DCDS
A generalized shadowing lemma
Shaobo Gan
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.
keywords: shadowing property Pseudo-orbit quasi-hyperbolic.
DCDS
Minimal non-hyperbolicity and index-completeness
Dawei Yang Shaobo Gan Lan Wen
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
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
DCDS
On the hyperbolicity of homoclinic classes
Christian Bonatti Shaobo Gan Dawei Yang
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.
keywords: hyperbolic time shadowing lemma. homoclinic class
JMD
On the singular-hyperbolicity of star flows
Yi Shi Shaobo Gan Lan Wen
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
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.
DCDS
$C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings
Shaobo Gan Kazuhiro Sakai Lan Wen
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
A robustly transitive diffeomorphism of Kan's type
Cheng Cheng Shaobo Gan Yi Shi

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

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