Estimation Of Hausdorff Measure Of Two Kinds Of Fractal Sets

Fractal geometry is developed in the mid1970s.It's a new science. It provides anew thought, methods and techniques to study the irregular set of the nature, and hasaroused great concern and interest. The famous American scientists J.A.Wheeler said,"Tomorrow, who are not familiar with fractal, who will not be called scientificintellectuals".Hausdorff measure and dimension of fractal geometry are two basic and importantconcepts, to compute and estimate them become one of the main problem of fractalgeometry. However, the calculation of a fractal set of Hausdorff measure and Hausdorffdimension is very difficult, especially in the calculation of the Hausdorff measure. Tomeet the conditions set of open since self-similar sets, its Hausdorff dimension has beenfully resolved, the Hausdorff dimension is equal to the self-similar dimension. But theHausdorff measure of the accuracy of calculation is still very difficult, only a few kindsof special and dimension of less than one been confirmed. For the fractal dimension isgreater than one, has not worked out its accurate Hausdorff measure.This paper mainly discusses the estimate of the Hausdorff measure of thePan-square flower and Sierpinski carpet, which are the self-similar sets. Our firstchapter introduces the research background of this paper. In the second chapterintroduces the knowledge of measure theory, the defines and property of Hausdorffmeasure and dimension, the theory of quality distribution and the defines and propertyof self-similar sets and OSC. In the third chapter, we discuss the upper and lowerboundary of Hausdorff measure of Pan-square. In the third chapter, we discuss theupper and lower boundary of Hausdorff measure of Sierpinski carpet. The innovation of this paper is to study similarity ratio of1/5in the Pan-squareflower, based on the fractal transform analysis, get the estimate of the upper boundaryformula, and applied this formula to draw a better estimation of the upper boundary.Using self-similar properties and the theory of quality distribution, through detailedcalculation to get a better lower boundary estimation. Studing the Sierpinski carpet,select a special coverage to get the upper boundary better than the literature [31], anduse the literature [27] results to get the lower boundary.