The Tangent Measure Distribution of Self-Conformal Fractals

We show that the tangent measure distribution of a self-conformal measure exists at almost all points of the support of the measure. Moreover, we prove, that it is the same for almost all points.     

Local Geometry of Fractals Given by Tangent Measure Distributions

 Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ. (Received 10 October 2000, in revised form 8 March 2001)     

Local Geometry of Fractals Given by Tangent Measure Distributions

 Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

The Average Density of Self-Conformal Measures

The paper calculates the average density of the normalized Hausdorffmeasure on the fractal set generated by a conformal iteratedfunction system. It equals almost everywhere a positive constantgiven by a truncated generalized s-energy integral, where sis the corresponding Hausdorff dimension. As a main tool a conditionalGibbs measure is determined. The appendix proves an appropriateextension of Birkhoff's ergodic theorem which is also of independentinterest.     

Harmonic Calculus on Fractals - A Measure Geometric Approach I

Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linear Cantor-type measures with arbitrary weights.     

Key Distribution Patterns Using Tangent Circle Structures

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In O'Keefe [5–7], examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in Quinn [12], examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in Quattrocchi and Rinaldi [10] and we give a comparison of the storage requirements.     

Lipschitz Equivalence of Self-Conformal Sets

The paper proves that any two dust-like invariant sets of conformaliterated function systems have the same Hausdorff dimensionif and only if they are nearly Lipschitz equivalent.     

基于小波分形理论的股价指数信息量测度研究

本把小波分析和分形理论引入到股价指数时间序列的分析中，给出了股价指数波动复杂性的信息量测度方法——信息熵和分形维方法。通过对上证综指和深证成指1994年1月3日至2002年3月4日期间的数据进行的实证分析显示，两种方法均能刻画股价指数波动的复杂程度，这对初步了解我国股市场的波动规律有一定的实际意义。     

Fractals Everywhere

Abstract

Like every other component of government these days, the U.S. federal statistical system is being reinvented, reengineered, and downsized. At this writing, several bills are before the U.S. Congress to restructure the system, relocate key agencies, and reallocate programs and budgets. Janet L. Norwood, Senior Fellow with The Urban Institute and former Commissioner of Labor Statistics, offers a trenchant commentary on the situation. Norwood identifies serious problems with the U.S. federal statistical system, as it is currently organized, that need to be addressed. At the same time, she notes the grave danger that solutions attempted by the U.S. Congress may make matters worse. She ends with a call to action for those who “care about data quality and who understand the important uses of the government's statistical output.”     

随机分形

本文概括了随机分形的主要结果，综述了随机分形的最新进展和目前的动态，提出了一些末解决的问题，全文共分为三部分：（１）由随机过程和随机场（如Ｌｅｖｙ过程，Ｇａｕｓｓ场，自相似过程等）产生的各种随机分形集（如象集、水平集、Ｋ重点集等）的Ｈａｕｓｄｏｒｆｆ维数、测度和ｐａｃｋｉｎｇ维数、测试；（２）随机Ｃａｎｔｏｒ型集和统计自相似集的维数和测试；（３）分形集（如Ｓｐｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ，ｎｅ     

强影响的分布与稳定性度量

本文考虑了数据集在样本空间的影响分布,并且指出数据集仅对与之强相关的数据点有较大的影响,因而Cook距离(?)中这一部分数据处的影响较突出,我们考虑用(?)作为影响度量将更合理,同时它也比Cook距离具有更好的稳定性。     

MoranFractals的推广及其多重分形分解

本文给出Ｍｏｒａｎｆｒａｃｔａｌｓ的一个推广并且研究了其多重分形分解问题。     

由表示系统生成的分形的维数

在这篇文章里,由Ｒｎ中点的表示系统所生成的自仿射集中,给出了自仿射集满足Ｍｏｒａｎ开集条件的一个新的判别方法和不满足开集条件的自相似集的Ｈａｕｓｄｏｒｆｆ维数的上界和下界,并根据两个实例估计出其上下界是相等的．     

Diophantine Approximations on Fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]², possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod 1}_{n ? \mathbb N}{\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}} are uniformly eventually bounded.     

Fractals and Bernstein polynomials

Self-similarity of Bernstein polynomials, embodied in their subdivision property is used for construction of an Iterative (hyperbolic) Function System (IFS) whose attractor is the graph of a given algebraic polynomial of arbitrary degree. It is shown that such IFS is of just-touching type, and that it is peculiar to algebraic polynomials. Such IFS is then applied to faster evaluation of Bézier curves and to introduce interactive free-form modeling component into fractal sets.     

A New Class of Harmonic Measure Distribution Functions

Let Ω be a planar domain containing 0. Let h _Ω (r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h _Ω is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X _n such that \(h_{X_{n}}\) converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h _Ω , where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X _n . Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h _Ω for a unique symmetric bounded simply connected domain Ω.