Plane Fractal Set Of Hausdorff Measure And Irregular

An important problem in the study of the fractal geometry is the estimation and calculation of the Hausdorff dimension and the Hausdorff measure of fractal sets. But this is a difficult task, by comparison, calculation of the Hausdorff measure is more difficult than calculation of the Hausdorff dimension.Up to now, there are not any general method to calculate the Hausdorff measure and the Huasdorff dimension of fractals. Currently we have got the Hausdorff measure of a few number of fractals which are of the Hausdorff dimension is not larger than 1.People have used some methods, such as net measures, mass distribution principles, special cover constructions, to calculate and estimate the Hausdorff measure and Hausdorff dimension of some special fractals.Another important problem in the study of the fractal geometry is a variety of irregular complicated geometrical structures, especially the local structure of s-sets. An S-set must be irregular unless S is an integer.But if S is an integer then the situation is more complicated.When S is an integer, the S-set maybe regular or maybe irregular. It is very important to judge whether an S-set is regular or irregular, moreover, the density theorem of an S-set is the useful tool to judge the regularity or irregularity of an S-set.This paper discusses the following two problems:1. For Sierpinski carpet, we estimate the upper bound of the Hausdorff measure of the Sierpiniski carpet and obtained a better upper bound.By using the symmetry of the Sierpiniski carpet and constructing a special cover and improving the result of paper [28], we got much better upper bound estimate.2. we prove irregularity of a planar 1-set.For a planar 1-set,we estimate the upper density and lower density of that planar 1-set,utilize the properties of upper and lower densities and prove the irregularity of that planar 1-set.