关于非B类整函数的Julia集和逃逸集的Hausdorff维数和Hausdorff测度问题的研究

For each entire function, the complex plane is split into two fundamentally different parts the Fatou set, where the behaviour of the iterates of the function is stable under local variation, and the Julia set, where it is chaotic. .Another key object of study is the escaping set which consists of the points that escape to infinity under iteration. This set plays a major role in complex dynamics since the Julia set is equal to the boundary of the escaping set. Much work in this area has focused on obtaining an understanding of the sizes of these sets and significant subsets as measured by their Hausdorff dimensions. This has led to fundamental insights into the nature of these sets with some results being completely unexpected. Recently, there are also many works on the Hausdorff measure, another important concept of franctal set, of the Julia set and escaping set. Almost all results on these topics concerning the class B..There are only comparatively few papers addressing topics such as Hausdorff dimension or measure of Julia sets for transcendental entire functions outside the B class. Recently, one such result is by Bergweiler and Karpinska who show that if the growth of a transcendental entire function of finite order is sufficiently regular, then the Julia set and the escaping set have Hausdorff dimension 2. We will aim to study the Hausdorff measure respect to some gauge function on these sets for functions from Bergweiler and Karpinska. Moreover, for entire function with infinite order which is outside of the class B, we will give some examples and obtain a more general setting, under assumptions concerning the regularity of growth, possibly also some regularity in the zero distribution.

对于超越整函数动力系统， Julia往往非常复杂也是研究的热点之一，另一重要的研究对象是逃逸集，即迭代序列趋于无穷的点构成的集合，这是由于它与Julia集关系密切．.许多的研究通过计算Julia集和逃逸集子集的Hausdorff维数来探索其几何性状．此外， Hausdorff测度也是分形几何中一个重要概念，也有许多关于Julia集和逃逸集的Hausdorff测度的研究，结果一定程度可以看做Hausdorff维数的推广．目前，这方面研究成果在B类函数范围内已经非常丰富．对非B类整函数，Bergweiler和Karpinska证明了满足正规增长的有穷级整函数的Julia集和逃逸集的Hausdorff测度为2．本项目的主要研究增长级为无穷的相关结果：首先构造无穷级整函数例子使得其Julia集和逃逸集的Hausdorff测度为2；其次利用正规增长条件或零点分布推广到更一般的无穷级整函数．

超越整函数动力系统中，Fatou集定义为迭代序列稳定点的集合，Julia作为其补集往往是个分形、比较复杂，也是研究的热点之一. 另一关键的研究对象是逃逸集，即迭代序列趋于无穷的点构成的集合，这是由于它与Julia集关系密切．许多的研究通过计算Julia集和逃逸集子集的Hausdorff维数来探索其几何性状．分形几何中的另一个重要概念是 Hausdorff，关于Julia集和逃逸集的Hausdorff测度的研究，结果一定程度可以看做Hausdorff维数的推广和细化．. 在本项目的研究周期内，我们主要做了如下四方面的研究工作：1.研究了一类正规增长的非Eremenko-Lyubich类整函数的逃逸集和Julia集的Hausdorff测度问题，证明了在一定的度规函数下，其测度为无穷；2.构造了一类非Eremenko-Lyubich类整函数且增长级为无穷，使得其逃逸集和Julia集的Hausdorff维数为二；3.证明了指数函数 Bergweiler-Wang的定理在参数空间中的对应定理，即证明了逃逸射线在一定度规函数下测度为零；4.给出了高维指数函数非逃逸点的Hausdorff维数的上界和下界，同时这个结果改进了指数函数的相应结果.