Stably weakly shadowing transitive sets and dominated splittings

集值映射及其逆极限的跟踪性与传递性

Shadowing property and transitivity of a set-valued map and its inverse limit

研究集值映射及其逆极限的跟踪性、传递性等性质，证明了满射上半连续集值映射与其逆极限的跟踪性等价，且广义逆极限上移位映射的传递性等性质能推出集值映射相应性质，还给出了具有跟踪性的集值映射相关性质的等价关系。

Hyperbolicity and Types of Shadowing for C1 Generic Vector Fields

We study various types of shadowing properties and their implication for C1 generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the proper…

双曲性、传递性与双侧极限伪轨跟踪性质

Hyperbolicity, transitivity and the two-sided limit shadowing property

本文刻画了闭流形上具有双侧极限伪轨跟踪性质的微分同胚，证明其C^1内部及通有情形均为传递Anosov微分同胚，从而将Anosov传递性猜想归结为该性质与Anosov性质的等价性。

No-shadowing for singular hyperbolic sets with a singularity

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

Chain Recurrence and Positive Shadowing in Linear Dynamics

We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed invariant subspace. We prove that every chain transitive linear dynamical system with the shadowing property is frequently hypercyclic, and as a corollary obtain that every shadowing hypercyclic linear dynamical system is frequently hypercyclic.

集值映射的周期跟踪性

Periodic Shadowing for Set-Valued Maps

研究集值动力系统的周期跟踪性质，证明在紧度量空间上正膨胀集值映射的跟踪性蕴含周期跟踪性，且链传递映射的周期跟踪性蕴含跟踪性和拓扑传递性。

跟踪性、传递性与 ω-混沌的一种变体

Shadowing, transitivity and a variation of omega-chaos

针对紧 Hausdorff 空间上的链传递连续自映射，研究 DSP 跟踪性及其基本性质，并据此给出统计型 ω-混沌的充分条件、拓扑通有性及不规则点分布结果。

具有跟踪性质的系统的完全不规则集研究

On the completely irregular set for systems with the shadowing property

该研究针对满足跟踪性质的非唯一遍历传递连续映射，证明了紧度量空间上完全不规则集的贝尔 generic 性及对应轨道的稠密性，还分析了传递性与跟踪性质的关联并探讨了结果适用范围。

存在奇点时的伪轨跟踪：定向伪轨跟踪与标准伪轨跟踪、熵及回复集的结构

Shadowing in the presence of singularities: oriented versus standard shadowing, entropy and the structure of recurrent sets

该研究区分了流的定向与标准两类伪轨跟踪性质，证明其在闭定向曲面上不同，关联标准伪轨跟踪与回复性、熵，还得到曲面流中定向伪轨跟踪结合有限奇点时链回复类为极小的结论。

关于具有双侧极限跟踪性质的同胚

On homeomorphisms with the two-sided limit shadowing property

本文证明双侧极限跟踪性质是已知最强的伪轨跟踪概念之一，并引入带间隙的双侧极限跟踪这一较弱概念，进而用它们在扩张同胚类中刻画拓扑传递性与混合性。

Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.

Shadowing, internal chain transitivity and $α$-limit sets

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescribed accuracy) by both the $α$-limit set and the $ω$-limit set of a full-trajectory. Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$ is equal to both the $α$-limit set and the $ω$-limit set of a full-trajectory. In particular this means that …

Partial Hyperbolicity for Symplectic Diffeomorphisms

We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.

Shades of Hyperbolicity for Hamiltonians

We prove that a C2 Hamiltonian system H in M is globally hyperbolic if any of the following statements holds: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stably weakly shadowable regular ene…

关于同胚传递性的充分条件

On sufficient conditions for the transitivity of homeomorphisms

本文推导了具有跟踪性质的同胚拓扑传递的充要条件（含稠密不变子集的重心性质），对比Anosov微分同胚的相关条件，描述$C^1$内部、讨论例子并给出应用，为动力学传递性研究提供新依据。

The omega-limit sets of quadratic Julia sets

In this paper we characterize $\w$-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin's symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an $\w$-limit set of a point if, and only if, it is internally chain transitive.

Shadowable Points for flows

A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a $G_δ$ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. We study the relations between shadowab…

Inverse pseudo orbit tracing property for robust diffeomorphisms

Let $M$ be a closed smooth Riemannian manifold $M$, and let $f:M\to M$ be a diffeomorphism. Herein, we demonstrate that (i) if $f$ has the $C^1$ robustly inverse shadowing property on the chain recurrent set $\mathcal{CR}(f)$, then $\mathcal{CR}(f)$ is hyperbolic and (ii) if $f$ has the $C^1$ robustly inverse shadowing property on a nontrivial transitive set $Λ\subset M$, then $Λ$ is hyperbolic for $f$. Especially, the item (ii) is a proof of the conjecture of Lee and Lee \cite{LL}.

Dynamics of continuous maps induced on the space of probability measures

For a continuous self-map $f$ on a compact interval $I$ and the induced map $\hat f$ on the space $\mathcal{M}(I)$ of probability measures, we obtain a sharp condition to guarantee that $(I,f)$ is transitive if and only if $(\mathcal{M}(I),\hat f)$ is transitive. We also show that the sensitivity of $(I,f)$ is equivalent to that of $(\mathcal{M}(I),\hat f)$. We prove that $(\mathcal{M}(I),\hat f)$ must have infinite topological entropy for any transitive system $(I,f)$, while there exists a transitive non-autonomous system $(I,f_{0,\infty})$ such that $(\mathcal{M}(I),\hat f_{0,\infty})$ has z…

拓扑超双曲系统ω-极限集的刻画

Characterizations of ω-Limit Sets of Topologically Hyperbolic Systems

本文研究拓扑超双曲系统中ω-极限集的性质，证明在多种系统中内部链传递性和弱不可压缩性特征一致，并扩展了有限型移位系统的相关结论。

局部凸空间上算子的广义双曲性、稳定性与扩张性

Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces

本文研究局部凸空间上算子的广义双曲性、拓扑稳定性与拓扑扩张性，建立其与有限跟踪、严格周期跟踪、混沌及传递性之间的关系，并刻画加权移位情形。

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of $ICT_f$ is equal to (resp. may be approximated by) the $α$-limit set and the $ω$-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.

On the omega-limit sets of tent maps

For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prov…

Some Properties of Recurrent Sets for Endomorphisms of Topological Groups

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, it is obtained that some dynamical properties induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property.