共形映像的Hausdorff测度及其算法

Hausdorff measure and its algorithm of conformal image

对于Rn中满足0<Hs(K)<∞的任意紧致集K,我们考虑其在共形映射f作用下的像集的Hausdorff测度Hs(f(K)).本文给出了下面结果:Hs(f(K))=Hs(K)·∫K|Dxf|sdμ(x),其中概率测度μ=Hs|K/Hs(K).给定满足开集条件的自相似集K,测度μ恰好是自相似测度,因此可以应用上述公式计算f(K)的Hausdorff测度,例如,K是λ-Sierpinski地毯,f(z)=z+εz2,其中0<λ1≤/4,复数ε满足|ε|≤0.1.而此刻f(K)恰好是自共形集,因此我们的算法能计算一类特殊的具有非线性结构的自共形集的Hausdorff测度.

Abstract For compact set K ∈R^n with 0 〈 H^8 （K） 〈 ∞ and any conformal bijection f, we obtain the Hausdorffmeasure of the conformal imageH^s（f（K））=H^s（K）·∫_K｜Dxf｜^s dμ（x）, whereμ=H^s｜K/H^s（K）is a probability measure. In particular, for any given self-similar set K satisfying the open set condition, the measure μ= is the corresponding self-similar measure, and thus we can calculate Hausdorff measure of f（K） by using the above formula. For example, K is the A-Sierpinski carpet with λ≤ 1/4 and f（z） = z ＋ εz^2 with ε ∈ C and ｜ε｜ ≤0.1 In this example, the image f（K） is a self-conformal set. Therefore, our algorithm can calculate the Hausdorff measures of a special class of self-conformal sets with nonlinear structure.