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### Open Access Journals

DCDS

In this paper, we prove a generalized shadowing lemma. Let $f \in$ Diff$(M)$.
Assume that $\Lambda$ is a closed invariant set of $f$ and there is a continuous invariant
splitting $T\Lambda M = E\oplus F$ on $\Lambda$. For any $\lambda \in (0, 1)$ there exist $L > 0, d_0> 0$ such
that for any $d \in (0, d_0]$ and any $\lambda$-quasi-hyperbolic d-pseudoorbit $\{x_i, n_i\}_{i=-\infty}^\infty$,
there exists a point $x$ which Ld-shadows $\{x_i, n_i\}_{i=-\infty}^\infty$. Moreover, if $\{x_i, n_i\}_{i=-\infty}^\infty$ is periodic, i.e., there exists an $m > 0$ such that $x_{i+m}= x_i$ and $n_{i+m} = n_i$ for all $i$,
then the point $x$ can be chosen to be periodic.

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.

DCDS

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admit two physical measures with intermingled basins. In particular, all these diffeomorphisms are not topologically mixing. Moreover, every such example exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

keywords:
Robustly transitive
,
physical measure
,
intermingled basin
,
Gibbs state
,
blender-horseshoe

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".
DCDS

We give a sufficient criterion for the hyperbolicity of a homoclinic
class. More precisely, if the homoclinic class $H(p)$ admits a
partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where
$E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all
periodic points homoclinically related with $p$ are

*uniformly $E^u$-expanding at the period*, then $H(p)$ is hyperbolic. We also give some consequences of this result.
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.

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