Upper Box Dimension And Quasisymmetric Minimality Of One-dimensional Moran Sets

Moran sets which is a class of important fractal sets in fractal geometry,is an important research object of many scholars of fractal geometry at home and abroad.In this paper,we mainly study two questions of one-dimensional Moran sets:upper box dimension of one-dimensional homogeneous Moran sets and quasisymmetric minimality of one-dimensional Moran sets.Fractal dimension is one of the main research contents of fractal geometry.S-ince the one-dimensional homogeneous Moran sets have simpler construction than general Moran sets and are more convenient to study,there are more classical con-clusions about fractal dimension of the one-dimensional homogeneous Moran set-s.In Chapter 3,we study the upper box dimension of one-dimensional homoge-neous Moran sets.We introduce a class of special one-dimensional homogeneous Moran sets:{mk}-homogeneous Moran sets which is defined by the connected com-ponents composed by the basic intervals,and prove that if E?M(I,{nk},{Ck})is a {Mk}-homogeneous Moran set,then the upper box dimension of E can get the minimum value of all the upper box dimension of the elements of M(I,{nk},{Ck})under certain conditions.Furthermore,we also obtain the range expression of the upper box dimension of the set E ?M{I,Ink},{Ck})under some conditions.We also prove under some specical conditions the expression of the upper box dimension of E E M{I,{nk},{ck}).Since quasisymmetric mappings can change the fractal dimension of the sets,which are different from bi-lipschitz mappings,how the quasisymmetric mappings affect and change the fractal dimension of the fractal sets is a hot topic in the in-terdisciplinary studies of fractal geometry and quasisymmetric mappings,and the quasisymmetric minimality of the fractal sets is an important research content of this topic.In Chapter 4,we study the quasisymmetric minimality for packing di-mension of a class of one-dimensional Moran sets on the real line.In order to get the lower bound of the dimension of the quasisymmetric image sets,we construct a Borel probability measure on the quasisymmetric image sets and a subset of the quasisymmetric image sets which has positive measure,then we use the mass distri-bution principle and prove that the above class of one-dimensional Moran sets whose conditions contain c*>0 is quasisymmetrically minimal for packing dimension.