Hausdorff Measure And Upper Convex Density Of A Class Self-similar Sets In R～3

On the fractal geometry, the Hausdorff measure of the self-similar sets had obtained extensive theoretic research. But the calculation and estimation of the Hausdorff measure is very difficult. At present, there are not any universal methods for calculating the Hausdorff measures, even if the extremely regular self-similar sets. Now we only can figure out the Hausdorff measures of few self-similar sets. The upper convex density is one of the important parameters to describe the local structure of fractal sets. It has close relation to the hausdorff measure. In this paper, the Hausdorff measures of a class of self-similar sets generated by cubes are studied based on the relationship between the upper convex density and the Hausdorff measure. Firstly, we present the Hausdorff measure and the hausdorff dimension in this paper. Secondly, we introduced the definition and natures of self-similar sets. Since then, the definition and nature of the upper convex density are given. Finally, we calculate accurately the values of the Hausdorff measures of a class of self-similar sets in R~3, the Sierpinski blocks, which satisfy the strong separation condition and have the Hausdorff dimensions less than 1. It is showed that the natural cover is the best one, as its inference, the Hausdorff measures of this kind sets are obtained. Meanwhile, some applications of this result are given in this paper.