Mardi après midi dynamique
Stable ergodicity beyond partial hyperbolicity.
We discuss a method for proving stable ergodicity of conservative
diffeomorphisms in its most general setting. Namely that of diffeomorphisms
with dominated splittings.
Entropy and decay of correlation for some partially hyperbolic systems.
We will focus on two simple families of partially hyperbolic maps:
suspension semi-flows and skew extensions of expanding maps. Assuming
real-analyticity of maps, we will show in both cases how to obtain
lower bounds
on correlations functions for well chosen observables. Those lower bounds
involve metric entropy and, among other things, imply existence of
essential
spectrum
for the associated Perron Frobenius operator.
Perturbed eigenvalues (Rare events and metastability: some precise formulas)
Coupled map lattices: a survey and some open problems
Rigidity of geodesic and magnetic flows on surfaces which preserve
highly regular codimension 1 foliations
Metric stability of deterministic random walks (with applications in renormalization theory) .
Consider deterministic random walks F : I x Z -> I x Z, defined
by F (x, n) = (f (x), q(x) + n), where f is an expanding Markov map on the
interval I and q : I -> Z (Z are the integers). We study the stability
of ergodic (for
instance, recurrence and transience), geometric and multifractal
properties in
the class of perturbations of the type F (x, n) = (f_n (x), q(x, n) +
n) which are
topologically conjugate with F and f_n are expanding maps exponentially
close
to f when |n| goes to infinity. We give applications of these results
in the study of the
regularity of conjugacies between (generalized) infinitely renormalizable
maps
of the interval. Joint work with C. G. Moreira.
Entropy of eigenfunctions of the Laplacian in
dimension 2
Given a compact Riemannian manifold M, semiclassical measures are a
particular family of probability measures invariant under the geodesic
flow on $S^*M$. They are constructed from the eigenfunctions of the
Laplacian on the manifold. Not so many things are known about them. In the
case of surfaces of Anosov type or of nonpositive curvature, I will try to
explain how we can show that their metric entropy is bounded from below by
half of the Ruelle upper bound.
Dynamical systems, roubst phenomena and the tangent map.
On concentration inequalities.
For 'nice' (e.g., iid, Markovian, Gibbsian, etc) stochastic
processes (X_k), a lot is known about the fluctuations of the
partial sum process S_n=X_1+X_2+...+X_n (CLT, large deviations,
etc).
Concentration inequalities allow to go far beyond this case and to
deal with ``complicated'' functions K(X_1,X_2,...,X_n) provided they
are Lipschitzian.
These inequalities were recently obtained for many `chaotic'
dynamical systems.
One way to prove such inequalities is coupling. I intend to present
this method in the case of Markov chains.
Quantum resonances for classically chaotic scattering systems