Exact Hausdorff measure and intervals of maximum density for Cantor sets
Exact Hausdorff measure and intervals of maximum density for Cantor sets
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DOI：
10.1090/s0002-9947-99-01982-0
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发表时间：
1999-01
影响因子：
1.3
通讯作者：
E. ayer;R. Strichartz
中科院分区：
文献类型：
--
作者：
E. ayer;R. Strichartz
考虑一个线性Cantor集K，它是一个线性迭代函数系统（i.f.s.）Sjx = pjx + bj，j = 1，.，m，在满足开集条件的直线上（其中开集是区间）。已知K具有Hausdorff维数ca，由方程FTL 1 pa = 1给出，并且7-t（K）是有限的和正的，其中7 H，x表示维数a的Hausdorff测度。我们给出了一个算法，计算7-t（K）正是作为一个有限的一组基本函数的参数的i.f.s.当pi = Pm时（或者更一般地，如果log p1和log pm是可交换的），该算法还给出了使密度d（I）= 7 'Ha（Kn I）/ II最大化的区间I。Hausdorff测度7-o，（K）不是i.f.s.的连续函数。参数我们还表明，给定收缩参数pj，可以选择平移参数bj，使得7-H（K）= IKIa，因此最大密度为1。本文中的大多数结果都是通过计算机实验发现的，但我们给出了传统的数学证明。
Consider a linear Cantor set K, which is the attractor of a linear iterated function system (i.f.s.) Sjx = pjx + bj, j = 1,... , m, on the line satisfying the open set condition (where the open set is an interval). It is known that K has Hausdorff dimension ca given by the equation FTL1 pa = 1, and that 7-t (K) is finite and positive, where 7H,x denotes Hausdorff measure of dimension a. We give an algorithm for computing 7-t (K) exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When pi = Pm (or more generally, if log p1 and log pm are commensurable), the algorithm also gives an interval I that maximizes the density d(I) = 7'Ha(K n I)/ II I. The Hausdorff measure 7-o,(K) is not a continuous function of the i.f.s. parameters. We also show that given the contraction parrameters pj, it is possible to choose the translation parameters bj in such a way that 7-H(K) = IKIa, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.