170 research outputs found
Transverse foliations on the torus \T^2 and partially hyperbolic diffeomorphisms on 3-manifolds
In this paper, we prove that given two foliations and
on which are transverse, there exists a non-null
homotopic loop in \diff^{1}(\T^2) such that
\Phi_t(\calF)\pitchfork \calG for every , and .
As a direct consequence, we get a general process for building new partially
hyperbolic diffeomorphisms on closed -manifolds. \cite{BPP} built a new
example of dynamically coherent non-transitive partially hyperbolic
diffeomorphism on a closed -manifold, the example in \cite{BPP} is obtained
by composing the time map, large enough, of a very specific
non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result
shows that the same construction holds starting with any non-transitive Anosov
flow on an oriented -manifold. Moreover, for a given transverse torus, our
result explains which type of Dehn twists lead to partially hyperbolic
diffeomorphisms.Comment: 34 pages, 7 figure
Existence of common zeros for commuting vector fields on -manifolds
In E. Lima proved that commuting vector fields on surfaces with non-zero
Euler characteristic have common zeros. Such statement is empty in dimension
, since all the Euler characteristics vanish. Nevertheless,
\cite{Bonatti_analiticos} proposed a local version, replacing the Euler
characteristic by the Poincar\'e-Hopf index of a vector field in a region
, denoted by ; he asked:
\emph{Given commuting vector fields and a region where
, does contain a common zero of and
?}
\cite{Bonatti_analiticos} gave a positive answer in the case where and
are real analytic.
In this paper, we prove the existence of common zeros for commuting
vector fields , on a -manifold, in any region such that
, assuming that the set of collinearity of
and is contained in a smooth surface. This is a strong indication that the
results in \cite{Bonatti_analiticos} should hold for -vector fields.Comment: Final version, to appear in Annales de L'Institut Fourie
Many intermingled basins in dimension 3
We construct a diffeomorphism of admitting any finite or
countable number of physical measures with intermingled basins. The examples
are partially hyperbolic with splitting and
can be made volume hyperbolic and topologically mixing.Comment: 20 pages, 4 figures. Some changes made after referee report. To
appear in Israel J. of Mat
Perturbation of the Lyapunov spectra of periodic orbits
We describe all Lyapunov spectra that can be obtained by perturbing the
derivatives along periodic orbits of a diffeomorphism. The description is
expressed in terms of the finest dominated splitting and Lyapunov exponents
that appear in the limit of a sequence of periodic orbits, and involves the
majorization partial order. Among the applications, we give a simple criterion
for the occurrence of universal dynamics.Comment: A few improvements were made, based on the referee's suggestion
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