The Topological Hausdorff Dimension Of A Class Of Fractal Squares

The topological Hausdorff dimension is a new dimension introduced by R.Balka,Z.Buczolich,M.Elekes in 2015.Its value is between the topological dimension and the Hausdorff dimension.Letn?2 be an integer and D={d1,d2,…,dm}(?){0,1,…,n-1}2.The unique nonempty compact set F satisfying the eqation F = 1/n(F+D)is called a fractal square.In this paper,we mainly discuss the topological Hausdorff dimension of fractal squares.It consists of three parts.In chapter 1,we recall the definition and some properties of the topological Haus-dorff dimension,in which the topological base,the topological dimension and the Haus-dorff dimension are involved.In chapter 2,we prove that the topological Hausdorff dimension of a fractal square F with n = 3,m<5 is 0 or 1.The idea is as follows:First we reduce the question to several special cases by using some results on the Lipschitz equivalence of fractal squares.Secondly,for every special case,we construct a sequence of topology bases of the related fractal square to show that its topological Hausdorff dimension is either 0 or 1.In chapter 3,we prove that the topological Hausdorff dimension of a disconnected fractal square with n=3,m = 6 is 1 or + log3/log2.In this case,the structure of fractal squares is more complicate and the construction of their bases is more difficult.It should be mentioned that the topological Hausdorff dimension of every connect-ed fractal square with n = 3,m = 6 is 1.