The Study Of The Pointwise Dimension Of Moran Measures And The Properties Of Doubling Measures

This thesis contains two parts.In the first part of this thesis (Chapter 3), we study the pointwise dimension of aclass of Moran measures.Under the strong separation condition, Geronino and Hardin [36] proved that thepointwise dimension of the self-similar measures supported on a self-similar set E equalsto a constant for almost all points of E. Strichartz [98] generalized this result under theopen set condition. Later, Cawley and Mauldin [12] considered the pointwise dimensionof the Moran measures supported on some Moran set. In their setting, the numberof contraction maps and contraction ratios are the same in every rank of the Moran set.Under some separation conditions, they obtained the formula for the pointwise dimensionof the Moran measures (in the sense of almost everywhere). In above researches, the basictools are the symbolic space and some properties of shift operator.In Chapter 3, we consider some more general Moran sets, in construction of theMoran sets, the number of contraction maps and contraction ratios may vary in di?erentrank of the Moran set. Thus, we can't use the symbolic space in our study as theydid. Instead of symbolic space, we use the Large Number Law and Zero-or-One Lawsin probability theory to discuss the piontwise dimension of the Moran measures. InChapter 3, we prove as follows results: (1) For the Moran set satisfying strong separationconditions and under the assumption that the contraction ratios are uniformly boundedaway from zero, we obtained the pointwise dimension of the Moran measures (in the senseof almost everywhere). (2) Under strong separation conditions, we obtained the pointwisedimension of the homogeneous Moran measures (in the sense of almost everywhere). (3)Under open set conditions, we prove that the pointwise dimension of the generalizedself-similar measure supported on a generalized self-similar set E equals to a constantfor almost all points of E. Also, we obtain formulas for the dimension of the Moranmeasures.In the second part of this thesis (Chapter 4), we study some properties of doublingmeasures on a compact set in Rn.Let X ? Rn be a compact set andμbe a doubling measure supported on X. LetE be the set of accumulation points of X and F the set of isolated points of X. We callμ|E the continuous part ofμandμ|F the atomic part ofμ. Kaufman and Wu [56] asked:do there exist a compact subset X in R1 and a doubling measureμon X, such that F isdense in X and that the restrictionμ|E is also a doubling measure on E? In Chapter 4, we answer this question completely. We prove that for every compactand nowhere dense subset E of Rn without isolated points and for every doubling measureμon E, there is a countable set F with E∩F = ? and a doubling measureνon E∪Fsuch thatν|E =μ.In [64], we have proved that there is a compact set X ? [0,1] with dimH X = 1such that all doubling measures on X are purely atomic. Naturally, we ask that is therea compact set X ? [0,1] of positive Lebesgure measure on which all doubling measuresare purely atomic. We give a negative answer to this question. In Chapter 4, we provethat every compact subset of Rn of positive Lebesgue measure carries a doubling measurewhich is not purely atomic.