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### Open Access Journals

DCDS

The spectra of Poincaré recurrences for two classes of
dynamical
systems are obtained in the framework of the
Carathéodory construction. One
class contains systems which are topologically
conjugate to subshifts with the
specification property, the other consists of minimal multipermutative symbolic systems. The spectra are shown
to be solutions of a non-homogeneous
Bowen equation, and their relationship with multifractal spectra of Lyapunov
exponents is exposed.

DCDS

Suppose that $f$ is a surface diffeomorphism with a
hyperbolic fixed point $\mathcal O$
and this fixed point has a transversal homoclinic orbit.
It is well known that in a
vicinity of this type of homoclinic there are hyperbolic
invariants sets. We introduce
smooth invariants for the homoclinic orbit which we call
the multipliers. As an
application, we study the influence of the multipliers
on numerical invariants of the
hyperbolic invariant sets as the vicinity becomes small.

DCDS-B

We prove, under some general assumptions, that master-slave synchronization
implies generalized synchronization, that is we show the existence and continuity
of the functional dependence between the “slave” coordinates and the “master”
ones. Then, we prove that this function may be Lipschitz continuous and even
less “smooth”, that is only Hölder continuous, depending on the coupling strength.
We go beyond the above mentioned assumptions by coupling two identical maps of
the interval that are neither continuous nor invertible to prove ‘almost-everywhere’
synchronization instead of global synchronization. Then we relate the Hausdorff dimension
and the dimension for Poincaré recurrence of the attractor of master and
slave systems. We provide some examples illustrating these results.

DCDS

We introduce pointwise dimensions and spectra associated with Poincaré
recurrences. These quantities are then calculated for any ergodic measure
of positive entropy on a weakly specified subshift.
We show that they satisfy a relation comparable to Young's formula for the
Hausdorff
dimension of measures invariant under surface diffeomorphisms.
A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length.
Examples are provided which show that this is not true for some systems with
zero entropy.
Similar results are obtained for special flows
and we get a formula relating spectra for measures
of the base to the ones of the flow.

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