Dimensions And Quasi-symmetric Geometry Of Planar Sets Defined Dy Digit Restrictions

How quasi-symmetric mappings affect the dimensions of fractal sets is a pop-ular direction in the research of quasi-symmetric mappings,and the developmen-t and results are gradually enriched and improved.Especially the research on quasi-symmetric minimal sets.The results of quasi-symmetric minimal sets on the line have been relatively rich.The 1-dimensional homogeneous uniform Cantor sets[18][38],a type of 1-dimensional Moran set[8],and a type of 1-dimensional sets defined by digit restrictions[7]are all quasi-symmetric minimal sets on the line.Full-dimensional sets are quasi-symmetrical minimal in high-dimensional spaces[16].And a class of 1-dimensional "antenna" self-similar sets on the plane are quasi-symmetrical minimal[4][42].This paper mainly studies the dimensions and quasi-symmetric geometry of a class of planar sets defined by digit restrictions.Firstly,we define a class of planar sets defined by digit restrictions,and prove its fractal dimension.Secondly,we construct the s*-dimensional planar sets defined by digit restrictions X and Y,and prove that Y is a quasi-symmetric minimal set,X is not a quasi-symmetric minimal but a locally quasi-symmetric minimal set.The structure and specific content of the full text are as follows:In Chapter 1,we give an overview of the background of the problems studied in this article,and present the research content of this article.In Chapter 2,we introduce the relevant preliminary knowledge involved in the re-search,such as the introduction of fractal dimension[41],quasi-symmetric mapping[34],conformal dimension[30],definition and properties of quasi-symmetric minimal set[4],etc.In Chapter 3,we define a kind of sets defined by digit restrictions F in the plane,give and prove the Hausdorff dimension[41],Upper box dimension[41],Packing dimension[41]and Assouad dimension[13].In Chapter 4,we construct sets defined by digit restrictions X and Y such as dim(X)=dim(Y)=s*in the plane.moreover,we prove Y is a quasi-symmetric minimal set and the conformal dimension of X is 0.In Chapter 5,we also prove that X constructed in Chapter 4 is a locally quasi-symmetric minimal set.In Chapter 6,we summarize the main results of this article,and give some ques-tions that needed to be further studied.