1,294 research outputs found
Partially hyperbolic dynamics in dimension 3
Partial hyperbolicity appeared in the sixties as a natural generaliza- tion
of hyperbolicity. In the last 20 years in this area there has been great
activity. Here we survey the state of the art in some topics, focusing
especially in partial hyperbolicity in dimension 3. The reason for this is not
only that it is the smallest dimension in which non-degenerate partial
hyperbolicity can occur, but also that the topology of 3-manifolds influences
this dynamics in revealing ways.Comment: Updated version, to appear in Ergodic Theory and Dynamical System
On m-minimal partially hyperbolic diffeomorphisms
We discuss about the denseness of the strong stable and unstable manifolds of
partially hyperbolic diffeomorphisms. In this sense, we introduce a concept of
m-minimality. More precisely, we say that a partially hyperbolic
diffeomorphisms is m-minimal if m-almost every point in M has its strong stable
and unstable manifolds dense in M. We show that this property has dynamics
consequences: topological and ergodic. Also, we prove the abundance of
m-minimal partially hyperbolic diffeomorphisms in the volume preserving and
symplectic scenario
A survey on partially hyperbolic dynamics
Some of the guiding problems in partially hyperbolic systems are the
following: (1) Examples, (2) Properties of invariant foliations, (3)
Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of
central foliations, (7) Transitivity and (8) Classification. Here we will
survey the state of the art on these subjects, and propose related problems.Comment: 57 pages, references adde
Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds
We study conservative partially hyperbolic diffeomorphisms in hyperbolic
3-manifolds. We show that they are always accessible and deduce as a result
that every conservative partially hyperbolic in a hyperbolic
3-manifold must be ergodic, giving an afirmative answer to a conjecture of
Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the
intermediary steps are also done for general partially hyperbolic
diffeomorphisms homotopic to the identity.Comment: 40 pages, 4 figures. New version takes into account the release of
arXiv:1908.06227 while the previous version made reference to an earlier (and
unavailable) version of that paper. Also adds new results on certain isotopy
classes on Seifert manifold
Some advances on generic properties of the Oseledets splitting
In his foundational paper [ICM 1983, Warzaw], Ma\~n\'e suggested that some
aspects of the Oseledets splitting could be improved if one worked under
C1-generic conditions. He announced some powerful theorems, and suggested some
lines to follow. Here we survey the state of the art and some recent advances
in these directions.Comment: 22 page
Robust transitivity implies almost robust ergodicity
In this paper we show the relation between robust transitivity and robust
ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive
systems are robustly ergodic. For the three dimensional case, we define it
almost robust ergodicity and prove that generically robustly transitive systems
are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also
show in higher dimensions, that under some conditions robust transitivity
implies robust ergodicity.Comment: 12 page
A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms
In this work we exhibit a new criteria for ergodicity of diffeomorphisms
involving conditions on Lyapunov exponents and general position of some
invariant manifolds. On one hand we derive uniqueness of SRB-measures for
transitive surface diffeomorphisms. On the other hand, using recent results on
the existence of blenders we give a positive answer, in the topology, to
a conjecture of Pugh-Shub in the context of partially hyperbolic conservative
diffeomorphisms with two dimensional center bundle.Comment: A Research Announcemen
Partial hyperbolicity and ergodicity in dimension three
In [15] the authors proved the Pugh-Shub conjecture for partially hyperbolic
diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism
are dense among the partially hyperbolic ones. In this work we address the
issue of giving a more accurate description of this abundance of ergodicity. In
particular, we give the first examples of manifolds in which all conservative
partially hyperbolic diffeomorphisms are ergodic.Comment: 14 page
On The Uniqueness of SRB Measures for Endomorphisms
In this paper we improve the results of \cite{MT} and show that a weak
hyperbolic transitivity implies the uniqueness of hyperbolic SRB measures. As
an important corollary, it arises the ergodicity of the system in a
conservative setting. It also arises the condition which implies the stable
ergodicity as well as the statistical stability for a general -regular
map
On the stable ergodicity of diffeomorphisms with dominated splitting
In this paper we obtain two criteria of stable ergodicity outside the
partially hyperbolic scenario. In both criteria, we use a weak form of
hyperbolicity called chain-hyperbolicity. It is obtained one criterion for
diffeomorphisms with dominated splitting and one criterion for weakly partially
hyperbolic diffeomorphisms. As an application of one of these criteria, we
obtain the -density of stable ergodicity inside a certain class of weakly
partially hyperbolic diffeomorphisms
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