Research On The Hausdorff Measure Of Some Self-Similar Sets

In this paper,we consider the exact computation of Hausdorff measure about two kinds of Fractals. Firstly, we expand the technique in paper [17] and discuss the Hausdorff measure of self-similar set generated by unit square (cube,tetrahedron). We suppose self-similar set satisfies strong separation condition and its Hausdorff dimension less than 1. Under some conditions,we prove the nature cover is the best cover by determining the biggest value of upper convex density, then, we obtain the Hausdorff measure about this kinds of self-similar sets.Secondly,considering the methods used in paper [11] and [14],we develop it to get the Hausdorff measure of homogeneous perfect set.There are three parts in this paper. We present the source of Fractal and some new results and developments in introduction. In the first part, we review the concepts and properties about Hausdorff measure and dimension. After that,we mention some skills (such as quality distribution principle)of computation about Hausdorff measure.In the second part,we elaborate self-similar condense system,open set condition,strong open set condition and strong separation condition systematically. In the last part,we investigate the Hausdorff measure of some special self-similar sets; in section one, we present the density theorem has important effect, then,we reconsider the square sierpinski carpet mentioned in paper [17] and dismiss the condition of λ_i < (1/3). in section three and four,we expand the methods to R~3,we discuss a kinds of homogeneous perfect set on the line in the last section.