Studies on complex dynamics of transcendental entire functions and the value distribution theory

超越整函数的复杂动力学及价值分布理论研究

基本信息

批准号：
12640182
负责人：
MOROSAWA Shunsuke
金额：
$ 2.11万
依托单位：
KOCHI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640182/
关键词：
complex dynamics Fatou set Julia set transcendental entire function value distribution theory semihyperbolicity wandering domain singular value 超越整関数超越整関数の位数ベーカー領域

项目摘要

The summary of research results is as follows.1. We consider semihyperbolic transcendental entire functions with professor Walter Bergweiler. Building on work by Mane, Carleson, Jones, and Yoccoz introduced the concept of semihyperbolicity for polynomials. It is a generalization of the concept today so-called hyperbolicity, which was introduced by Fatou. It has a close relationship with behavior of singular orbits. In the case of polynomials or rational functions, only critical values are singular values, whose sets are always finite. On the other hand, in the case of transcendental entire functions, asymptotic values are also singular values and sets of singular values may be infinite. First, we show a characterization of semihyperbolic transcendental entire functions are different from that of polynomials by constructing an example. Furthermore, transcendental entire functions may have wandering domains. We show that semihyperbolic transcendental entire functions never have wandering … More domains where the limit functions of iterate are finite. Moreover, we give a sufficient condition that the Julia sets of semihyperbolic transcendental entire functions are locally connected.2. We consider the family of transcendental entire functions {f(z ; a,b,c) = abz + e^<bz> + c}. Functions of this family are of infinite singular type. Hence, they may have Baker domains or wandering domains. We give a range of real parameters where the Fatou sets are coincide with a Baker domains. Furthermore, we show that the area of the Julia set are equal to zero in that case. We also give an integral representation of the functions of the family.3. In dynamics of polynomials, there never exist Baker domains nor wandering domains. To the contrary, in that of transcendental entire functions, there may exists those. On the other hand, any transcendental entire function can be approximated by some sequences of polynomials in the sense of locally uniformly convergence. We consider the Caratheodory convergence of Fatou sets and the Hausdorff convergence of Julia sets of such sequences of polynomials to transcendental entire functions which have Baker domains or wandering domains.4. We consider subhyperbolic rational functions. In particular, we consider the boundaries of Fatou components of such rational functions which are simply connected. We construct an example of a subhyperbolic rational function whose Julia set has a topological property called Sierpinski carpet. Less

研究结果的摘要如下1。我们考虑与Walter Bergweiler教授的半节日性先验全部功能。在Mane，Carleson，Jones和Yoccoz的工作的基础上，引入了多项式的半透明度概念。这是对当今所谓的双曲线的概括，这是Fatou引入的。它与奇异轨道的行为有着密切的关系。对于多项式或有理函数，只有临界值是单数值，其集合始终是有限的。另一方面，在整个函数的情况下，不对称值也是奇异值，而单数值的集合可能是无限的。首先，我们通过构造一个示例来显示半透明的超验整个功能的表征与多项式的表征不同。此外，先验的整个功能可能具有流浪的域。我们表明，半色重的先验整个功能从来没有流浪……更多的域，其中迭代的极限函数是有限的。此外，我们给出了足够的条件，即局部连接了半色重的超验整个功能的朱莉娅集。2。我们考虑了整个函数的家族{f（z; a，b，c）= abz + e^<bz> + c}。这个家族的功能是无限的奇异类型。因此，他们可能有面包师域或流浪域。我们提供了一系列实际参数，其中FATOU集合与面包师域一致。此外，我们表明，在这种情况下，朱莉娅集的面积等于零。我们还给出了家庭功能不可或缺的代表。3。在多项式的动力学中，从来没有贝克域也没有徘徊域。与之形成鲜明对比的是，在整个功能的超验功能方面，可能存在这些功能。另一方面，从局部统一收敛的意义上，某些序列的多项式序列都可以通过任何先验的整个函数近似。我们考虑FATOU集合的Caratheodory收敛性和此类多项式序列的Julia序列的Hausdorff收敛到具有贝克域或徘徊域的超验整个功能。4。我们考虑了亚液压合理功能。特别是，我们考虑了这种合理函数的FATOU组成部分的边界，这些函数是简单地连接的。我们构建了一个尺寸有理函数的示例，其朱莉娅集具有称为Sierpinski地毯的拓扑特性。较少的