Mukta Garg, Ruchi Das
In this paper, we introduce and study notions of average chain transitivity, average chain mixing, total average chain transitivity and almost average shadowing property. We also discuss their interrelations.
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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。
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, 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。
Limit sets, internal chain transitivity and orbital shadowing of tree-shifts defined on Markov-Cayley trees
本文研究了 Markov-Cayley 树上树移位的 $ω$-极限集与伪轨道，通过引入基于完全前缀集的修正定义及投影伪轨道概念，推广了 $d$-树上的相关结论并探究了其跟踪性质。
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 …
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。
This paper is concerned with the study of fuzzy dynamical systems. Let (X;M; *) be a fuzzy metric space in the sense of George and Veeramani. A fuzzy discrete dynamical system is given by any fuzzy continuous self-map defined on X. We introduce the various fuzzy shadowing and fuzzy topological transitivity on a fuzzy discrete dynamical systems. Some relations between this notions have been proved.
In this paper we study relations between almost specification property, asymptotic average shadowing property and average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples that compactness is a necessary condition for these implications to hold. As a consequence of our methodology we also obtain a proof that limit shadowing in chain transitive systems implies shadowing.
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。
In this paper we present a systematic study of shadowing properties with average error in tracing such as (asymptotic) average shadowing, $\underline{d}$-shadowing, $\overline{d}$-shadowing and almost specification. As the main tools we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center. Next, we show that always (e.g. without assumption that the map is onto) almost specification implies asymptotic average shadowing,…
Sensitivity, local stable/unstable sets and shadowing
本文研究紧致度量空间上具有伪轨跟踪性质的灵敏同胚，证明其局部稳定/不稳定集必含紧致完备子集，并据此推广了正可数膨胀同胚只能定义在可数空间中的既有结论。
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…
Let $(X,G,Φ)$ be a dynamical system, where $X$ is compact Hausdorff space, and $G$ is a countable discrete group. We investigate shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property, fix some finite subset $S\subset G$. We prove that if $X$ is totally disconnected, then $Φ$ has $S$-shadowing property if and only if $(X,G,Φ)$ is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric space (may be not totally disconnected), we prove that if $Φ$ has $S$-sha…
Topological Stability of Random $cw$-Expansive Systems with Shadowing
研究随机动力系统的拓扑稳定性，通过随机连续扩张性和跟踪性概念扩展经典结果，证明相关系统随机拓扑稳定，还建立性质共轭下拓扑不变性并分析拓扑传递性与周期跟踪性的关系。
In this paper we show the relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive systems are robustly ergodic. For the three dimensional case, we define it almost robust ergodicity and prove that generically robustly transitive systems are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also show in higher dimensions, that under some conditions robust transitivity implies robust ergodicity.
We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform con…
Shadowing, average shadowing and transitive properties of multiple mappings
该研究从集值视角定义多重映射的多类动力学性质，证明单映射的对应性质不必然传递给其构成的多重映射，给出多重映射满足传递等性质的充分条件，明确跟踪类性质的迭代不变性及相关性质间的推导关系。
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。
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 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。
Equivalence of Variants of Shadowing of Free Semigroup Actions
本文证明了有限生成自由半群作用中六类平均型跟踪性及参数化 $M_\alpha$-跟踪性相互等价，从而肯定回答了 Kulczycki 等人提出的公开问题。
A saturation theorem for distinct-degree polynomials and an application to joint transitivity
本研究在几乎一对一扩张模下证明极小$\boldsymbol{\rm Z}^k$系统的拓扑有理Kronecker因子是不同次数多项式交换变换的拓扑特征因子，并给出这类多项式迭代联合传递性的充要条件。
Topological dynamics for the endograph metric II: Extremely radical properties
研究动力系统\((X,f)\)的扎德扩展\((\mathcal{F}(X),\hat{f})\)在终图度量\(d_{E}\)下的拓扑动力学性质。证明该度量在多种性质上表现极端激进，解决了文献中开放问题并产生新结果。
For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost 1-1 extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\varepsilon>0$ the collection of ergodic measures (supported on almost 1-1 extensions of odometers) with entropy between $c$ and $c…
Shadowing and the basins of terminal chain components
本文从替代视角重新证明并强化了关于跟踪性的结论，刻画终端链分支吸引域的稠密性，并在局部连通及链回归集完全不连通条件下推出映射几乎链连续。
We find sufficient conditions for commutative non-autonomous systems on certain metric spaces to be topologically stable. In particular, we prove that (i) Every mean equicontinuous, mean expansive system with strong average shadowing property is topologically stable. (ii) Every equicontinuous, recurrently expansive system with almost shadowing property is topologically stable. (iii) Every equicontinuous, expansive system with shadowing property is topologically stable.
Dynamics of the Takagi function and the shadowing property
研究塔克函数生成的离散动力系统，证明几乎所有点的轨道收敛至2/3，并探讨塔克映射家族的影子性质，展示塔克函数具有影子性质，并提出两种技术以确定参数γ使映射失去影子性质。
Minimal and proximal examples of $\bar{d}$-stable and $\bar{d}$-approachable shift spaces
本文研究移位空间的 $\bar{d}$-跟踪与 $\bar{d}$-可逼近性，证明其蕴含 $\bar{d}$-稳定性并建立两类 Hausdorff 距离的等价性，同时构造极小与近距反例。
The Euler non-mixing made easy
本文指出任意维Euler方程中无额外约束的非传递性几乎是显然的，并可通过Morse理论推导得出。
On Equicontinuity and Related Notions in Nonautonomous Dynamical Systems
本研究针对交换同胚族生成的非自治动力系统，纠正了已有文献中极小性充要条件的错误，推导了等度连续性、传递性等相关性质的新结论，实现了自治系统经典结果的非自治类比推广。
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