一类Sierpinski垫的Hausdorff测度

The Hausdorff Measure of a Class of Sierpinski Gaskets

中文关键词:  自相似集  Sierpinski垫  Hausdorff测度.

英文关键词:self-similar set  Sierpinski gasket  Hausdorff measure.

中文摘要:

      设$S_{\lambda}$ 为压缩比为$\lambda~(\lambda\leq\frac{1}{3})$ 的一类Sierpinski垫, $s=-\log_{\lambda}3$为 $S_{\lambda}$的Hausdorff维数, $N$为产生$S_{\lambda}$的所有基本三角形的集合.本文使用网测度方法,获得了$S_{\lambda}$ 的$s$-维Hausdorff测度的精确值$H^{s}(S_{\lambda})=1$,同时证明了$H^{s}(S_{\lambda})$ 可由 $S_{

英文摘要:

      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.