Xiao Wen, Lan Wen
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.
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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。
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…
Bi-shadowing of Quasi-semi-hyperbolic Pseudo-orbit
本文引入准半超双曲伪轨道的概念，证明在误差足够小时，准半超双曲性蕴含准双曲性，并展示了有限和无限的准双曲伪轨道均具有双影子性质，从而确立了周期性。
We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with "jumps" along the central foliation. The proof is based on the Tikhonov-Shauder fixed point theorem.
Let M be a closed, symplectic connected Riemannian manifold, f a symplectomorphism on M. We prove that if f is C1-stably weakly shadowing on M, then the whole manifold M admits a partially hyperbolic splitting.
We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.
In this note, we introduce the notion of $r$-homoclinic points. We show that an operator on a Banach space is hyperbolic if and only if it is shadowing and has no nonzero $r$-homoclinic points. We also solve invariant subspace problem (ISP for brevity) for shadowing operators on Banach spaces. Afterwards, we verify that the set of generalized hyperbolic operators is invariant under $λ$-Aluthge transforms for every $λ\in \left( 0,1 \right)$. Next, the Aluthge iterates of invertible operators converge to hyperbolic operators only if the initial operators are hyperbolic. Finally, we prove that th…
We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in [5] for Hamiltonian systems.
J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fréchet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the Shadowing lemma in the same context.
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…
A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $τ$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $τ$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When…
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。
Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit $\{x_k\}_{k\in \mathbb{Z}}$, there is a sequence $\{y_k\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the local center leaf of $f(y_k)$ for any $k\in \mathbb{Z}$. $f$ is called topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $π$ and a motion $τ$ along the center foliation such that $π\circ g=τ\circ f\circπ$. In this paper we prove that if $f$ is dynamically coherent then it has quasi-shadowing and topologi…
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。
A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result…
A note on Bohr chaos and hyperbolic sets
研究动力系统中跟踪现象与玻尔混沌的关系，给出基于跟踪的玻尔混沌充分条件，结合跟踪引理得出关于玻尔混沌和双曲集的结果，突出双曲集结构性质。
Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy
本文证明具有联合连续稳定/不稳定叶层的cw-双曲同胚满足周期阴影性质，且在拓扑混合时满足周期规范性质，并讨论将Bowen技术应用于获得最大熵测度的困难，给出Walter伪Anosov微分同胚和Sierpiński地毯上同胚的唯一最大熵测度。
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。
In hyperbolic dynamics, a well-known result is that every hyperbolic attracting set, have a finite pseudo-orbit tracing property (FPOTP). It's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics; Komuro in [Lorenz attractors do not have the pseudo-orbit tracing property], provides a negative answer for this question, by proving that the geometric Lorenz Attractor doesn't have a FPOTP. In this paper, we generalized the result of Komuro, we prove that every codimension one sectional-hyperbolic attractor set with a unique singularity Lorenz-like, which is of bounda…
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.
Parameterized shadowing for nonautonomous dynamics
该研究针对含参数的非自治非线性微分与差分方程，给出参数化$C^k$跟踪性的充分条件，适用于线性部分非双曲情形，还在双曲情形下得到参数化Hyers-Ulam稳定性相关结果。
On the Shadowableness of Flows With Hyperbolic Singularities
本文证明了闭流形上含指标为一的附着双曲奇点的链回归流不满足伪轨跟踪性质，并将结论推广到非紧流形上的重标度伪轨跟踪性质。
Shadowing maps
本文针对紧度量空间同胚的跟踪性质，提出跟踪映射的理论与层级体系，结合Bowen的经典方法从双曲括号构造自调谐跟踪映射，丰富了跟踪性相关研究。
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 of actions of hyperbolic groups on their boundaries
本文证明任意双曲群在其 Gromov 边界上的典范作用具有跟踪（伪轨跟踪）性质，并由此重新得到 Mann 等人所述此类作用拓扑稳定的结论。
A shadowable chain recurrent set with an attached hyperbolic singularity
本文证明了除收缩至奇点的闭轨道外单射的因子映射保持标准跟踪性质，并据此在四维球面上构造了含附着双曲奇点却具标准跟踪性质的 $C^\infty$ 流，反驳了 Arbieto 等人的猜想。
We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering set consisting of the finite number of critical elements (i.e., singularities or closed orbits). Moreover, we prove that each isolated singularity of a topological flow on a closed surface with the oriented shadowing property is either asymptotically stable, backward asymptotically stable, or admits a neighborhood which splits into two or four hyperbolic sectors.
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.
Resolving the generalized hyperbolicity conjecture for shadowing
该研究通过构造反例证明，关于跟踪性的广义双曲性猜想在一般巴拿赫空间上不成立，但在可分希尔伯特空间上成立，明确了两类空间的差异源于满射与右可逆的间隙。
Shadowing, chain transitive sets with nonempty interior and attractor boundaries
本文从阴影角度研究$C^1$动力学中的现象，改进超双曲集结果并探讨吸引子边界的稳定性。
Adjoint shadowing directions in hyperbolic systems for sensitivity analysis
本文研究了双曲系统中伴随影子方向的定义与唯一性，提出了一种用于伴随灵敏度分析的新方法，为长时平均目标如NILSAS的高效伴随灵敏度方法提供了理论基础。
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…
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。
We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient d…
Shadowing Endomorphisms of Compact Groups
该研究刻画了紧Hausdorff群连续自同态的跟踪性，通过约化到紧连通满自同态情形，分别给出了阿贝尔群和一般紧连通群自同态具有跟踪性的等价条件。
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems. This consists in the following steps: (i) computation of the scattering map and of the transition map for a flow, (ii) reduction to the scattering map and to the tran…
Backpropagation in hyperbolic chaos via adjoint shadowing
本文提出伴随跟踪算子 S，将反向传播推广到离散与连续时间双曲混沌，并统一刻画线性响应中的跟踪与不稳定贡献。
L-shadowing lemma for the Cauchy equation
本文证明 Banach 空间中 Cauchy 问题的双曲性与 L-阴影性质在有限维情形下等价，推广了 Ombach 关于线性同胚的已有结果，并给出若干应用。
A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expa…
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…
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。
We study the renormalization operator of circle homeomorphisms with a break point and show that it possesses a hyperbolic horseshoe attractor.
Continuum-wise hyperbolicity
本文提出连续统意义下的双曲性概念，证明cw-双曲同胚的阴影引理与谱分解，引入cwN-双曲性并研究其周期点有限性等动力学性质，还证明由环面拓扑双曲同胚诱导的球面同胚是cw-双曲且拓扑共轭于线性cw-Anosov微分同胚。
For any $θ, ω> 1/2$ we prove that, if any $d$-pseudotrajectory of length $\sim 1/d^ω$ of a diffeomorphism $f\in C^2$ can be $d^θ$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured by Hammel-Grebogi-Yorke that for $θ= ω= 1/2$ this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof we introduce the notion of sublinear growth property for inhomogenious linear equations and prove that it implies exponential dichotomy.
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。
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。
We prove that for any closed manifold of dimension 3 or greater that there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly,we review some results due to Anosov that concern premaximality.
We prove that small nonlinear perturbations of random linear dynamics admitting a tempered exponential dichotomy have a random version of the shadowing property. As a consequence, if the exponential dichotomy is uniform, we get that the random linear dynamics is Hyers-Ulam stable. Moreover, we apply our results to study the conservation of Lyapunov exponents of the random linear dynamics subjected to nonlinear perturbations.
The hyperbolic umbilic singularity in fast-slow systems
本文针对含三个慢变量、快变量具梯度结构的快慢系统，采用爆破技术将双曲脐点爆破为5维球面，分析其附近局部动力学，证明非退化条件下吸引慢流形过奇点时跃迁快区并扇形展开，同时关联了突变理论等经典结果。
In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)
Uniform Hyperbolicity and Symbolic Dynamics: Markov Partitions, Shadowing, and the Coding of Axiom A Diffeomorphisms
本文研究均匀双曲集的几何理论，通过定量界证明五个主要定理，包括稳定流形定理、谱分解定理、影子引理、马尔可夫划分的存在性以及符号映射的构造，揭示了双曲动力系统的结构与性质。
For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\ < J\ > = \underset{n\rightarrow\infty}{\lim} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares sensitivity method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $d\ < J\ > \over ds$ from the solution. This paper proves that as the size of the least squares problem approaches infinity, the computed approximation converges to the true derivative.
A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains a singularity. So, there is an upper bound for the number of attractors obtained from such perturbations. Furthermore, every three-dimensional flow $C^r$ clos…
Fast differentiation of hyperbolic chaos
本文提出了一种用于双曲混沌系统快速求导的算法，通过非侵入性阴影算法和归一化二阶切线方程计算参数导数，并在21维例子中验证了其有效性。
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}.
We study a class of slow-fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At $% ε=0$ the slow dynamics is frozen. We assume that the frozen system (i.e. the unperturbed fast dynamics) has families of hyperbolic periodic orbits with transversal heteroclinics. For each periodic orbit we define an action $J.$ This action may be viewed as an action Hamiltonian (in the slow variables). It has been shown in (N. Brännström, V.Gelfreich (2008)) that there are orbits of the full dynamics which shadow any \emph{finite} combination o…
We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain non\-auto\-nomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to …
A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly simple formula.
We study the omega-limit sets $ω_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $ \{x\in U:ω_Y(x)$ contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the persistence of singular-hyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor \cite{gw} and the example in \cite{mpu}.
On Shadowing and Chain Recurrence in Linear Dynamics
本文研究线性动力学中的跟踪与链回归，证明巴拿赫空间上跟踪与有限跟踪等价，刻画多类空间中相关算子的混合性、分布混沌性等，得到加权移位链回归的完整刻画及广义双曲性与周期跟踪的关系。
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