117 research outputs found
Conservative flows with various types of shadowing
In the present paper we study the C1-robustness of the three properties:
average shadowing, asymptotic average shadowing and limit shadowing within two
classes of conservative flows: the incompressible and the Hamiltonian ones. We
obtain that the first two properties guarantee dominated splitting (or partial
hyperbolicity) on the whole manifold, and the third one implies that the flow
is Anosov.Comment: 13 page
Stable weak shadowable symplectomorphisms are partially hyperbolic
Let M be a closed, symplectic connected Riemannian manifold and f a symplectomorphism on M. We prove that if f is C1-stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.info:eu-repo/semantics/publishedVersio
Variational equalities of entropy in nonuniformly hyperbolic systems
In this paper we prove that for an ergodic hyperbolic measure of a
diffeomorphism on a Riemannian manifold , there is an
-full measured set such that for every invariant
probability , the metric
entropy of is equal to the topological entropy of saturated set
consisting of generic points of :
Moreover, for every nonempty, compact and connected subset of
with the same hyperbolic rate, we
compute the topological entropy of saturated set of by the following
equality:
In particular these results can be applied (i) to the nonuniformy hyperbolic
diffeomorphisms described by Katok, (ii) to the robustly transitive partially
hyperbolic diffeomorphisms described by ~Ma{\~{n}}{\'{e}}, (iii) to the
robustly transitive non-partially hyperbolic diffeomorphisms described by
Bonatti-Viana. In all these cases
contains an open subset of .Comment: Transactions of the American Mathematical Society, to appear,see
http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06780-X
Dominated Pesin theory: convex sum of hyperbolic measures
In the uniformly hyperbolic setting it is well known that the set of all
measures supported on periodic orbits is dense in the convex space of all
invariant measures. In this paper we consider the converse question, in the
non-uniformly hyperbolic setting: assuming that some ergodic measure converges
to a convex combination of hyperbolic ergodic measures, what can we deduce
about the initial measures?
To every hyperbolic measure whose stable/unstable Oseledets splitting
is dominated we associate canonically a unique class of periodic
orbits for the homoclinic relation, called its \emph{intersection class}. In a
dominated setting, we prove that a measure for which almost every measure in
its ergodic decomposition is hyperbolic with the same index such as the
dominated splitting is accumulated by ergodic measures if, and only if, almost
all such ergodic measures have a common intersection class.
We provide examples which indicate the importance of the domination
assumption.Comment: final version, new co-author, to appear in: Israel Journal of
Mathematic
Structurally Stable Homoclinic Classes
In this paper we study structurally stable homoclinic classes. In a natural
way, the structural stability for an individual homoclinic class is defined
through the continuation of periodic points. Since the homoclinic classes is
not innately locally maximal, it is hard to answer whether structurally stable
homoclinic classes are hyperbolic. In this article, we make some progress on
this question. We prove that if a homoclinic class is structurally stable, then
it admits a dominated splitting. Moreover we prove that codimension one
structurally stable classes are hyperbolic. Also, if the diffeomorphism is far
away from homoclinic tangencies, then structurally stable homoclinic classes
are hyperbolic.Comment: arXiv admin note: substantial text overlap with arXiv:1410.430
The Relation between Approximation in Distribution and Shadowing in Molecular Dynamics
Molecular dynamics refers to the computer simulation of a material at the
atomic level. An open problem in numerical analysis is to explain the apparent
reliability of molecular dynamics simulations. The difficulty is that
individual trajectories computed in molecular dynamics are accurate for only
short time intervals, whereas apparently reliable information can be extracted
from very long-time simulations. It has been conjectured that long molecular
dynamics trajectories have low-dimensional statistical features that accurately
approximate those of the original system. Another conjecture is that numerical
trajectories satisfy the shadowing property: that they are close over long time
intervals to exact trajectories but with different initial conditions. We prove
that these two views are actually equivalent to each other, after we suitably
modify the concept of shadowing. A key ingredient of our result is a general
theorem that allows us to take random elements of a metric space that are close
in distribution and embed them in the same probability space so that they are
close in a strong sense. This result is similar to the Strassen-Dudley Theorem
except that a mapping is provided between the two random elements. Our results
on shadowing are motivated by molecular dynamics but apply to the approximation
of any dynamical system when initial conditions are selected according to a
probability measure.Comment: 21 pages, final version accepted in SIAM Dyn Sy
- …