## Journals

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DCDS

I show that the dynamical determinant, associated to an
Anosov diffeomorphism, is the Fredholm
determinant of the corresponding Ruelle-Perron-Frobenius transfer
operator acting on appropriate Banach spaces.
As a consequence it follows, for example, that the zeroes of the dynamical determinant
describe the eigenvalues of the transfer operator and the Ruelle
resonances and that, for $\C^\infty$ Anosov diffeomorphisms, the
dynamical determinant is an entire function.

DCDS

We present an approach to the investigation
of the statistical properties of weakly coupled map lattices that
avoids completely cluster expansion techniques. Although here it
is implemented on a simple case we expect similar strategies to be
applicable in a much larger class of situations.

DCDS

I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.

DCDS

As the field of ergodic theory has branched out in considerably many
directions it has become harder, even for the expert, to keep a
unitary vision of the field.

From May 17th, 2004 to May 28th, 2004 the following were held in Marseille, CIRM; first a one week school on ergodic theory devoted to the presentation of several topics of the field, second, a one week conference focused on the recent advances in the study of non-uniformly hyperbolic dynamical systems, in connection with smooth ergodic theory.

The idea then emerged of the edition of a special issue collecting contributions to this two week event, providing a vast panorama on ergodic theory especially addressed to people working in concrete smooth dynamical systems.

For more information please click the “Full Text” above.

From May 17th, 2004 to May 28th, 2004 the following were held in Marseille, CIRM; first a one week school on ergodic theory devoted to the presentation of several topics of the field, second, a one week conference focused on the recent advances in the study of non-uniformly hyperbolic dynamical systems, in connection with smooth ergodic theory.

The idea then emerged of the edition of a special issue collecting contributions to this two week event, providing a vast panorama on ergodic theory especially addressed to people working in concrete smooth dynamical systems.

For more information please click the “Full Text” above.

keywords:
dynamics.

JMD

By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure.

JMD

We present some of the results and techniques due to Dolgopyat. The
presentation avoids technicalities as much as possible while trying to focus on
the basic ideas. We also try to present Dolgopyat's work in the context of a
research program aimed at enlightening the relations between dynamical systems
and nonequilibrium statistical mechanics.

JMD

We provide abstract conditions which imply the existence of a robustly
invariant neighborhood of a global section of a fiber bundle flow. We
then apply such a result to the bundle flow generated by an Anosov
flow when the fiber is the space of jets (which are described by local
manifolds). As a consequence we obtain sets of manifolds
(e.g., approximations of stable manifolds) that are left invariant

*for all*negative times by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper*Smooth Anosov flows: correlation spectra and stability*[2] by the present authors.
DCDS

We establish a quenched Central Limit Theorem (CLT) for a smooth
observable of random sequences of iterated linear hyperbolic maps on the torus.
To this end we also obtain an annealed CLT for the same
system. We show that, almost surely, the variance of the quenched
system is the same as for the annealed system. Our technique is the
study of the transfer operator on an anisotropic Banach space
specifically tailored to use the cone condition satisfied by the maps.

## Year of publication

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