Topological Hausdorff Dimension Of Fractal Squares And Its Application To Lipschitz Equivalence Classification

One of the basic but fundamental topics of fractal geometry is the calculation of fractal dimension.The topological Hausdorff dimension(dim_tH)of a set,introduced by Balka et al in 2015,seems to be the latest fractal dimension.This dimension is defined by the Hausdorff dimension of the boundaries of a topological basis of the set and it is defined by a very natural combination of the definition of the topological dimension and the Hausdorff dimension.The present thesis aims at calculating the topological Hausdorff dimension of a class of planar fractal sets known as fractal squares and in-vestigates the application of topological Hausdorff dimension to Lipschitz classification Since the Hausdorff,box,packing and Assouad dimension for any fractal square are always equal each other except for the topological Hausdorff dimension,the topological Hausdorff dimension is a new important Lipschitz invariant of fractal squaresIn order to obtain a lower estimate of dim_tH F,the method in this thesis is to seek a comb-like subset of F.A set(?)R2 is called comb-like,if there is a linear map f:R2?R2 such that f(E)=X×[0,1]with X(?)R.Balka et al proved that dim_tH F>dim_tH E for any comb-like subset E of F.In this thesis,we construct comb?like subsets by virtue of the periodic extension H=F+Z2 of F.Set ??={?;l?,?(?)H}where l?,? denotes the line with slope ? and y-interception ?.Then we get a lower estimate of dim_tH F by using dimH??.Moreover,we show that ?? has a self-similar structure and hence its Hausdorff dimension can be easily computed.In particular,when ?=0 or ?,dimH QT can be computed by using the numbers of vertical bars and horizontal bars of D.As for upper estimates of dim_tH F,our basic idea is to construct a topological basis of F by defining certain self-similar curves.Given a curve ?[0,1]?R2,the slope of the straight line connecting ?(0)and ?(1)is called the slope of the curve?.The quadrilateral which is formed by two curves with different slope is called a curve parallelogram.Select proper curves with different slope such that F is included in the interior of the corresponding curve parallelogram,then we can construct the topological basis of F by using the images of the curve parallelogram under similarity transformations.Here the curve parallelogram plays a role like the unit circle for general topology.To obtain self-similar curves,we introduce an iterated function system(IFS)with condensation and investigate the connectedness of the invariant set K generated by an IFS with condensation.In this thesis,with the help of Hata graph constructed by the IFS with condensation of K,we prove that if the Hata graph of K is connected,then K is connected.Moreover,if K is connected then it must be locally connected by virtue of the well-chained property and S property introduced by Whyburn.Finally,by using Hahn-Mazurkiewicz theorem we show that if K is both connected and locally connected it must be arcwise connected,that is,K must be a Peano continuum which contains the desired curves.To get a good enough upper bound,we hope that the intersection of the curves and H has Hausdorff dimension as small as possible.In this thesis,we first consider the construction of a vertical curve:(1)construct a vertical graph according to D;(2)seek one of the shortest paths from the top to the bottom in the vertical graph;(3)using the path we can construct an IFS with condensation whose invariant set is precisely a Peano continuum providing the desired vertical curve.In a similar way,a horizontal graph and curve can be constructed.Thus,we can get a topological basis of F by using these curves.Next,we go further to design a method to construct vertical curves if a horizontal curve already exists,and vice versa,and obtain a better upper bound of dim_tH F.Finally,we construct a topological basis of F by using periodic curves.A curve ?:[0,1]?R2 is called periodic if ? satisfying ?(1)-?(0)?Z2\{0}·Still using IFS with condensation,we prove that a vertical curve ?v can always be constructed if a periodic curve ? already exists and thus the upper bound of its Hausdorff dimension can be obtained from dimH(?? H)and How to construct'nice'periodic curves in any direction will be a problem we will study in the future,where'nice' means the intersection of the curves and H has small Hausdorff dimension.