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

研究结果总结如下：1。我们与Walter Bergweiler教授一起研究了半双曲超越全函数。在Mane、Carleson、Jones和Yoccoz的工作基础上，引入了多项式的半双曲性概念。它是对今天所谓的双曲性概念的概括，双曲性是由法头提出的。它与奇异轨道的行为有密切的关系。对于多项式或有理函数，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，并且奇异值的集合可以是无限的。首先，我们通过构造一个例子证明了半双曲超越全函数不同于多项式的特征。此外，超越的整个函数可能有漫游域。我们证明了半双曲超越的整个函数在迭代的极限函数是有限的情况下永远不会有游荡的定义域。并给出了半双曲超越全函数的Julia集是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^<bz> + c}。该类函数具有无限奇异型。因此，它们可能具有贝克域或漫游域。我们给出了一个实参数范围，其中Fatou集与Baker域重合。进一步，我们证明了在这种情况下Julia集合的面积等于零。我们也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，在先验的全函数中，可能存在这些函数。另一方面，任何超越整函数都可以用多项式序列在局部一致收敛的意义上逼近。3 .考虑了这类多项式序列的Fatou集的Caratheodory收敛性和Julia集对具有Baker定域或游荡定域的超越整函数的Hausdorff收敛性。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法头分量的边界。我们构造了一个次双曲有理函数的例子，它的Julia集具有称为Sierpinski地毯的拓扑性质。少

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