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

• 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

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

研究成果总结如下：1.研究成果综述。我们与沃尔特·伯格韦勒教授一起研究了半双曲超越整函数。在Mane、Carleson、Jones和Yoccoz工作的基础上，引入了多项式的半双曲性的概念。它是今天所谓的双曲性概念的推广，它是由Fatou引入的。它与奇异轨道的行为有着密切的关系。在多项式或有理函数的情况下，只有临界值是奇异值，其集合总是有限的。另一方面，在超越整函数的情况下，渐近值也是奇异值，奇异值的集合可以是无穷的。首先，通过构造一个例子说明了半双曲超越整函数不同于多项式的一个刻画。此外，超越整函数可能有游荡域。证明了半双曲超越整函数不存在游荡…迭代的极限函数是有限的更多区域。此外，还给出了半双曲超越整函数的Julia集局部连通的一个充分条件。我们考虑超越整函数族{f(z；a，b，c)=abz+e^&lt；bz&gt；+c}。这个族的函数是无穷奇异型的。因此，它们可能具有Baker域或游荡域。我们给出了Fatou集与Baker域重合的一系列实参数。此外，我们还证明了在这种情况下Julia集的面积等于零。我们还给出了族函数的积分表示。在多项式动力学中，既不存在贝克域，也不存在游荡域。相反，在先验的整体功能中，可能存在那些。另一方面，在局部一致收敛的意义下，任何超越整函数都可以用多项式序列逼近。我们考虑了Fatou集的Caratheodory收敛和这类多项式序列的Julia集对具有Baker域或游荡域的超越整函数的Hausdorff收敛。我们考虑次双曲有理函数。特别地，我们考虑了这类单连通有理函数的Fatou分支的边界。我们构造了一个次双曲有理函数的例子，它的Julia集具有一个称为Sierpinski地毯的拓扑性质。较少

项目成果

E.Fujikawa,H.Shiga & M.Taniguchi: "Discreteness of the mapping class group for Riemann surfaces of infinite analytic type"Proc.of the 2nd ISAAC Congress. 2. 1015-1023 (2000)

藤川E、志贺H

T.Ohtsubo, K.Toyonaga: "Optimal policy for minimizing risk models in Markov decision processes"J. Mathematical Analysis and Applications. (印刷中).

T. Ohtsubo，K. Toyonaga：“马尔可夫决策过程中最小化风险模型的最优策略”J. 数学分析和应用。

S. Morosawa: "The Caratheodory convergence of Fatou components of Polynomials to Baker domains or wandering domains"Proceedings of the Second Congress ISAAC, Kluwer Acad, Pub.. Vol.1. 347-356 (2000)

S. Morosawa：“多项式 Fatou 分量的 Caratheodory 收敛到 Baker 域或漫游域”第二届大会 ISAAC 会议记录，Kluwer Acad，Pub.. Vol.1。

S.Morosawa: "Julia sets of subhyperbolic rational functions"Complex Variables. 41. 151-162 (2000)

S.Morosawa：“朱莉娅亚双曲有理函数集”复变量。

J.Heittokangas, R.Korhonen, I.Laine, J.Rieppo, K.Tohge: "Complex difference equations of Malmquist type"Computational Methods and Function Theory. (印刷中).

J. Heittokangas、R. Korhonen、I. Laine、J. Rieppo、K. Tohge：“Malmquist 型复杂差分方程”计算方法和函数理论（出版中）。