Topological Structure And Lipschitz Classification Of Fractal Sets

In this thesis,we devote to the study of the topological classification and Lipschitz classification of self-similar sets with overlaps,the connectedness of self-similar sets and self-affine sets.The main content of this thesis is divided into the following seven chapters.In Chapter one,we introduce the background and the recent progresses on the topological study of fractal sets,we state the main results of this thesis here.In Chapter two,we recall some basic concepts and theorems of iterated function systems（IFS）,self-similar/affine sets,symbolic space,Gromov hyperbolic graphs induced by IFS and the matrix rearrangeable condition（MRC）.In Chapter three,we consider a class of planar self-similar sets with overlapping structure.Precisely,given an integer m≥2 and a digit set （?）（（?）²m）（?）[0,m-1]²,then there exists a self-similar set F（?）² satisfying the set equation F=（F+）m,which we call the fractal square.We show that there are only three types of the topology of F:1)F is totally disconnected;2)all non-trivial components of F are parallel line segments;3)F contains a non-trivial component that is not a line segment.Moreover,this classification can also be extended to the self-affine fractals.In Chapter four,based on the result of previous chapter,we further point out that the Lipschitz equivalence of a kind of totally disconnected self-similar sets only depends on the number of overlaps occurring in the first iteration.The main methods used are the Gromov hyperbolic boundary theory and the matrix rearrangeable condition introduced by Lau and Luo^[54-56].In Chapter five,we study two classes of planar self-similar sets with a shifting parameter.The first one is a class of self-similar tiles by shifting x-coordinates of some digits,we give a detailed discussion on the disk-likeness in terms of the parameter.We also prove that the self-similar tiles determines a quasi-periodic tiling if and only if the parameter is rational.The second one is a class of self-similar sets by shifting diagonal digits.We give a necessary and sufficient condition for the self-similar sets to be connected.In Chapter six,we use the radix expansion method to solve the connection problem of a class of planar self-affine sets.By letting an expanding matrix A=diag（n,m）and a digit set （?）={0,1,...,n-1}X{0,1,...,m-1}.Trivially,the pair（A,（?））determines a self-affine set F（A,（?））which is a unit square.If we move one digit from （?）to another place,then the structure of the resulting self-affine set will become very interesting.In this chapter,we give a complete characterization for the connectedness of the self-affine set in terms of the moving domain of the digit.Finally,in the last chapter,we summarize the main results of the thesis and provide some remarks and open questions for further consideration.