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DCDS

In the study of systems which combine slow and fast motions
which depend on each other (fully coupled setup) whenever the averaging
principle can be justified this usually can be done only in the sense of
$L^1$-convergence on the space of initial conditions. When fast motions are
hyperbolic (Axiom A) flows or diffeomorphisms (as well as
expanding endomorphisms) for each
freezed slow variable this form of the averaging principle was derived in
[19] and [20] relying on some large deviations arguments which
can be applied only in the Axiom A or uniformly expanding case. Here we give
another proof which seems to work in a more general framework, in particular,
when fast motions are some partially hyperbolic or some nonuniformly
hyperbolic dynamical systems or nonuniformly expanding endomorphisms.

DCDS

I consider discretized random perturbations
of hyperbolic dynamical systems
and prove that when perturbation
parameter tends to zero invariant measures
of corresponding Markov chains converge to
the Sinai-Bowen-Ruelle measure of the dynamical system. This
provides a robust method for computations of
such measures and for visualizations of some hyperbolic
attractors by modeling randomly perturbed
dynamical systems on a computer.
Similar results are true for
discretized random perturbations
of maps of the interval satisfying the Misiurewicz
condition considered in [KK].

DCDS

We introduce a relative Gurevich pressure for random countable
topologically mixing Markov shifts. It is shown that the relative variational
principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables
us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous
functions. This is accomplished via a new construction of an equivariant
family of fiber measures using Crauel's relative Prohorov theorem. Some
properties of the Gibbs measures are discussed as well.

DCDS

We correct a flaw in the proof of Proposition 6.3 in [1].

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