Jorge Iglesias, Aldo Portela
We will construct an action $Φ$, C0 and C1-stable and we will prove that every C0-stable action acting in a manifold of dimensions greater or equal to two, have the shadowing property.
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We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. As a corollary, we show that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov.
S-limit shadowing implies the denseness of chain components with the shadowing property
本文证明紧致度量空间连续自映射中s-极限跟踪蕴含具有跟踪性质的链分支稠密性，部分回答Moothathu的问题，并证明闭微分流形同胚空间中s-极限跟踪非通有。
We explore connections among the regional proximal relation, the asymptotic relation and the distal relation for a topological dynamical system with the shadowing property, and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.
In this paper we consider the shadowing property for iterated function systems,(IFS). Some important result about shadowing property are extended to iterated function systems. For example, we define topological conjugacy for IFS and prove that if F and G are two topological conjugate IFS then F has the shadowing property iff so does G. More- over, we introduce some examples and investigate the relationship between the original shadowing property and shadowing property for IFS.
We study the relation between the shadowing property and the limit shadowing property. We prove that if a continuous self-map $f$ of a compact metric space has the limit shadowing property, then the restriction of $f$ to the non-wandering set satisfies the shadowing property. As an application, we prove the equivalence of the two shadowing properties for equicontinuous maps.
In this paper we generalize the notion of asymptotic average shadowing property to parameterized IFS and prove some related theorems on this notion. Specially, this is proved that every uniformly contracting IFS has the asymptotic average shadowing property. As an important result, we show that if a continuous surjective IFS F on a compact metric space X has the asymptotic average shadowing property then F is chain transitive. Moreover, we introduce some examples and investigate the relationship between the original asymptotic average shadowing property and asymptotic average shadowing propert…
We consider shadowing properties for vector fields corresponding to different type of reparametrisations. We give an example of a vector field which has the oriented shadowing properties, but does not have the standard shadowing property.
We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property and the asymptotic average shadowing property in the conservative case.
Shadowing property for set-valued map and its inverse limit
本文研究集值映射的跟踪性质与其逆极限系统的关系，证明在扩张且开放条件下逆极限系统具有跟踪性质，并建立两者跟踪性质的等价性。
In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.
Contractive shadowing property of dynamical systems
该文研究紧度量空间上拓扑动力系统的压缩跟踪性质，证明其基本性质，指出非退化同胚不具备该性质，还探讨超度量空间情形及链回归集的谱分解并辅以例证。
In this paper we show that every homeomorphism of the plane with the topological shadowing property has a fixed point. Also, we show that a linear isomorphism of an Euclidean space has the topological shadowing property if and only if the origin is an attractor or a repeller.
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.
For the free group $F_2$ acting in $S^{1}$, we will prove that if the minimal set for the action is not a Cantor set, then the action does not have the shadowing property. We will also construct an example, whose minimal set is a Cantor set, that it has the shadowing property.
Positively n-expansive homeomorphisms and the L-shadowing property
本文研究了具有L-阴影性质的n-扩张同胚的动力学，证明了正n-扩张同胚若具有L-阴影性质则只能定义在有限度量空间上。
We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate for this class of spaces, that in order to determine whether a system has shadowing, it is sufficient to check that continuously generated pseudo-orbits can be shadowed.
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。
Beyond topological hyperbolicity: the L-shadowing property
本文研究紧空间动力系统的L-阴影性质，证明结构稳定微分同胚和部分伪Anosov微分同胚满足该性质，并借助谱分解、拓扑半马蹄及正有限可扩性给出其刻画与推论。
This paper is concerned with the study of various notions of shadowing of dynamical systems induced by a sequence of maps, so-called time varying maps, on a metric space. We define and study the shadowing, h-shadowing, limit shadowing, s-limit shadowing and exponential limit shadowing properties of these dynamical systems. We show that h-shadowing, limit shadowing and s-limit shadowing properties are conjugacy invariant. Also, we investigate the relationships between these notions of shadowing for time varying maps and examine the role that expansivity plays in shadowing properties of such dyn…
We consider low-dimensional systems with the shadowing property. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $ε$-transitive class, and in contrast we provide an example of a $C^\infty$ Kupka-Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally we consider the case of transitive endomorphisms of the circle, and we prove that the $α$-Hölder shadowing property with $α>1/2$ implies that the system is conjugate to an expanding map.
The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The "gluing" construction used in the analysis of the last property turns out to be the key point of this theory.
We call that a vector field has the oriented shadowing property if for any $\varepsilon>0$ there is $d>0$ such that each $d$-pseudo orbit is $\varepsilon$-oriented shadowed by some real orbit. In this paper, we show that the $C^1$-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the $Ω$-stability.
In this note we study some properties of topological entropy for noncompact non-metrizable spaces.
We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing homeomorphisms with bounded variation.
Product Anosov diffeomorphisms and the two-sided limit shadowing property
本文用双侧极限跟踪性质刻画乘积阿诺索夫微分同胚，证明其等价于万有覆盖上提升的唯一双侧极限跟踪性质，并通过巴拿赫空间中映射的不动点关联跟踪轨道。
It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known as generalized hyperbolic operators. This class of operators seems to be an important bridge between hyperbolicity and the shadowing property. In this article, we show that for a large natural c…
Topology of global attractors for homeomorphisms with the topological shadowing property in $\mathbb{R}^m$
研究非紧空间中动力系统的行为，探讨全局吸引子和影子性质，证明在特定动力系统中，全局吸引子的拓扑影子性质当且仅当吸引子是平凡的。
Suspensions of homeomorphisms with the two-sided limit shadowing property
本文研究连续流的两侧极限伪轨跟踪性质，证明同胚的悬浮流保持该性质，并揭示流与同胚在该性质上的关键差异，包括圆周上的例子、带间隙版本的等价性以及奇异悬浮流不满足该性质。
Topological $(\mathscr{F},\mathscr{G})-$shadowing property
本文在一致空间上引入拓扑 $(\mathscr{F},\mathscr{G})-$ 跟踪性，证明紧致链回归系统中若干跟踪性等价，并给出极小点不稠密时系统不具有 $(\mathscr{P}(\mathbb{N}),\mathscr{F}_{ps})-$ 跟踪性的结论。
N-expansive homeomorphisms with the shadowing property
本文研究紧度量空间上具有阴影性质的n-扩张同胚，构造了非(n-1)-扩张且有无穷多链回归类的例子，并利用局部稳定集证明极限阴影性质，为拓扑混合n-扩张同胚的存在性问题提供方向。
Homoclinic classes with shadowing property
本文证明了对于$C^1$通有微分同胚，孤立同宿类具有跟踪性质当且仅当其为双曲基本集。
Shadowing for local homeomorphisms, with applications to edge shift spaces of infinite graphs
本文发展了度量局部紧空间上局部同胚的跟踪性质理论，通过路径集合条件完全刻画了有向图Deaconu-Renault系统的跟踪性质，并应用于无限图边移位空间与C*-代数相关系统。
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,…
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.
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 Bi-Shadowing for Semi-Hyperbolic Systems
本文研究半超双曲系统中的影子性质，定义了三种影子性质类别，并在特定条件下证明半超双曲族具有L^p双影子性、极限双影子性和渐近双影子性。证明策略是将影子问题转化为固定点问题，再应用布劳威尔固定点定理进行验证。
Shadowing property on hyperspace of continua induced by Morse gradient system
本文研究了由Morse梯度系统诱导的连续体超空间的影子性质，证明了此类映射不具有影子性质。
Countably and entropy expansive homeomorphisms with the shadowing property
本文研究具有伪轨跟踪性质的可数膨胀与熵膨胀同胚，证明传递可数膨胀同胚在传递点集上膨胀，并证明在曲面情形下可数膨胀和熵cw-膨胀均等价于拓扑共轭于阿诺索夫微分同胚。
The notion of stochastic shadowing property is introduced. Relations to stochastic stability and standard shadowing are studied. Using tent map as an example it is proved that, in contrast to what happens for standard shadowing, there are significantly non-uniformly hyperbolic systems that satisfy stochastic shadowing property.
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…
We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag-Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally we show that any linear action of a non-abelian free group does not have the shadowing property.
Shadowing in the hyperspace of continua
本文研究连续统超空间上诱导映射的跟踪性，证明Anosov微分同胚等不满足该性质，而树状单调映射的跟踪性等价于其超空间映射，并构造了万有树状上的跟踪同胚。
In this paper, subsystems with shadowing property for $\mathbb{Z}^{k}$-actions are investigated. Let $α$ be a continuous $\mathbb{Z}^{k}$-action on a compact metric space $X$. We introduce the notions of pseudo orbit and shadowing property for $α$ along subsets, particularly subspaces, of $\mathbb{R}^{k}$. Combining with another important property "expansiveness" for subsystems of $α$ which was introduced and systematically investigated by Boyle and Lind, we show that if $α$ has the shadowing property and is expansive along a subspace $V$ of $\mathbb{R}^{k}$, then so does for $α$ along any sub…
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.
We study conditions under which a piecewise affine mapping has the Lipschitz shadowing property. As an application, we show that there exists a homeomorphism with a nonisolated fixed point having the Lipschitz shadowing property.
We investigate a shadowing property which appears naturally in the study of piecewise monotonic maps of the interval. It turns out to be a weak form of the rank one property, a well-known notion in abstract ergodic theory. We show that this new property is implied by finite or even local rank, but that it is logically independent of loose Bernoulliness. We give (counter)examples, including L.B. systems with arbitrarily high-order polynomial complexity. The shadowing property defines a small subset of all zero-entropy systems, in the sense that it defines a closed set with empty interior with r…
This paper aims at formulating definitions of topological stability, structural stability, and expansiveness property for an iterated function system( abbrev, IFS). It is going to show that the shadowing property is necessary condition for structural stability in IFSs. Then, it proves the previous converse demonstration with the addition of expansiveness property for IFSs. It asserts that structural stability implies shadowing property in IFSs and presents an example to reject of the converse assertion.
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.
L-shadowing for the induced hyperspace homeomorphism
本文研究了诱导超空间同胚的L-shadowing性质，证明了同胚与其诱导同胚之间L-shadowing性质的等价性，并探讨了其在局部稳定/不稳定集上的收缩统一性问题。
We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.
In this paper we show that generic continuous Lebesgue measure preserving circle maps have the s-limit shadowing property. In addition we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.
We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant m…
Quasi-shadowing property for nonuniformly partially hyperbolic systems
本文研究非一致部分双曲系统，建立自适应伪轨道运动的拟跟踪性质，并考察拟 specification 与拟闭合性质，进而将 Katok 周期轨道增长结果推广到任意遍历测度。
Local Product Structure of Expansive Flows with Shadowing Property
该研究证明定义在全移位映射上的悬挂流不存在局部乘积结构，否定了Gelfert-Kwietniak-Lima近期提出的相关公开问题。
We consider inverse periodic shadowing properties of discrete dynamical systems generated by diffeomorphisms of closed smooth manifolds. We show that the $C^1$-interior of the set of all diffeomorphisms having so-called inverse periodic shadowing property coincides with the set of $Ω$-stable diffeomorphisms. The equivalence of Lipschitz inverse periodic shadowing property and hyperbolicity of the closure of all periodic points is proved. Besides, we prove that the set of all diffeomorphisms that have Lipschitz inverse periodic shadowing property and whose periodic points are dense in the nonwa…
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。
We study various weaker forms of inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called Ergodic Inverse Shadowing property (Birhhoff averages of continuous functions along the exact trajectory and the approximating one are close). We demonstrate that this property implies continuity of the set of invariant measures in Hausdorff metrics. We show that the class of systems with Ergodic Inverse Shadowing is quite broad, it includes all diffeomorphisms with hyperbolic nonwandering sets. Secondly, we study the so-called Individual In…
The shadowing property for piecewise monotone interval maps
本文针对由传递分段单调区间映射（特别是β-变换）驱动的非连续动力系统，给出简短证明表明其不具有阴影性质，从而补充并推广了Chen和Portela的已有结果。
The orbital shadowing property (OSP) of discrete dynamical systems on smooth closed manifolds is considered. Nondensity of OSP with respect to the C^1-topology is proved. The proof uses the method of skew products developed by Yu.S. Ilyashenko and A.S. Gorodetski.
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