The Fatness And Thinness Of General Cantor Sets For Doubling Measures And The Correlation And Local Dimensions Of Borel Measures

This dissertation contains three parts.In the first part of this dissertation (Chapter 3), we study the fatness and thinness of general Cantor sets for doubling measures.Buckley, Hanson and MacManus [8] discussed the middle interval Cantor sets for doubling measures, and they gave the necessary and sufficient conditions for the fatness and thinness of such sets. Further, Han, Wang and Wen [36], Peng and Wen [81] com-pletely characterized the fatness and thinness of homogeneous Cantor sets for doubling measures. Notice that either the middle interval Cantor sets or the homogeneous Cantor sets, their structures are very symmetric and the length of the same basic interval at the same rank is equal. The research object of this dissertation is a kind of generalized Cantor sets, whose structure is more complex than the homogeneous Cantor sets. We give the necessary and sufficient conditions to characterize the fatness and thinness of such Cantor sets for doubling measures under suitable conditions.In the second part of this dissertation (Chapter 4), we study the correlation dimen-sion of measures on metric space.The correlation dimension was introduced by Procaccia, Grassberger and Hentschel [82]. In this dissertation, we consider the correlation dimension of measures on metric space. Firstly, we describe the correlation dimension of measures by integral form and discrete form respectfully, and this shows that the correlation dimension of measures and L2-spectrum are equal. Secondly, we estimate the correlation dimension and also give a sufficient condition for the correlation dimension equals to the Hausdorff dimen-sion. Finally, we point out that the correlation dimension of measures is quasi-Lipschitz invariant.In the third part of this dissertation (Chapter 5), we study the local dimension of measures on Moran construction.Under the strong separation condition, Geronimo and Hardin [31] proved that the local dimension of the self-similar measure supported on a self-similar set equals to a constant for almost all points of this self-similar set. Strichartz [93] generalized this result to the open set condition. Later, Cawley and Mauldin [9] considered the local dimension of the Moran measure supported on some Moran set. In this setting, the number of contraction maps and contraction ratios are the same in every rank of this Moran set. They obtained the formula for the local dimension of the Moran measures in the sense of almost everywhere under the strong separation condition. Recently, Lou and Wu [67] considered the local dimension on some more general Moran sets. In the construction of this Moran sets, the number of contraction maps and contraction ratios may vary in different rank. They obtained the formula for the local dimension of the Moran measures on such Moran sets in the sense of almost everywhere under the strong separation condition. Li and Wu [63] proved that this result also holds under the open set condition. In Chapter 5, we consider more general Moran construction and obtain the formula for the local dimension under weaker separation condition. Thus, the results on local dimension in this dissertation generalize the above no matter from the research object or the condition.