DCDS
A uniform $C^1$ connecting lemma
Lan Wen
Discrete & Continuous Dynamical Systems - A 2002, 8(1): 257-265 doi: 10.3934/dcds.2002.8.257
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.
keywords: $C^1$ connecting lemma $C^1$ perturbation $\epsilon$-kernel transition.
DCDS
The selecting Lemma of Liao
Lan Wen
Discrete & Continuous Dynamical Systems - A 2008, 20(1): 159-175 doi: 10.3934/dcds.2008.20.159
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.
keywords: quasi-hyperbolic string dominated splitting selecting shadowing.
DCDS
On the preperiodic set
Lan Wen
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 237-241 doi: 10.3934/dcds.2000.6.237
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
$C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings
Shaobo Gan Kazuhiro Sakai Lan Wen
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 205-216 doi: 10.3934/dcds.2010.27.205
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.
keywords: dominated splitting partially hyperbolic. shadowing Weak shadowing pseudo-orbit homoclinic class chain component chain recurrent set
DCDS
Minimal non-hyperbolicity and index-completeness
Dawei Yang Shaobo Gan Lan Wen
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1349-1366 doi: 10.3934/dcds.2009.25.1349
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.
keywords: Dimension theory multifractal analysis. Poincaré recurrences
DCDS
Robustly transitive singular sets via approach of an extended linear Poincaré flow
Ming Li Shaobo Gan Lan Wen
Discrete & Continuous Dynamical Systems - A 2005, 13(2): 239-269 doi: 10.3934/dcds.2005.13.239
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".
keywords: star flow. Robustly transitive set extended linear poincaré flow partial hyperbolicity
JMD
On the singular-hyperbolicity of star flows
Yi Shi Shaobo Gan Lan Wen
Journal of Modern Dynamics 2014, 8(2): 191-219 doi: 10.3934/jmd.2014.8.191
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.
keywords: Lyapunov stable class Singular-hyperbolicity star flow shadowing.
DCDS
Indices of singularities of robustly transitive sets
Shengzhi Zhu Shaobo Gan Lan Wen
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 945-957 doi: 10.3934/dcds.2008.21.945
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.
keywords: Strongly homogenous robustly transitive singular hyperbolic.

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