О хаусдорфовой мере однородного треугольного $(c,\theta)$-ковра Серпинского

N. Yu. Svetova

Problemy analiza, no. 18 (2011), pp. 61-69

Résumé

The generalized homogeneous Sierpinski $(c,\theta)$-gasket is considered. It has received that the $s$-dimensional Hausdorff measure of $(c, \theta)$-gasket for $c\in (0; 1/3]$ is equal $H^{s}(D_{c,\theta})=(2\sin\frac{\theta}{2})^{s}$, for $\theta\in [\frac{\pi}{3}, \pi)$ and $(\frac{2\sin \theta}{\sqrt{5-4\cos \theta}})^{s}\le H^{s}(D_{c-\theta}) \le 1$ for $\theta\in (0,\frac{\pi}{3})$. As a consequence the $s$-dimensional Hausdorff measure for a generalized homogeneous Pascal triangle is received, it is equal $2^{s/2}$.