Fractal Dimension And Measure Of A Class Of Self-similar Sets And Moran Sets

In this dissertation, we mainly study the dimension and the measure of a class of self-similar sets and their translations , the local dimension of Moran measures satisfying the strong separation condition and the Hausdorff measure of the attractor of an iterated function system with parameter.In Chapter 1, we first briefly review the birth, develop and the current situation of fractal geometry.In Chapter 2, we give some foundation concepts and properties of the fractal and measure theory.In Chapter 3, we discuss that the dimension and measure of the intersection of the triadic Cantor sets. In fact, we give a very brief calculation formula of the measure which attained by the triadic expansion of the translate length can be used for classifying the sets with the same dimension.In Chapter 4, we generalize the results in Chapter 2 to three dimensional spaces, and get the similar properties.In Chapter 5, we investigate the local dimension of measures on a class of Moran sets. The measure, called Moran measure, is an extension of the self-similar measure. We give the exact formulae of the local dimension and the Hausdorff dimension of the Moran measur satisfying the strong separation condition.In Chapter 6 , we calculate a Hausdorff measure of a fractal set with parameter. We obtained the exact value of the Hausdorff measure of theattractor—when the parameterθ∈[π/3,π] by elementary method.