广义Cantor集关于加倍测度胖瘦性及Borel测度的关联和局部维数

发布时间：2018-04-11 10:04

本文选题：加倍测度 + 广义Cantor集　；参考：《华南理工大学》2016年博士论文

【摘要】：本论文主要研究了分形几何中的三个方面问题。论文的第一部分,即第三章,研究广义Cantor集关于加倍测度的胖性和瘦性。Buckley, Hanson和MacManus [8]研究了中间区间Cantor集关于加倍测度的胖性和瘦性,给出了判定中间区间Cantor集关于加倍测度是胖集和瘦集的充要条件。进一步,Han, Wang(?)Wen [36], Peng和Wen[81]刻画了齐次Cantor集关于加倍测度的胖性和瘦性。注意到,无论是中间区间Cantor集还是齐次Cantor集,它们的结构都有很强的对称性,其同阶基本区间和间隔的长度相等。对于一般的Moran集,关于加倍测度的胖性和瘦性的刻画是困难的。本文考虑的广义Cantor集,其基本区间和间隔的长度可不同,我们引入了(αk)-正则,好的和相当好的概念,分别在这些条件下,给出了广义Cantor集关于加倍测度的胖性和瘦性刻画。论文的第二部分,即第四章,研究了Borel测度的关联维数。测度关联维数的概念由Procaccia, Grassberger和Hentschel [82]于1982年引入。测度的关联维数是通过能量表示,在随机动力系统中应用广泛。本文考虑度量空间中Borel测度的关联维数。首先,用积分形式和离散形式分别刻画了测度的关联维数,其离散形式表明测度的关联维数和测度的L2-谱是相等的；然后,研究了测度的关联维数和测度的Hausdorff维数之间的关系,并给出了判定二者相等的一个充分条件；最后,指出测度的关联维数是拟-Lipschitz不变量。论文的第三部分,即第五章,研究了Moran结构集上测度的局部维数。对于自相似集上的自相似测度,Geronimo和Hardin [31]在强分离条件下证明了其局部维数几乎处处为一个常数。Strichartz [93]则将这个结果推广到开集条件。进一步,Cawley和Mauldin [9]研究了一类特殊Moran集上的Moran测度,在这类Moran集在构造中,逐阶压缩映射个数和压缩比不变,他们在强分离条件下给出了这类Moran测度在几乎处处意义下局部维数的公式。Lou和Wu[67]研究了一类更广的Moran集上的Moran测度的局部维数,在该类Moran集的构造中,逐阶压缩映射个数和压缩比均不同,她们在强分离条件下给出了这类Moran测度在几乎处处意义下局部维数的公式。后来,Li和Wu[63]将这一结果推广到开集条件。本文第五章考虑了更一般的Moran结构集,在更弱的分离性条件下,我们给出了Moran结构集上测度的下、上局部维数的刻画。因此,本文研究的Moran结构集上测度的局部维数结果无论是从研究对象上还是条件上都在一定程度上推广了前述结果。
[Abstract]:This paper mainly studies three aspects of fractal geometry.In the first part of the paper, chapter 3, we study the fatness and thinness of generalized Cantor sets on doubling measures. Hanson and MacManus [8] study the fatness and thinness of intermediate interval Cantor sets on doubling measures.A necessary and sufficient condition for determining that the intermediate interval Cantor set is a fat set and a thin set is given.Further, Han, Wang(?)Wen [36], Peng and Wen [81] characterize the fatness and thinness of homogeneous Cantor sets with respect to doubling measures.It is noted that both the intermediate interval Cantor set and the homogeneous Cantor set have strong symmetries and the length of the same order basic interval and interval is equal.For a general Moran set, it is difficult to characterize the fatness and thinness of doubling measures.In the second part, chapter 4, we study the correlation dimension of Borel measure.The concept of measure correlation dimension was introduced by Procacia, Grassberger and Hentschel [82] in 1982.The correlation dimension of measure is widely used in stochastic dynamic systems by energy representation.In this paper, we consider the correlation dimension of Borel measure in metric space.A sufficient condition for determining the equality of the two is given, and finally, it is pointed out that the correlation dimension of the measure is quasi-Lipschitz invariant.In the third part, chapter 5, we study the local dimension of measure on Moran structure set.For the self-similar measure and Hardin [31] on the self-similar set, it is proved that the local dimension of the measure is almost everywhere a constant. Strichartz [93] is extended to the open set condition.Further, Cawley and Mauldin [9] study the Moran measure on a special Moran set. In the construction of this kind of Moran set, the number of contractive mappings and the ratio of contractions are invariant.Under the condition of strong separation, they gave the local dimension formula of the Moran measure in almost every sense. Lou and Wu [67] studied the local dimension of Moran measure on a class of Moran sets. In the construction of this kind of Moran set,The number of contractive mappings and the ratio of contractions are different from each other. Under the condition of strong separation, the formulas of the local dimension of this kind of Moran measure in almost everywhere sense are given.Later, Li and Wu [63] extended this result to the open set condition.In the fifth chapter, we consider the more general set of Moran structures. Under the condition of weaker separation, we give the characterization of the lower and upper local dimensions of the measure on the set of Moran structures.Therefore, the results of the local dimension of measures on the set of Moran structures studied in this paper generalize the above results to a certain extent, both on the object of study and on the condition.
【学位授予单位】：华南理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O174.12

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