Making the Tent function complex

This note can be used to illustrate to the student such concepts as periodicity in the complex plane. The basic construction makes use of the Tent function which requires only that the student have some working knowledge of binary arithmetic.     

On the predictability of discrete dynamical systems III

Given an integer n?2, a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that “most” homeomorphisms are non-sensitive μ-almost everywhere on X. Moreover, we also prove that for “most” homeomorphisms the non-wandering set Ωf has μ-measure zero.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of C~n

In this paper,the authors establish distortion theorems for various subfamilies H_k(B)of holomorphic mappings defined in the unit ball in C~n with critical points,where k is any positive integer.In particular,the distortion theorem for locally biholomorphic mappings is obtained when k tends to ∞.These distortion theorems give lower bounds on|det f′(z)|and Re det f′(z).As an application of these distortion theorems,the authors give lower and upper bounds of Bloch constants for the subfamiliesβ_k(M)of holomorphic mappings.Moreover,these distortion theorems are sharp.When B is the unit disk in C,these theorems reduce to the results of Liu and Minda.A new distortion result of Re det f′(z)for locally biholomorphie mappings is also obtained.     

On the use of the topological degree theory in broken orbits analysis

Dynamical systems in are studied. Let be a bounded open set. We will be interested in those periodic orbits such that at least one of its points lies inside and at least one of its points lies outside ; the orbits with this property are called -broken. Information about the structure of the set of -broken orbits is suggested; results are formulated in terms of topological degree theory.

     

A Note on Schwarz-Pick Lemma for Bounded Complex-Valued Harmonic Functions in the Unit Ball of Rn

In this paper, the authors prove a Schwarz-Pick lemma for bounded complexvalued harmonic functions in the unit ball of $\mathds{R}^n$.     

Devaney混沌的随机性质

讨论了Devaney混沌的随机性质,证明了如果度量空间上的连续变换f是弱混合的,那么f是拓扑传递的,并且若f的周期点稠密,则f还是初值敏感的.     

Remarks on the three ingredients of chaos

This article is a collection of remarks about the three ingredients – transitivity, denseness of periodic points and sensitivity – in one of the definitions of chaos in topological dynamics.     

On the absence of cycles for unimodal mappings of the square

Unimodal 2-endomorphisms of a special kind of the unit square are considered. Conditions for the absence of periodic points are presented. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 370–378, March, 1998.     

On the periodic points of functions on a manifold

In a conference on fixed point theory, B. Halpern of Indiana University considered the problem of reducing the number of periodic points of a map by homotopy. He also asked whether the number of periodic points of a function could be increased by a homotopy. In this paper, we will show that for any map on a closed manifold, an arbitrarily small perturbation can always create infinitely many periodic points of arbitrarily high periods.

     

PL Homeomorphisms of the circle which are piecewise <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> conjugate to irrational rotations

For a PL homeomorphism f with irrational rotation number , the following properties are equivalent(i) f is conjugate to the rotation by through a piecewise C ¹ homeomorphism,(ii) the number of break points of f ⁿ is bounded by some constant that doesnt depend on n,(iii) f is conjugate to an affine 2-intervals exchange transformation (with rotation number ) through a PL homeomorphism,(iv) f is conjugate to the rotation by through a piecewise analytic homeomorphism.     

Points of Weak-Norm Continuity in the Unit Ball of Banach Spaces

Using the M-structure theory, we show that several classical function spaces and spaces of operators on them fail to have points of weak-norm continuity for the identity map on the unit ball. This gives a unified approach to several results in the literature that establish the failure of strong geometric structure in the unit ball of classical function spaces. Spaces covered by our result include the Bloch spaces, dual of the Bergman space L¹_a and spaces of operators on them, as well as the space C(T)/A, where A is the disc algebra on the unit circle T. For any unit vector f in an infinite-dimensional function algebra A we explicitly construct a sequence {f_n} in the unit ball of A that converges weakly to f but not in the norm.     

C^n中单位球上的强拟凸映照

若记B^n为C^n中的单位球,本文研究B^n上的强拟凸映强,（1）强拟凸映照与星形映照及凸映照的关系,（2）强拟凸映照的二次项系;（3）单位多圆柱上的强拟凸映照。     

周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

Independence of almost periodicity upon the topology of the acting group

Answering a question first explicitly stated by de Vries in 1993, we observe that for an arbitrary topological dynamical system the property of being an almost periodic point does not depend on the topology of the acting group. In other words, the traditional distinction made between the notions of an almost periodic point and of a discretely almost periodic point is unnecessary.     

Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

Random Dynamics of the Boussinesq System with Dynamical Boundary Conditions

Abstract

A coupled system of the two-dimensional Navier–Stokes equations and the salinity transport equation with spatially correlated white noise on the boundary as well as in fluid is investigated. The noise affects the system through a dynamical boundary condition. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain asymptotic estimates on system evolution, and then show that the long time dynamics is captured by a random attractor.     

单位球上的对偶Toeplitz算子

本文研究了单位球Bergman空间的直交补上的对偶Toeplitz算子的代数性质,首先我们给出了对偶Toeplitz算子的有界性和紧性的完全刻画,然后给出对偶Toeplitz算子的谱性质,最后证明了不存在以有界全纯或者反全纯函数为符号的拟正规对偶Toeplitz算子.