Periodic orbits and expansiveness

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles.     

Expanding maps of solenoids

We prove that compact metric groups which admit expanding maps must be solenoidal groups, and that every expanding map on a solenoidal group is topologically conjugate to a positively expansive group endomorphism. This first was studied by Shub for expanding differentiable maps of tori and by Manning for Anosov diffeomorphisms of tori.     

Positively expansive differentiable maps

We study the space of positively expansive differentiable maps of a compact connected C ∞ Riemannian manifold without boundary. It is proved that （i） the C1-interior of the set of positively expansive differentiable maps coincides with the set of expanding maps, and （ii） Cl-generically, a differentiable map is positively expansive if and only if it is expanding.     

Expansive measures for flows

We extend the concept of expansive measure [2] from homeomorphism to flows. We prove for continuous flows on compact spaces that every expansive measure has no singularities in the support, is aperiodic, is expansive with respect to time-T maps (but not conversely), remains expansive under topological equivalence, vanishes along the orbits and is natural under suspensions. We apply these properties to prove that there are no expansive flows (in the sense of [26]) of any closed surface.     

Dynamical properties of expansive one-sided cellular automata

To every one-sided one-dimensional cellular automatonF with neighbourhood radiusr we associate its canonical factor defined by considering only the firstr coordinates of all the images of points under the powers ofF. Whenever the cellular automaton is surjective, this factor is a subshift which plays a primary role in its dynamics. In this article we study the class of positively expansive one-sided cellular automata, i.e. those that are conjugate to their canonical factors. This class is a natural generalisation of the toggle or permutative cellular automata introduced in [He]. We prove that the canonical factors of positively expansive one-sided cellular automata are mixing subshifts of finite type that are shift equivalent to full shifts. Moreover, the uniform Bernoulli measure is the unique measure of maximal entropy forF. Consequently, their natural extensions are Bernoulli. We also describe a family of non-permutative positively expansive cellular automata. This research was partially supported by action E9314 of ECOS-CONICYT.     

超凸度量空间中非扩张可交换映象族的几乎不动点

获得了关于超凸度量空间中非扩张可交换映象族不动点存在性的更一般结果.证明了可数个非扩张可交换映象的公共ε-几乎不动点集的超凸性,并在一定条件下证明了任意个非扩张可交换映象的公共ε-几乎不动点集的超凸性.还讨论了Baillon的两个重要结果间的等价关系.     

Compactification of moduli space of harmonic mappings

We introduce the notion of harmonic nodal maps from the stratified Riemann surfaces into any compact Riemannian manifolds and prove that the space of the energy minimizing nodal maps is sequentially compact. We also give an existence result for the energy minimizing nodal maps. As an application, we obtain a general existence theorem for minimal surfaces with arbitrary genus in any compact Riemannian manifolds. Received: 1 April 1997; revised: 15 April 1998.     

Shift equivalence and the Conley index

In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

     

Nondegenerate -biresolving textile systems and expansive automorphisms of onesided full shifts

We study nondegenerate, -biresolving textile systems and using properties of them, we prove a conjecture of Boyle and Maass on arithmetic constraints for expansive automorphisms of onesided full shifts and positively expansive endomorphisms of mixing topological Markov shifts. A similar result is also obtained for expansive leftmost-permutive endomorphisms of onesided full shifts.

     

Existence for VT-harmonic maps from compact manifolds with boundary

We consider a kind of generalized harmonic maps, namely, the VT-harmonic maps. We prove an existence theorem for the Dirichlet problem of VT-harmonic maps from compact manifolds with boundary.

     

A Global Existence Result For The Heat Flow of Harmonic Maps

We consider the heat flow for harmonic maps from R^m to a compact manifold N. When the L^m norm of the gradient of the initial data is small, we prove the existence of a global solution. We prove a similar result for the boundary value problem, when the boundary of the manifold M maps into a point.     

Recent Results on the Dynamics of Higher-dimensional Hénon Maps

We investigate different aspects of chaotic dynamics in Hénon maps of dimension higher than 2. First, we review recent results on the existence of homoclinic points in 2-d and 4-d such maps, by demonstrating how they can be located with great accuracy using the parametrization method. Then we turn our attention to perturbations of Hénon maps by an angle variable that are defined on the solid torus, and prove the existence of uniformly hyperbolic solenoid attractors for an open set of parameters.We thus argue that higher-dimensional Hénon maps exhibit a rich variety of chaotic behavior that deserves to be further studied in a systematic way.     

Nonlinear equations with expansive nonlinearities

Summary We prove existence and multiplicity theorems for nonlinear equations at resonance with expansive nonlinearities.

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Riassunto Si provano teoremi di esistenza e molteplicità per equazioni nonlineari in risonanza con nonlinearità espansiva.

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These results were obtained while the second author was visiting the University of Ferrara through a grant of C.N.R.

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Supported by C.N.R., G.N.A.F.A.     

Banach空间中多类1-集-压缩映象的逼近定理和不动点定理

本文证明了 Ｋｙ Ｆａｎ定理［１］对定义在 Ｂａｎａｃｈ空间中含有内点的有界闭凸子集上的１－集－压缩映象正确．１－集－压缩映象类包含凝聚映象、非扩张映象、半压缩映象、局部几乎非扩张映象和其它映象．作为定理的应用,得到了一些非自映象在许多众知边界条件下的不动点定理,改进和推广了许多作者的最近结果．     

On multiresolution analysis (MRA) wavelets in ℝ n

We prove that for any expansive n×n integral matrix A with |det A|=2, there exist A-dilation minimally supported frequency (MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition |det A|=2 was known to be necessary, and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result for A-dilation MRA subspace wavelets.     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

The classification of topologically expansive lorenz maps

We show that topologically expansive Lorenz maps can be described up to topological conjugacy by their kneading invariants. We also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive Lorenz map. A simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described.     

变分方法与反向上下解

本文在非线性方程的下解不小于上解这一条件（即反向上下解条件）下研究了其解的存在性．我们证明了，如果非线性算子方程有变分结构，有一对反向上下解，对应的算子映某个锥入锥并且满足一定的辅助条件，那么这一方程在锥中至少有两个解．另外，本文还研究了反向上下解条件下非线性椭圆边值问题正解的存在性．     

SHIFTLIKE APPROXIMATIONS AND TOPOLOGICAL MIXING

Abstract

We show that every map in the group G of self-homeomorphisms of the bisequence space can be approximated by homeomorphisms which “look like” the shift map and are expansive. By removing a certain open set of maps from G, we obtain a closed subspace M which contains all mixing maps. If φ · M then any shiftlike approximation to φ is topologically strong mixing. Thus the strong mixing expansive maps are dense in M. Further the weak mixing maps form a dense Gδ sets in M.     

The maximum principle and the Dirichlet problem for Dirac-harmonic maps

We establish a maximum principle and uniqueness for Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold N. Then we prove an existence theorem for a boundary value problem for Dirac-harmonic maps.