两三分康托尘的交集的分形维数与测度

The Fractal Dimension and Measure of Intersection of Two Cantor Dust

一个三分康托尘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(x,y,z)的三进制展开式,就能得到两个三分康托尘的交集I(x,y,z)的分形维数以及此维数下的Hausdorff测度.具体地,当(x,y,z)能有限展开且它的所有系数之和(∑ki=1xi,∑ki=1yi,∑ki=1zi)为偶数时,其交集I(x,y,z)在维数log8/log3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log8/log3下Hausdorff测度为零.

It is discovered that the dimension and measure of the intersection of triadic Cantor dust with their translates are related to the translate length. The fractal dimension and measure of I （ x, y, z） , the intersection of two Cantor dust, are attained by the triadic expansion of the translate length （ x, y, z） . That is, when （ x, y, z） has a finite triadic expansion and （∑i=1^kxi,∑i=1^kyi,∑i=1^kZi）is an even,the Hausdorff measure at dimension log8/log3 is nonzero. A very brief caleulation formula of the measure is given and can be used for classifying the sets with the same dimension. Otherwise, the Hausdorff measure at dimension log8/log3 is zero.