Some Studies About Doubling Measures And Packing Measures On Uniform Cantor Sets

In the dissertation, we study doubling measures, packing measures, and pre-packing measures on uniform Cantor sets. For doubling measures on uniform Cantor sets, we mainly discuss the following questions:(1) Which measures are doubling on a given uniform Cantor set?(2) Which doubling measures on a given uniform Cantor set can be extended to dou-bling measures on the unit interval?We give a complete answer to question (1), and obtain a necessary and sufficient con-dition for a probability measure on an arbitrarily given uniform Cantor set to be doubling.. For question (2), we consider ultimately uniform measures and prove that such a measure can be extended to a doubling measure on [0,1] if and only if the underlying uniform Cantor set is of positive Lebesgue measure.For packing measures and pre-packing measures on uniform Cantor sets, we study the following questions:(3) Whether or not, for every doubling gauge, there is a Cantor set with positive finite Hausdorff, packing, and pre-packing measures simultaneously?(4) When does a uniform Cantor set, after adding an appropriate countable set, give a set of different values of positive finite packing and pre-packing measures?We give a positive answer to question (3), i.e. for any doubling gauge g, there is a positive integer n and a Cantor set" such that0<H9{E)≤P9(E)≤P09(E)<∞.. For question (4), we give a class of uniform Cantor sets and prove that, for each set E in this class, there is a countable set F and a doubling gauge g, such that E U F is compact and has different values of positive finite packing and pre-packing measures.The dissertation consists of five chapters. Chapter1is devoted to a summary on background and motivation of this paper. Chapter2contains notions and known results related to our work. In Chapter3, we consider self-similar sets on the real line, with equally contracting ratio and without the strong separation condition. We prove that a self-similar measure on such a self-similar set is doubling if and only if its first weight is equal to the last one. In Chapter4, we obtain a necessary and sufficient condition for a probability measure on an arbitrarily given uniform Cantor set to be doubling. Also, we discuss the extension question of doubling measures on uniform Cantor sets. In Chapter5, we prove that for every doubling gauge g, there exists a Cantor set E, such that g is simultaneously a Hausdorff, packing, and pre-packing gauge of E. For question (4), we fail to prove that, for every uniform Cantor set, there exists a countable set F and a doubling gauge g, such that E U F is compact and has different values of positive finite packing and pre-packing measures. However, we prove that for every s∈[0.1/2], there exists a uniform Cantor set of dimp E=s with this property. In Chapter5, we also prove that for any uniform Cantor set, the distribution function of the uniform distribution is Hausdorff, packing, and also pre-packing gauge. At the end of this chapter, we give a further discussion about the packing and pre-packing measures of the uniform Cantor set.