自相似集的Hausdorff测度与连续性

Hausdorff Measure of Self-Similar Sets and Continuity

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.

For any F Rn, denote by dimF and HdimF(F) the Hausdorff dimension and the dim F-dimensional Hausdorff measure of F. Suppose T is the attractor of the IFS {f1,..., fm} consisting of similitudes. If fi(T) ∩ fj(T) = for i≠ j, then for any ε > 0 there exists some δ > 0 such that |Hdim T(T) -Hdim D(D)| < ε whenever D is the attractor of the IFS {g1,...,gm} where g1,...gm are similitudes with sup{||/fk(x) -gk(x)||: ||x|| ≤1,1≤k≤m}<ε.