弱分离条件下的自共形迭代函数系统

开集条件是分形几何的一个重要概念,弱分离条件(WSC)在研究有重叠的迭代函数系统(IFS)中扮演着重要角色.本文考虑满足弱分离条件的自共形迭代函数系统,并给出确定其不变集的Hausdorff维数的一种方式.     

Graphs induced by iterated function systems

For an iterated function system (IFS) of similitudes, we define two graphs on the representing symbolic space. We show that if the self-similar set \(K\) has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic; moreover the hyperbolic boundaries are homeomorphic to the self-similar sets.     

A generalized finite type condition for iterated function systems

We study iterated function systems (IFSs) of contractive similitudes on Rd with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655-672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811-829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix.     

Fractal Tilings from Iterated Function Systems

A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transformation on the entire space upon which the IFS acts.     

关于IVIFS算子与IFS算子关系的讨论

Atanassov教授在给出区间教直觉模糊集算子的定义以后曾给出断语:直觉模糊集算子满足的结论,都可以推广到相应的区问数直党模糊集算子.本文详细分析多条直觉模糊集算子的性质与相应区间数直党模糊集算子的性质之间的关系,证明这些直觉模糊集算子的性质都不能推广到区间数直党模糊集算子,因而对Atanassov的论断进行了澄清.     

Remarks on limit sets of infinite iterated function systems

If an iterated function system (IFS) is finite, it is well known that there is a unique non-empty compact invariant set K and that K?=???(I ^??), where ?? is the coding map. For an infinite IFS, there are two different sets generalising K, namely ??(I ^??) and its closure ${\overline{\pi(I^\infty)}}$ . In this paper we investigate the relations between these sets and their Hausdorff dimensions. In particular, we show how to construct an IFS for any pair of prescribed dimensions for ??(I ^??) and ${\overline{\pi(I^\infty)}\setminus \pi(I^\infty)}$ . Moreover, we investigate a set which depends only on the first iteration of an IFS, and characterise its relation to the abovementioned sets. This also extends and clarifies recent results by Mihail and Miculescu, who investigated the coding map for an infinite IFS and a condition for this map to be onto. Finally, we study the special case of one-dimensional IFS and show that in terms of the relations of the abovementioned sets these systems exhibit some very special features which do not generalise to higher dimensional situations.     

Ruelle operator with nonexpansive IFS on the line

Ruelle operator defined by weakly contractive iterated function systems (IFS) satisfying the open set condition was discussed in the paper [K.S. Lau, Y.L. Ye, Ruelle operator with nonexpansive IFS, Studia Math. 148 (2001) 143-169]. There, one of our theorems gave a sufficient condition for the possession of the Perron-Frobenius property. In this paper we consider Ruelle operator defined by nonexpansive IFS on the line instead of by weakly contractive one. And we prove, under the same condition, that the newly defined Ruelle operator has the Perron-Frobenius property. It extends the Ruelle-Perron-Frobenius theorem partially to the nonexpansive IFS.     

Self-similar sets satisfying the common point property

A self-similar set is a fixed point of iterated function system (IFS) whose maps are similarities. We say that a self-similar set satisfies the common point property if the intersection of images of the attractor under the maps of the IFS is a singleton and this point has a common pre-image, under the maps of the IFS, and the pre-image is in the attractor.Self-similar sets satisfying the common point property were introduced in Sirvent (2008) in the context of space-filling curves. In the present article we study some basic topological and dynamical properties of self-similar sets satisfying the common point property. We show examples of this family of sets.We consider attractors of a sub-IFS, an IFS formed from the original IFS by removing some maps. We put conditions on this attractors for having the common point property, when the original IFS have this property.     

Real Projective Iterated Function Systems

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points, as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.     

Boundedness of the Hardy–Littlewood Maximal Operator Along the Orbits of Contractive Similitudes

In this note we obtain results regarding the preservation of homogeneity properties along the whole orbit of a given iterated function system (IFS). We have essentially two types of results. The first class of them contains negative results: it is possible for a classical IFS to have a complete non-homogeneous sequence of spaces along the orbit, starting from very classical homogeneous spaces such as those defined by Muckenhoupt weights. The second class contains positive results which can be summarized here by saying that the sequence of spaces defined by the orbit of contractive similitudes starting at a normal space in the sense of Ahlfors, Macías, and Segovia, preserves doubling. As a consequence of these results we conclude boundedness properties of the Hardy–Littlewood maximal operator along the orbits.     

Self-similar groups in the sense of an iterated function system and their properties

The notion of self-similarity in the sense of iterated function system (IFS) for compact topological groups is given by ?. Koçak in Definition 3. In this work, first we give the definition of strong self-similar group in the sense of IFS. Then, we investigate the main properties of these groups. We also obtain the relations between profinite groups and strong self-similar groups in the sense of IFS. Finally, we construct some examples of these groups.     

随机迭代函数系统的仿射变换

对分形图形的一种构造方法——随机迭代函数系统,给出了确定一个随机迭代函数系统的原图像经过仿射变换后得到的新图像所对应的随机迭代函数系统的具体步骤,最后用平移、旋转、拉伸和对称变换的例子作了详细的说明.     

Affine Mappings in Iterated Function Systems

Today the reproduction of pictures in journals and books is no longer possible without “information compression”. This important process needs “Iterated Function Systems”-in short IFS. Doing so, affine mappings and their properties play a fundamental role. In this paper we distinguish between distance contracting and area contracting mappings and we investigate the connections between. The main goal is to find a criterion which allows to decide if the iteration process of an IFS converges or not.     

Uniform perfectness of the attractor of bi-Lipschitz IFS

In this paper, we prove that the attractor of C1,a bi-Lipschitz IFS in R is uniformly perfect if it is not a singleton. Then we construct an example to show that this does not hold for C1 bi-Lipschitz IFS in Rn.     

On the structures of generating iterated function systems of Cantor sets

A generating IFS of a Cantor set F is an IFS whose attractor is F. For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in under the open set condition (OSC). If dim_HF<1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study.     

三角剖分上的仿射迭代函数系统分形插值曲面

对二维平面上三角形区域进行三角剖分,构造仿射变换,由二元分形插值函数引入第三维的值,构成迭代函数系统(IFS).利用此IFS构造了一类山状分形插值曲面.通过数值实验对比分析表明:它比人们以往用矩形剖分得出的分形图形效果更逼真,更接近于自然.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of \mathbbR^d{{\mathbb{R}}^d} . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.     

Valuation theory,generalized IFS attractors and fractals

Using valuation rings and valued fields as examples, we discuss in which ways the notions of “topological IFS attractor” and “fractal space” can be generalized to cover more general settings.     

A new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and IFS invariant measures

In this paper we present a new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and invariant measures associated with iterated function systems (IFS). A multinomial measure is shown to be generated as the invariant measure of an associated IFS. Then, the multifractal spectrum of the measure is determined by a couple of parametric implicit equations. This analysis generalizes some results previously obtained for the case of single-scaled multinomial measures (e.g., the binomial measure). A geometric interpretation of this new framework working in the space of codes of the IFS gives new insight into the nature of the multifractal formalism. This paper extends the results presented in Gutiérrez et. al. Fractals 4, (1996) 17–27.     

基于一类新的胞腔排除遗传算法求解迭代函数系逆问题

提出求解迭代函数系（IFS）逆问题的一类有效遗传算法,该算法基于新发展的可拼接/可分解编码,并结合使用胞腔排除技巧,对于典型图像的应用表明;该方法可有效应用于基于矩匹配表示的IFS逆向题求解,从而为IFS逆问题的数值方法研究提供了一条新颖途径。