两康托集的交子集的分形维数与测度  

Fractal dimension and measure of intersection of two Cantor sets

作　　者：戴美凤[1] 刘德华[1] 田立新[1] 

机构地区：[1]江苏大学非线性科学研究中心,江苏镇江212013

出　　处：《江苏大学学报（自然科学版）》2006年第6期552-555,共4页Journal of Jiangsu University：Natural Science Edition

基　　金：国家自然科学基金资助项目(10071033);江苏大学高级专业人才科研启动基金资助项目(05JDG041)

摘　　要：一个三分康托集与它的平移集的交集的维数与测度均与平移的长度相关.通过此平移长度t的三进制展开式,就能得到两个三分康托集的交集I(t)的分形维数以及此维数下的Hausdorff测度.具体的,当t能有限展开t=[0.t1t2…tn]3且它的所有系数之和∑ni=1ti为偶数时,其交集I(t)在维数log32下Hausdorff测度非零,并且给出了一个非常简便的测度计算公式,此计算公式可用于相同维数下分形集的分类;其余情况均得到在此维数log32下Hausdorff测度为零.It is discovered that the dimension and measure of the intersection of triadic Cantor sets with their translates are related to the translate length. The fraetal dimension and Hausdorff measure of I（ t）, the intersection of two Cantor sets, are attained by the triadic expansion of the translate length t, That is, when t has a finite triadic expansion t = [0.t1,t2…tn] 3 and ∑i-1^ntiis an even, the Hausdorff measure at dimension log32 is nonzero, A very brief calculation formula of the measure is given and can be used for classifying the sets with the same dimension. Otherwise, the Hausdorff measure at dimension log32 is zero.

关 键 词：三分康托集 交集 分形维数 HAUSDORFF测度 三进制展开 

分 类 号：O174.12[理学—数学]