The Dimensions Of Several Different Fractals & The Upper And Lower Densities Of A Kind Of Non-symmetric Cantor Set

This dissertation consists of four parts.The first part introduces some basic concepts and the background of our research work as well as the work done by others.The second part is to study the dimensions of a class of subset of Moran fractals related to frequencies of their codings.The research of dimensions of non-differentiability points of a class of Cantor function is arranged at the third part.While the fourth part is devoted to the solution of the concrete upper and lower densities of a kind of non-symmetric Cantor set.In chapter 2,we study the dimensions of a class of subset of Moran fractals induced by the mixed group frequencies in the codings.Generally,to prove the Hausdorff and Packing dimensions of a given set,we need to guess its dimension formula,which is usually hard to get.However,for some specific subsets of Moran fractals,we can treat their Hausdorff and Packing dimensions in a unified manner.The advantage of our approach is that the Hausdorff and Packing dimensions do not guess to be a priori.In chapter 3,we study the dimensions of non-differentiability points of a class of Cantor function.As far as we know,the condition 'for all i,p_i > a_i' is required in all papers about the non-differentiability points of Cantor function,where P_i is the i-th component of a given probability vector and a_i is the contraction ratio of the i-th function of the Iterated Function System which generates the Cantor set.However, if there exists at least one i,such that p_i < a_i,how to guess and prove the fractal dimensions of non-differentiability points of a Cantor function is a difficult question.To crack this problem,we analyzed the structure of non-differentiability points of Cantor functions,and finally ingeniously used the conclusions of[43,45]to achieve our goal.In chapter 4,we study the concrete upper and lower densities of a kind of non-symmetric Cantor set.Feng,Hua and Wen[16]got the upper and lower densities of Cantor sets based on the requirement that the target Cantor set is symmetric.However, we derived some concrete results about a kind of non-symmetrical Cantor set.