一类Sierpinski垫的Hausdorff测度

设S_λ为压缩比为λ(λ≤1/3)的一类Sierpinski垫,s=-log_λ3为S_λ的Hausdorff维数,N为产生S_λ的所有基本三角形的集合.本文使用网测度方法,获得了S_λ的s-维Hausdorff测度的精确值H~s(S_λ)=1,同时证明了H~s(S_λ)可由S_λ关于网N的s-维Hausdorff测度H_N~s(S_λ)确定,获得了S_λ的非平凡的最佳覆盖.

The Hausdorff Measure of a Class of Sierpinski Gaskets

Let $S_{\lambda}$ be a class of Sierpinski gaskets with compression ratio $\lambda~ (\lambda\leq\frac{1}{3})$, $s=-\log_{\lambda}^{3}$ be the Hausdorff dimension of $S_{\lambda}$, and $N$ be the set of all the basic triangles to produce $S_{\lambda}$. In the paper, by the method of net measure, the exact value of the Hausdorff measure of $S_{\lambda}$, $H^{s}(S_{\lambda})=1$, is obtained, the fact that the Hausdorff measure of $S_{\lambda}$ can be determined by net measure $H^{s}_{N}(S_{\lambda})$ is shown, and the best coverings of $S_{\lambda}$ that are nontrivial are obtained.