Some Problems Related To The Dimensions Of Homogeneous Moran Sets

As a kind of typical fractal set.s,Moran sets are important in the development and application in many aspects and are attracted wide attention.Due to the complexity of Moran sets,homogeneous Moran set is an important part on the research of the Moran sets.,One of the key problems of fractal geometry is to obtain many forms of dimensions of fractal sets.The dimension can be used as a measure of fractal set and crack,as reflecting the ability of the set to fill the space,as an important parameter to describe the fractal characteristics.This thesis is divided into seven chapters,we mainly study some problems related to the dimensions of homogeneous Moran sets.In the first chapter we begin with a brief review of the development course and the present research of fractal geometry,then introduce the main research results and research actuality of Moran sets and homogeneous Moran sets and their dimensions,introduce the research background,finally we state the main results of this paper.In the second chapter,we introduce the preliminaries which are involved in this paper.We first introduce the relevant concepts and properties of several common dimensions in fractal geometry——Hausdorff dimension,box dimension and packing dimension,then introduce the concept of iterated function system and related results.At last,the paper introduces the concept and properties of symbolic space.In the third chapter,we review the produce,development and research status of Moran sets,introduce the concepts and dimension results of Moran sets and homo-geneous Moran sets.Specially,in the one-dimensional case,we provide a new result of the sufficient conditions for Hausdorff dimension of Moran set to reach the upper bound,and provide a new proof for a existing conclusion only using the mass distri-bution principle.Compared with the original proof,the new proof is more concise and readable.The next three chapters are the main parts of this paper.In the fourth chapter,we consider the structure and Hausdoff dimension es-timate of a class of special homogeneous Moran sets-{mk}-Moran set.Moreover,we explore the construction and properties of {mk|-quasi homogeneous Cantor set whose dimension can reach the upper bound of {mk}-Moran sets.In the fifth chapter we first use {mk}-quasi homogeneous Cantor set in the fourth chapter to prove the intermediate value theorem of Hausdorff dimension for the homogeneous Moran set constructively.Furthermore,in the case of mk>1((?)k?1)we obtain the packing dimension of {mk}-quasi homogeneous Cantor set.Then on this basis,we prove the intermediate value theorem of packing dimension for the homogeneous Moran set constructively.At the end of this chapter,we deduce the sufficient conditions for the dimension of homogeneous Moran set to reach the minimum value.In the sixth chapter,we extend the results in the fifth chapter to higher di-mensions and prove the intermediate value theorem of Hausdorff dimension for the d(d>2)-dimensional homogeneous Moran set.In the subsequent section we discuss a special class of homogeneous Moran sets in the plane,that is,the collection of Cartesian products of two one-dimensional homogeneous Moran sets with the corre-sponding order ratios ck=ck'((?)k?1).we obtain the lower bound of their packing dimensions.In the last chapter,we combine the Moran structure with some classical fractal sets,and get the Hausdorff dimension,packing dimension and upper box dimension of Moran-Sierpinski carpet and Moran-Sierpinski sponge.