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。     

Stability of orbit spaces of endomorphisms

In this paper it is first proved that, for a hyperbolic set of aC ¹ (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC ¹-small perturbations and is semi-stable underC ⁰-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC ¹-small perturbations and is semi-Θ-stable and semi-R-stable underC ⁰-small perturbations. This research is supported by the National Natural Science Foundation of China     

线性系统的极限跟踪性

给出了Rⁿ上的线性同构和线性流具有极限跟踪性的特征：线性同构具有极限跟踪性当且仅当其对应的矩阵为双曲的;线性流具有极限跟踪性当且仅当其对应矩阵的所有特征根均具有非零实部.     

两类具有极限跟踪性的双曲系统

本文讨论了紧度量空间上连续满射及同胚的一类特殊的跟踪性—极限跟踪性的几个基本性质,并证明了R~n上双曲自同构及环面T~n上双曲自同态(n≥1)具有极限跟踪性。     

Vector fields with stable shadowable property on closed surfaces

We prove that the set of vector fields satisfying the C1 stable shadowable property on closed surfaces is characterized as the set of Morse-Smale vector fields. Hence, the vector fields satisfying shadowing property on closed surfaces are C1 dense.     

<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.     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.     

强链回归集与强跟踪性

为了研究强跟踪性，本文给出了强链回归集的定义．证明了：若度量空间上的一个连续自映射有强跟踪性，则其强链回归集与极限集相同．     

Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles

The C ¹ density conjecture of Palis asserts that diffeomorphisms exhibiting either a homoclinic tangency or a heterodimensional cycle are C ¹ dense in the complement of the C ¹ closure of hyperbolic systems. In this paper we prove some results towards the conjecture.* Work supported by the National Natural Science Foundation and the Doctoral Education Foundation of China, and the Qiu Shi Science and Technology Foundation of Hong Kong.     

具有跟踪性可扩流的基本集

本文研究了具有跟踪性可扩流的谱分解中基本集的整体性质,其中包括稳定集和不稳定集的性质、无环性及汇和源的存在性.     

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system.The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of orderε| ln ε|. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.     

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC ¹ functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian. Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province (105129) Research supported by Academy of Finland     

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.     

DIFFEOMORPHISMS WITH VARIOUS C1 STABLE PROPERTIES

Let M be a smooth compact manifold and Λ be a compact invariant set.In this article,we prove that,for every robustly transitive set Λ,f|Λ satisfies a C1-genericstable shadowable property (resp.,C1-gene...     

Theory of pseudo-orbit shadowing in dynamical systems

This is a survey of the main results obtained in the theory of pseudo-orbit shadowing in dynamical systems in the first decade of the 21st century. The main directions are shadowing and structural stability, C ¹-interiors of sets of systems with the shadowing property, shadowing properties equivalent to the structural stability, and the denseness problem.     

Weak shadowing for discrete dynamical systems on nonsmooth manifolds

In J. Math. Anal. Appl. 189 (1995) 409-423, Corless and Pilyugin proved that weak shadowing is a C⁰ generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set.     

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.     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

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.     

Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

Shadowing trajectories are one of the most powerful ideas of modern dynamical systems theory, providing a tool for proving some central theorems and a means to assess the relevance of models and numerically computed trajectories of chaotic systems. Shadowing has also been seen to have a role in state estimation and forecasting of nonlinear systems. Shadowing trajectories are guaranteed to exist in hyperbolic systems, but this is not true of nonhyperbolic systems, indeed it can be shown there are systems that cannot have long shadowing trajectories. In this paper we consider what might be called shadowing pseudo-orbits. These are pseudo-orbits that remain close to a given pseudo-orbit, but have smaller mismatches between forecast state and verifying state. Shadowing pseudo-orbits play a useful role in the understanding and analysis of gradient descent noise reduction, state estimation, and forecasting nonlinear systems, because their existence can be ensured for a wide class of nonhyperbolic systems. New theoretical results are presented that extend classical shadowing theorems to shadowing pseudo-orbits. These new results provide some insight into the convergence behaviour of gradient descent noise reduction methods. The paper also discusses, in the light of the new results, some recent numerical results for an operational weather forecasting model when gradient descent noise reduction was employed.