Studies on complex dynamics of transcendental entire functions

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The summary of research results is as follows.1. First, we consider the family of transcendental entire function {fィイD2λィエD2(z) =λz exp(z)}, which is a one complex parameter family. In the case that λ is a real parameter, we study its dynamical property by using the theory of unimodal maps. On this way we proved that there exists many λ for which the dynamics of fィイD2λィエD2 is chaotic.2. In the case that λ is a complex parameter, we considered the parameter space. In particular, we decided a domain of the parameter space where the Fatou set of the dynamics corresponding to a point consists of bounded components. The method of proof is to study a property of certain curves in the Julia set. By the argument similar to this, we constructed a transcendental entire function whose Julia set is a Sierpinski carpet.3 . One of the difference between the dynamics of transcendental entire functions and the dynamics of rational functions is that Fatou sets of the dynamics of transcendental entire functions may have Baker domains and/or wandering domains. We constructed a transcendental entire function whose Fatou set contains cyclic Baker domains whose period is greater than one.4. A transcendental entire function is always considered as a limit function of locally uniform convergence of a sequence of polynomials. We considered how the convergence of sequences of Fatou sets and Julia set. In particular, we considered it in the case that the Fatou set of a transcendental entire function has Baker domains or wandering domains. In general, locally uniform convergence of a sequence of polynomial does not imply convergence of a sequence of Fatou sets or Julia sets. Under some conditions, we showed a sequence of components of Fatou sets converges to a Baker domain. We also showed that if a sequence of components of Fatou sets satisfies certain conditions then the Fatou sets of the limit function contains a wandering domain.