The Hausdorff Measure Of General Sierpinski Gasket

The concepts of Hausdorff dimension and measure have been introduced for near one hundred years. There are lots of works have been done on the calculation and estimation of Hausdorff dimension, such as the result of self-similar set which satisfies open set condition. But, there are not many results about calculation of Hausdorff measure. By far, except of few, the Hausdorff measure of many fractals can not be worked out, including that of self-similar sets [2]. In this thesis, the problem of calculating the Hausdorff measure for general Sierpinski gasket is discussed.We start from an unit equilateral triangle S0 in R 2. Three small equilateral triangles are made in the three angles of S0 with edge length a (a<1/2). The union set of the three small triangles is denoted as S1. Then for each small triangle, this process is repeated to get nine small equilateral triangles. The union set of these nine small triangles is denoted as S2. Then let this process to go infinite. And we get . then∩S Sn is called generaizedlized Sierpinski gasket made from S0. The Hausdorff dimension of S is -log3/loga. If a=1/2, S is the normal Sierpinski gasket. Its exact Hausdorff measure has not obtained although many researchers studied it using many methods. In this thesis, the problem of calculating the Hausdorff measure for generalized Sierpinski gasket with a∈(1/4,1/2) is discussed using the method of Zuolin Zhou and Zengxi Zhang.In Chapter 1, we introduce the definitions of Hausdorff dimension and Hausdorff measure, and some related definitions and theorems. In Chapter 2, we introduce self-similar sets and open set condition. In Chapter 3, we introduce our main results. When the similitude ratio a∈[1/4,1/3], we denote s=-log3/loga as the Hausdorff dimension of the general Sierpinski gasket. It is proved that the Hausdorff measure of the general Sierpinski gasket is equal to 1. In addition, when the similitude ratio a∈(1/3,1/2), a good upper bound of the Hausdorff measure of general Sierpinski gasket is given.