Research On Several Kinds Of Fractal Sets And Fractal Measures

We study several theoretical and practical problems in fractal geometry.The main work of the dissertation is summarized as follows:1.There are plenty of well-known fractal sets in fractal geometry,for example middle-third Cantor set,Moran set,Sierpi?nski carpet,Sierpi?nski gasket,and so on,which have been fully discussed in the subject.Using the theories in hyperbolic geometry,we study the hy-perbolization of the above fractal sets in Chapter 2.We shall construct a Gromov hyperbolic space with respect to some given fractal set,and prove the set is isometric to the Gromov hyperbolic boundary of the Gromov hyperbolic space.Thus we obtain a new representation of the fractal set.Besides,we prove the above constructed Gromov hyperbolic space is strongly hyperbolic,and construct several strongly hyperbolic metrics on Ptolemy spaces.2.Given a fractal set,using the theories in graph theory and complex networks,we construct a sequence of weighted networks in Chapter 3.The induced class of weighted networks shares several properties with the fractal set,thus it could be a approximate repre-sentation of the original set.We study several topological properties of the networks in this chapter,including the average weighted geodesic distance,the node strength distribution and the clustering coefficient,and prove the Weighted fractal networks is small-world and scale-free.3.Quasisymmetric mappings play important role in function theory,by which the notion of quasisymmetric minimality is induced.Discussing the quasisymmetric minimality of various fractal sets is a major issue in fractal geometry.In Chapter 4,we study the quasisymmetric minimality of homogeneous perfect sets,and prove that the homogeneous perfect sets with Hausdorff dimension 1 are 1-dimensional quasisymmetrically minimal under some conditions.4.Self-affine sets are important fractal sets.Self-affine measures on the self-affine set provide an alternative way to studying self-affine sets.In Chapter 5,we study the Beurling dimension of Bessel spectra and frame spectra of self-affine measures on R~d,and obtain nontrivial bounds of the Beurling dimension.Moreover,we study a class of spectra of the Sierpi?nski-type self-affine measures and show that the exact upper bound of their Beurling dimension is the Hausdorff dimension of the general Sierpi?nski carpets.The self-affine measure is a kind of linear measure,whereas inhomogeneous self-similar measures are nonlinear measures.At the end of this chapter,we shall study the Fourier transformations of inhomogeneous self-similar measures,and calculate the infinity lower Fourier dimension of the measures.