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

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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