The Hausdorff Measure of a Class of Sierpinski Gaskets

Abstract:

      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.