Quasisymmetric Minimality And Hausdoff Dimension Of One-dimensional Moran Sets

Fractal sets always have self-similarities,and their whole and parts are similar in some way.Self-similar sets can be defined by these self-similarities.By generalizing non-overlapping self-similar sets by weakening the requirements on compression ratio and relative position,a class of important fractal sets which is called Moran sets can be obtained.Moran sets have been studied extensively by many fractal geometry scholars at home and abroad.In this paper,we study the quasisymmetric minimality on packing dimension for one-dimensional Moran sets and the Hausdorff dimension of one-dimensional homogeneous Moran sets.For studying the quasisymmetric minimality on packing dimension for onedimensional Moran sets,we study the quasisymmetric minimality on packing dimension for two classes of one-dimensional Moran sets in chapter 3.Firstly,a probability measure μd supported on the quasisymmetric image is defined.Then,we get the estimates of the range of μd.Next,the lower bound of packing dimension of the image sets is estimated by the mass distribution principle.Finally,we obtain two conclusions:(1)the decreasing Moran sets on line of packing dimension 1 with supk {nk}<+∞,(?)and(?) are all quasisymmetrically packing-minimal;(2)the decreasing Moran sets on line of packing dimension 1 with supk {bk}<+∞ and 0<infk Dk≤supk Dk<1 are all quasisymmetrically packing-minimal.For studying the Hausdorff dimensions of one-dimensional Moran sets,we introduce a class of special one-dimensional homogeneous Moran sets called {mk}quasi-homogeneous perfect sets and study the Hausdorff dimension of this kind of sets in chapter 4.This kind of sets is defined by the basic intervals and their gaps which are formed by the connected components,and it is more extensive than the homogeneous perfect set since it is not necessary that the gaps between the basic intervals of the same order are fixed.In this paper,we obtain that the expression of Hausdorff dimension of {mk}-quasi-homogeneous perfect sets is dim_H (?) under three kinds of conditions by using the mass distribution principle and the structure of {mk}-quasi-homogeneous perfect sets.