无穷迭代函数系统的遍历定理

度量空间的压缩映射的一个集合称为一个迭代函数系统．凝聚迭代函数系统可以被看成无穷迭代函数系统．研究了紧度量空间上的无穷迭代函数系统．利用Banach极限的特性和均匀压缩性,证明了紧度量空间上无穷迭代函数系统的随机迭代算法满足遍历性．于是,凝聚迭代函数系统的随机迭代算法也满足遍历性．     

The Existence of the Attractor of Countable Iterated Function Systems

The theory of iterated function systems (IFS) and of infinite iterated function systems consisting of contraction mappings has been studied in the last decades. Some extensions of the spaces and the contractions concern many authors in fractal theory. In this paper there are described some results in that topic concerning the existence and uniqueness of nonempty compact set which is a set ”fixed point” of a countable iterated function system (CIFS). Moreover, some approximations of the attractor of a CIFS by the attractors of the partial IFSs are given.     

Edelstein's fixed point theorem in topological spaces

An extension to topological spaces of a wellknown fixed point theorem of M. Edelstein for contractive mappings on metric spaces is presented. Results based on the generalized Edelstein's theorem are also established concerning the existence of fixed points of continuous selfmaps on a topological space. As a special case a compact starshaped subset of a linear topological space is considered. The results extend the fixed point theoremsfor nonexpansive mappings on a compact metric space of L.F.Guseman, Jr. and B.C. Peters, Jr.     

Invariant Measure Associated with a Generalized Countable Iterated Function System

Generalized countable iterated function systems (GCIFS) are an extension of countable iterated function systems by considering contractions from X × X into X instead of contractions on the compact metric space X into itself. For a GCIFS endowed with a system of probabilities we associate an invariant and normalized Borel measure whose support is just the attractor of the respective GCIFS, extending the classical Hutchinson’s construction.     

Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.     

Cone metric spaces and fixed point theorems in diametrically contractive mappings

In this paper, some topological concepts and definitions are generalized to cone metric spaces. It is proved that every cone metric space is first countable topological space and that sequentially compact subsets axe compact. Also, we define diametrically contractive mappings and asymptotically diametrically contractive mappings on cone metric spaces to obtain some fixed point theorems by assuming that our cone is strongly minihedral.     

Overlapping iterated function systems on a segment

Overlapping iterated function systems generate families of injective mappings from the attractor onto shift-invariant subsets of the code space. In this paper we consider an example of such a family for the uniformly linear systems of iterated functions on the unit segment.     

Isometries of spaces of compact or compact convex subsets of metric manifolds

The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.     

σ-porosity of the set of strict contractions in a space of non-expansive mappings

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uniform convergence. We show that the set of strict contractions is a σ-porous subset. If the underlying Banach space is separable, we exhibit a σ-porous subset of the space of non-expansive mappings outside of which all mappings attain the maximal Lipschitz constant one at typical points of their domain.     

Typical property of the topological entropy of continuous mappings of compact sets

We show that the topological entropy viewed as a functional on the space of continuous mappings of a metric compact set into itself with the uniform topology is a function of the second Baire class and is lower semicontinuous at a Baire typical point. In particular, we show that the topological entropy is zero at a Baire typical point of the space of continuous mappings of the Baire space of sequences of zeros and units.     

Generic Existence of Fixed Points for Set-Valued Mappings

We first consider a complete metric space of nonexpansive set-valued mappings acting on a closed convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then establish analogous results for two complete metric spaces of set-valued mappings with convex graphs.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

SOME NEW ITERATED FUNCTION SYSTEMS CONSISTING OF GENERALIZED CONTRACTIVE MAPPINGS

Iterated function systems (IFS) were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle.In 2010,D.R.Sahu and A.Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings.In this paper,following Hutchinson,D.R.Sahu and A.Chakraborty,we present some new iterated function systems by using the so-called generalized contractive mappings,which will also cover a large range of mappings.Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.     

On different forms of self similarity

Fractal geometry is mainly based on the idea of self-similar forms. To be self-similar, a shape must able to be divided into parts that are smaller copies, which are more or less similar to the whole. There are different forms of self similarity in nature and mathematics. In this paper, some of the topological properties of super self similar sets are discussed. It is proved that in a complete metric space with two or more elements, the set of all non super self similar sets are dense in the set of all non-empty compact sub sets. It is also proved that the product of self similar sets are super self similar in product metric spaces and that the super self similarity is preserved under isometry. A characterization of super self similar sets using contracting sub self similarity is also presented. Some relevant counterexamples are provided. The concepts of exact super and sub self similarity are introduced and a necessary and sufficient condition for a set to be exact super self similar in terms of condensation iterated function systems (Condensation IFS’s) is obtained. A method to generate exact sub self similar sets using condensation IFS’s and the denseness of exact super self similar sets are also discussed.     

Topological classification of scattered IFS-attractors

We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal δ   we show that a countable compact space of height δ+1$\delta +1$ can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

Extension of functions preserving the modulus of continuity

The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 236–245, February, 1997. Translated by N. K. Kulman     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.     

关于序列覆盖紧映射

本文利用了ｃｓ网、序列邻域网、序列开网和弱基的概念,讨论了空间中点正则覆盖,一致覆盖和点有限覆盖的点星网之间的关系．建立了度量空间在几类序列覆盖（紧）映射下象空间的特征,特别地证明了度量空间的序列覆盖（或１序列覆盖）紧映象等价于具有点正则ｃｓ网的空问,回答了Ｔａｎａｋａ等提出的一个问题．     

一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.