Study of functional equations by applying value distribution theory and complex dynamics

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This research project has been carried out as planned, and each researcher who took responsibility for part of this project has achieved satisfactory results as follows :(0)Tohge studied some complex differential or functional equations and their solutions given as meromorphic functions in the plane. A new knowledge for the existence of those solutions was obtained and known estimates on the value distribution of those functions were sharpened considerably. He also considered how the facts obtained in this research relate to other subjects and found a guide for further researches.(1)Taniguchi studied the complex dynamics of entire functions, especially those of structurally finite entire functions in detail and succeeded to geometrize them combinatorically. He also investigated the structures of the spaces of Bell representations and the modified spaces of structurally finite entire functions, and studied related covering structure and dynamical structure of those functions.(2)Shimomur … More a studied function theoretic properties of the Painleve transcendents and gave the sharp estimates for the growth orders in both directions. He also observed the value distribution of fourth Painleve PIV concerning its moving targets.(3)Morosawa generalized the concept of semi-hyperbolic for rational functions into transcendental entire functions and also introduced the "complex" error function in order to investigate its dynamical properties at large and obtain new information on the figure of Julia sets, non-existence of wandering domains and Baker domains of those functions. He also obtained a result concerning the convergence of the sequence of Fatou sets of polynomials converging uniformly to a transcendental entire function.(4)Ishizaki studied various types of functional equations, which possess meromorphic solutions in the whole complex plane. Especially, he proved the existence of those solutions to linear difference equations and q-difference equations, as well as an exact estimate for their growth order. Then a new method, which seems to be applicable widely, was developed in the process.(5)Kisaka concentrated on the study of complex dynamics of entire functions, and succeeded to construct an example of transcendental functions, which possess a doubly-connected wandering domain by the method of quasi-conformal surgery. He also gave an example of those functions with a wandering domain of arbitrary assigned connectivity and settled this existence problem posed by I.N.Baker about 40 years ago Less