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.     

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.     

Analysis of μ R,D -Orthogonality in Affine Iterated Function Systems

Let μ _R,D be a self-affine measure associated with an expanding integer matrix R∈M _n (ℤ) and a finite subset D⊆ℤ ⁿ . In the present paper we study the μ _R,D -orthogonality and compatible pair conditions. We also show that any set of μ _R,D -orthogonal exponentials contains at most 3 elements on the generalized plane Sierpinski gasket and the number 3 is the best.      

A Contractivity Condition for Iterated Function Systems

We prove a condition for long-term contractivity and the existence of a unique invariant measure for iterated function systems. We also give an intuitive interpretation of the condition in terms of weighted derivatives and weighted Wasserstein metrics. We use our condition in order to show some results for stochastic population models based on the logistic and Ricker maps.     

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．     

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.     

Exceptional Parameters for Iterated Function Systems with Overlaps

Periodica Mathematica Hungarica -     

Rigidity of Conformal Iterated Function Systems

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conformal, satisfying the Open Set Condition, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative d/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure m _F to the equivalence class of the conformal measure m _G , then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.     

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.     

Non-Linear Iterated Function Systems with Overlaps

Periodica Mathematica Hungarica -     

迭代函数系统的吸引子的维数估计

某些迭代函数系统（ＩＦＳ）的吸引子的Ｈａｕｓｄｏｒｆ维数的经典估计通常由求解两个非线性方程的解得到．在相同条件下，本文给出了一些显式不等式估计，使经典估计是其中的特殊情形．而且某些ＩＦＳ的吸引子的维数可以通过求解满足一定条件的非线性方程而得出     

Iterated Function Systems with Overlaps and Self-Similar Measures

The paper considers the iterated function systems of similitudeswhich satisfy a separation condition weaker than the open setcondition, in that it allows overlaps in the iteration. Suchsystems include the well-known Bernoulli convolutions associatedwith the PV numbers, and the contractive similitudes associatedwith integral matrices. The latter appears frequently in waveletanalysis and the theory of tilings. One of the basic questionsis studied: the absolute continuity and singularity of the self-similarmeasures generated by such systems. Various conditions to determinethe dichotomy are given.     

Diophantine Analysis of Conformal Iterated Function Systems

 A formula for the Hausdorff dimension of a subset of the limit set of a conformal infinite iterated function system is derived and expressed as a unique zero of the topological pressure of corresponding strongly H?lder family of functions. Received August 10, 2001; in revised form February 23, 2002     

Duality for Curves in Affine Plane Geometry

The dual of a plane curve in equiaffine geometry is defined as a curve in the space of conics. The image and inverse of this duality are described. It is shown that the duality transforms equiaffine vertices into cusps. As an application, an analogue of Kneser's lemma for osculating conics is proved.     

Ergodic Theorems for Iterated Function Systems Controlled by Regenerative Sequences

Iterated function systems are considered, where the function to iterate in each step is determined by a regenerative sequence. Ergodic theorems of distributional and law of large numbers types are obtained under log-average contractivity conditions.     

Fixed Point Theorems of the Iterated Function Systems

In this paper, we present some fixed point theorems of iterated function systems consisting of α-ψ-contractive type mappings in Fractal space constituted by the compact subset of metric space and iterated function systems consisting of Banach contractive mappings in Fractal space constituted by the compact subset of generalized metric space, which is also extensively applied in topological dynamic system.     

Generalized Iterated Function Systems with Place Dependent Probabilities

In the present paper we introduce the concept of generalized iterated function system with place dependent probabilities (GIFSpdp). In this framework, we provide sufficient conditions under which the Markov operator associated to a GIFSpdp is Lipschitz. We also prove, under certain conditions, the existence of an analogue of Hutchinson measure associated to a GIFSpdp and study its properties.     

The Box counting Dimension of Invariant Sets for Iterated Function Systems

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定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.