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- Advances in Mathematics of Communications
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DCDS

We prove there are uniform bounds for quantities that guarantee
C1 connecting of orbits. Here the uniformity is with respect to all systems
in a $C^1$ neighborhood of the given system, and with respect to certain set of
accumulation points.

DCDS

The selecting lemma of Liao selects, under certain
conditions of a non-hyperbolic setting, a special
kind of orbits of finite length, called quasi-hyperbolic strings, which can be
shadowed by true orbits. In this article we give an exposition on this lemma,
and illustrate some recent applications.

DCDS

A point is called $C^r$ preperiodic if it can be made
periodic via arbitrarily
small $C^r$ perturbation.
We discuss some general properties of the $C^r$
preperiodic set,
and prove that the $C^1$ preperiodic set contains no
obstruction points if and only
if the system is Axiom A plus no-cycle.

keywords:
Preperiodic set.

DCDS

We study a problem raised by Abdenur et. al. [3] that asks,
for any chain transitive set $\Lambda$ of a generic diffeomorphism
$f$, whether the set $I(\Lambda)$ of indices of hyperbolic periodic
orbits that approach $\Lambda$ in the Hausdorff metric must be an
"interval", i.e., whether $\alpha\in I(\Lambda)$ and $\beta\in
I(\Lambda)$, $\alpha<\beta$, must imply $\gamma\in I(\Lambda)$ for
every $\alpha<\gamma<\beta$. We prove this is indeed the case if, in
addition, $f$ is $C^1$ away from homoclinic tangencies and if
$\Lambda$ is a minimally non-hyperbolic set.

DCDS

Morales, Pacifico and Pujals proved recently that every robustly
transitive singular set for a
3-dimensional flow must be partially hyperbolic. In this paper we
generalize the result
to higher dimensions. By definition, an
isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is
called

*robustly transitive*if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called*singular*if it contains a singularity. The set $\Lambda$ is called*strongly homogeneous*of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincaré flow "to singularities".
JMD

We prove for a generic star vector field $X$ that if, for every chain
recurrent class $C$ of $X$, all singularities in $C$ have the same
index, then the chain recurrent set of $X$ is singular-hyperbolic. We
also prove that every Lyapunov stable chain recurrent class of a
generic star vector field is singular-hyperbolic. As a corollary, we
prove that the chain recurrent set of a generic 4-dimensional star
flow is singular-hyperbolic.

DCDS

It was proved recently in [4] that any robustly transitive singular
set that is strongly homogenous must be partially hyperbolic, as
long as the indices of singularities and periodic orbits satisfy
certain condition. We prove in this paper that this index-condition
is automatically satisfied under the strongly homogenous condition,
hence can be removed from the assumptions. Moreover, we prove that a
robustly transitive singular set that is strongly homogenous is in
fact singular hyperbolic.

DCDS

Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$.
In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting.
Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.

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