自相似集的一类维函数

设E是由满足开集条件的IFS{fi}in=1生成的自相似集,其中fi为相似映射,相似比为ci,0ci1.已经知道,E的Hausdorff维数,填充维数,盒维数及相似维数相等,而且E具有正有限s维Hausdorff测度及预填充测度.将要证明若g为维函数且满足条件:(1)∑ni=1g(ci)=1;(2)对于由数字{1,2,…,n}生成的任意一个k项序列σ=i1…ik,有g(ci1…cik)=g(ci1)…g(cik),则E具有正有限g-Hausdorff测度及g-预填充测度.

A class of dimensional functions of self-similar sets

Let EM be the self-similar set generated by an IFS f1,…,fn satisfying the OSC,where fi are similarity mappings with contracting ratios ci,0〈ci〉1.It is known that for the self-similar set E its Hausdorff,packing,box,and similarity dimension are all equal,and E has finite and positive s-dimensional Hausdorff and prepacking measure.It will prove that E has finitie and positive g-Hausdorff and g-prepacking measure,provided g is a dimensional function satisfing the conditions：（1）∑i=1^ng（ci）=1;（2）g（ci1…cik）=g（ci1）…g（cik） for every k-term sequence i1…ik of digits 1,2,…,n.