Whitney集与图递归弧

通过构造具有有向图结构的迭代函数系通的子系通,证明了HausdorfF维数大于1的图递归弧均为Whitney集,该结果不需要有向图满足传递条件．     

自相似集的滑动

利用有向图结构与鞅论讨论了自相似集滑动过程中的Lipschitz等价问题.     

自相似集的Hausdorff测度——Koch曲线

讨论满足开集条件的自相似集 .对于这样一个分形 ,用定义估计它的Haus dorff测度只能得到上限 ,因而如何判断某一个上限是否就是它的准确值是一个重要问题.给出了一个否定判据 .作为应用 ,否定了Marion关于Koch曲线的Hausdorff测度的猜测.     

广义自相似集的维数研究

广义自相似集的维数研究华苏（清华大学应用数学系，北京１０００８４）ＯＮＴＨＥＤＩＭＥＮＳＩＯＮＯＦＧＥＮＥＲＡＬＩＺＥＤＳＥＬＲ－ＳＩＭＩＬＡＲＳＥＴＳ￥ＨＵＡＳＵ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉ...     

随机次自相似集的表示

本文引进了随机次自相似集与随机推移集的概念,讨论了随机次自相似集的结构,并证明了任一随机集是随机次自相似集的充分必要条件是：该随机集可以表为某一个随机推移集的某个像集。     

统计自相似集的重分形分解

设K是由Graf定义的统计自相似集.本文构造了一支撑在K上的随机测 度p,并研究了结合P的K的重分形分解,得到了分形谱函数f(α)的表达式.     

关于自相似集的Hausdorff测度的一个判据及其应用

讨论了满足开集条件的自相似集。对于此类分形，用自然覆盖类估计它的Hausdorff测度只能得到一个上限，因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用，统一处理了一类自相似集，得到了平面上的一个Cantor集－Cantor尘的Hausdorff测度的准确值，并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。     

｛1,4,5｝自相似集的摄动和强分离参数集

对于自相似集Eλ=Eλ/5U(Eλ/5+λ/5)∪(Eλ/5+4/5),本文讨论了λ靠近3(对应于{1,4,5}集)时, Eλ的维数逼近性质,强分离性, Lipschitz等价性等.此外,还探讨了参数集{λ:Eλ不满足强分离条件}的几何结构.     

关于自相似集的Hausdorff测度

得到了 Ｈａｕｓｄｏｒｆｆ容度与 Ｈａｕｓｄｏｒｆｆ测度相等的集的充分必要条件．对于满足开集条件的自相似集,验证了它的Ｈａｕｓｄｏｒｆｆ容度与Ｈａｕｓｄｏｒｆ测度相等并给出了它的Ｈａｕｓｄｏｒｆｆ测度的一个便于应用的公式．作为例子,给出了均匀康托集的Ｈａｕｓｄｏｒｆｆ测度的一种新的计算方法,对于Ｋｏｃｈ曲线的Ｈａｕｓｄｏｒｆｆ测度的上限也作了讨论．     

点集Bézier曲线

提出了点集Bézier曲线的概念,给出了点集Bézier曲线的性质及细分算法.按照点集算术的定义,当点集是长方形闭域或圆盘时,点集Bézier曲线就是区间Bézier曲线或圆盘Bézier曲线,因此,点集Bézier曲线是对区间Bézier曲线和圆盘Bézier曲线的推广.     

Computability of Self-Similar Sets

We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar set is a recursive compact set. A further result on the case that the self-similar set forms a curve is also discussed.     

自相似集的Hausdorff测度与连续性

对集合F Rn,以dim F和Hdim F(F)分别表示F的Hausdorff维数和dim F维Hausdorff测度.设T=T(f1,...,fm)为Rn中的自相似集,即由相似压缩组成的迭代函数系统{f1...,fm)的吸引子.假如fi(T)∩fj(T)= (i≠j),那么,对任意ε>0,存在δ>0,若D=D(g1,...,gm)为Rn中的自相似集并且sup{||fk(x)-gk(x)||:||x||≤1,1≤k≤m}<δ,则1HdimT(T)-Hdim D(D)|<ε.     

A counterexample concerning Whitney reversible properties

Two properties of continua are shown which are strong Whitney reversible while not sequential strong Whitney reversible. This answers an old question of S. B. Nadler, Jr.

     

Gap Property of Bi-Lipschitz Constants of Bi-Lipschitz Automorphisms on Self-similar Sets

For a given self-similar set E ∪→ R^d satisfying the strong separation condition, let Aut（E） be the set of all bi-Lipschitz automorphisms on E. The authors prove that {f ∈ Aut（E） ： blip（f） = 1} is a finite group, and the gap property of bi-Lipschitz constants holds, i.e., inf{blip（f） ≠ 1： f ∈ Aut（E）} 〉 1, where lip（g） =sup x,y∈E x≠y ｜g（x）-g（y）｜/｜x-y｜ and blip（g） =max（lip（g）, liP（g^-1））.     

Whitney numbers of matroid extensions and co-extensions

Crapo found that single-element extensions of matroid are equivalent to modular cuts. This paper will study the single-element extension and co-extension of a matroid and establish the upper semi-continuity for the signless coefficients of Whitney polynomials and Whitney numbers on both extensions via its extension lattice, a lattice of all non-empty modular cuts.     

Doubling measures on Whitney modification sets

A Whitney modification set in R is a set of the form E∪F(W), where E is a nonempty closed set in R, W is a Whitney decomposition of R?E, and F(W) consists of centers of intervals in W. We prove that a measure on a Whitney modification set is doubling if and only if it is the Whitney modification of a doubling measure on R.     

平面上一类自相似集的Hausdorff测度与上凸密度

考虑平面单位正方形内生成的一类自相似集的Hausdorff测度的计算问题.在满足强分离条件及维数小于1的条件下,当相似比满足某些条件时,证明了自然覆盖为其实现上凸密度1计算的最好形状,因而自然覆盖即是最好覆盖.而作为它的直接推论得到该类自相似集的Hausdorff测度的精确值为(2s)~/(1/2),其中s为其Hausdorff维数.     

Whitney towers and gropes in 4-manifolds

Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to -solvability, -cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.