Periodic shadowing and Ω-stability

We show that the following three properties of a diffeomorphism f of a smooth closed manifold are equivalent: (i) f belongs to the C ¹-interior of the set of diffeomorphisms having the periodic shadowing property; (ii) f has the Lipschitz periodic shadowing property; (iii) f is Ω-stable.     

<Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript> transversality and shadowing properties

Let f be an Axiom A diffeomorphism of a closed smooth two-dimensional manifold. It is shown that the following statements are equivalent: (a) f satisfies the C ⁰ transversality condition, (b) f has the shadowing property, and (c) f has the inverse shadowing property with respect to a class of continuous methods.     

Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

The structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C ¹-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.     

Inverse shadowing and related measures In Memory of Professor Shantao Liao

We study various weaker forms of the inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property(Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing(any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory).We demonstrate that this property is closely related to structural stability and ?-stability of diffeomorphisms.     

SHADOWING,EXPANSIVENESS AND SPECIFICATION FOR C~1-CONSERVATIVE SYSTEMS

We prove that a C~1-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,as in[10,27],we prove that the C~1-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.     

$\mathbb{Z}^{k}$-作用一维子系统的Lipschitz跟踪性

本文主要研究了$\mathbb{Z}^{k}$-作用一维子系统的跟踪性质. 文中运用两种等价的方式引入了$\mathbb{Z}^{k}$-作用一维子系统的伪轨以及跟踪性的概念. 对于一个闭黎曼流形上的光滑$\mathbb{Z}^{k}$-作用$T$, 我们通过诱导的非自治动力系统提出了Anosov方向的概念. 借助Bowen几何的方法, 我们证明了$T$沿着任意Anosov方向具有Lipschitz跟踪性.     

Lipschitz shadowing and structural stability of flows

Let ? be the flow generated by a smooth vector field X on a smooth closed manifold. We show that the Lipschitz shadowing property of ? is equivalent to the structural stability of X and that the Lipschitz periodic shadowing property of ? is equivalent to the Ω-stability of X.     

On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.     

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

On bi-Lyapunov stable homoclinic classes

For a C ¹ generic diffeomorphism if a bi-Lyapunov stable homoclinic class is homogeneous then it does not have weak eigenvalues. Using this, we show that such homoclinic classes are hyperbolic if it has one of the following properties: shadowing, specification or limit shadowing.     

具有跟踪性质的连续半流

在这篇论文中，我们给出了连续半流的跟踪性质与其逆极限的跟踪性质之间的一些等价条件，并且做为应用，我们证明了具有强跟踪性质的连续半流在其游荡集上的限制也具有强跟踪性质以及连续半流的谱分解定理。     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

General hyperbolicity conditions for set-valued mappings

A definition of hyperbolicity for dynamical systems generated by set-valued mappings of general form in terms of local selectors is given. It is shown that a system hyperbolic in this sense has the shadowing and inverse shadowing properties. It is also shown that the hyperbolicity property holds true for a certain class of set-valued mappings where images of points are convex polytopes. Bibliography: 13 titles.     

From the Boundary of the Convex Core to the Conformal Boundary

If N is a hyperbolic 3-manifold with finitely generated fundamental group, then the nearest point retraction is a proper homotopy equivalence from the conformal boundary of N to the boundary of the convex core of N. We show that the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse and that one may bound the Lipschitz constants in terms of the length of the shortest compressible curve on the conformal boundary.     

Diffeomorphisms as time one maps

Summary. We investigate the inverse ODE problem of finding a vector field such that the time one map associated to its flow coincides with a given diffeomorphism. Using a constructive approach we solve this problem for a class of diffeomorphisms having a globally attracting fixed point. Furthermore we consider how the solution fields depend on the diffeomorphism. As an example we show that for certain parameters, the Hénon map is the time one map of a two dimensional flow.     

Generic points for dynamical systems with average shadowing

We study the Besicovitch pseudometric \(D_B\) for compact dynamical systems. The set of generic points of ergodic measures turns out to be closed with respect to \(D_B\). It is proved that the weak specification property implies the average asymptotic shadowing property and the latter property does not imply the former one nor the almost specification property. Furthermore an example of a proximal system with the average shadowing property is constructed. It is proved that to every invariant measure \(\mu \) of a compact dynamical system one can associate a certain asymptotic pseudo orbit such that any point asymptotically tracing in average that pseudo orbit is generic for \(\mu \). A simple consequence of the theory presented is that every invariant measure has a generic point in a system with the asymptotic average shadowing property.     

具有渐近平均跟踪性质的系统

简记渐近平均跟踪性质为AASP.对于紧致度量空间上的连续映射f,证明了:(1)f有AASP当且仅当其逆极限空间上的移位映射有AASP;(2)若f有AASP且是等度连续的,则f是极小同胚.此外,讨论了AASP的拓扑共轭不变性.     

Rn上无奇点C1流的强极限跟踪性

本文证明了Rⁿ上无奇点C¹流的双曲集附近具有强极限跟踪性。     

Chaos and Shadowing Around a Heteroclinically Tubular Cycle With an Application to Sine-Gordon Equation

The theory of chaos and shadowing developed recently by the author is amplified to the case of a heteroclinically tubular cycle. Specifically, let F be a   C ³  diffeomorphism on a Banach space. F has a heteroclinically tubular cycle that connects two normally hyperbolic invariant manifolds. Around the heteroclinically tubular cycle, a Bernoulli shift dynamics of submanifolds is established through a shadowing lemma. As an example, a sine-Gordon equation with a chaotic forcing is studied. Existence of a heteroclinically tubular cycle is proved. Also proved are chaos associated with the heteroclinically tubular cycle, and chaos cascade referring to the embeddings of smaller-scale chaos in larger-scale chaos.     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.