rely heavily on
the smoothness of the stable foliation for the flow.
In this paper, we show that many statistical properties hold for singular
hyperbolic flows with no smoothness assumption on the stable foliation. These
properties include existence of SRB measures, central limit theorems and
associated invariance principles, as well as results on mixing and rates of
mixing. The properties hold equally for singular hyperbolic flows in higher
dimensions provided the center-unstable subspaces are two-dimensional.Comment: Accepted version. To appear in Advances in Mat
Random perturbations of non-uniformly expanding maps
We give both sufficient conditions and necessary conditions for the
stochastic stability of non-uniformly expanding maps either with or without
critical sets. We also show that the number of probability measures describing
the statistical asymptotic behaviour of random orbits is bounded by the number
of SRB measures if the noise level is small enough.
As an application of these results we prove the stochastic stability of
certain classes of non-uniformly expanding maps introduced in \cite{V} and
\cite{ABV}.Comment: 44 pages, 2 figure
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