Let f(z) denote a transcendental entire function, F(f) and J(f) are the Fatou set and Julia set of f respectively. In this thesis, we shall mainly investigate the dynamical properties of transcendental entire functions. In complex dynamics, there are some basic and important problems, the first (and classical) one is to study the behavior of points in C under iterations of f(z); another one is to study the analytic and geometric properties of the sets F(f) and J(f). Two of the prevailing research topics in complex dynamics are: (i) The dynamics of two permutable transcendental entire functions. Julia proved in 1920’s that if f and g are two permutable rational functions then J(f) = J(g). I.N. Baker (1960’s) extended this to a certain class of transcendental entire functions. (ii) Geomet...[ Read more ]

Let f(z) denote a transcendental entire function, F(f) and J(f) are the Fatou set and Julia set of f respectively. In this thesis, we shall mainly investigate the dynamical properties of transcendental entire functions. In complex dynamics, there are some basic and important problems, the first (and classical) one is to study the behavior of points in C under iterations of f(z); another one is to study the analytic and geometric properties of the sets F(f) and J(f). Two of the prevailing research topics in complex dynamics are: (i) The dynamics of two permutable transcendental entire functions. Julia proved in 1920’s that if f and g are two permutable rational functions then J(f) = J(g). I.N. Baker (1960’s) extended this to a certain class of transcendental entire functions. (ii) Geometric properties of Julia sets and Fatou sets, that is, are there buried points or buried components in Julia sets? Are all the Fatou components bounded?

In the first Chapter, we will give a simple introduction of Nevanlinna’s value distribution theory (which is the main tools in our investigations), factorization of meromorphic functions, minimum modulus, maximum modulus, Poincare metric theory, the classical function theory, the definition of dynamical theory of transcendental entire functions and some key lemmas that will be used throughout this thesis. Then in Chapter 2, we recall four different kinds of sets on which fⁿ(z) (the nth iterate of f) goes to ∞ in four different ways, and these sets have a close relationships with the Julia set J(f) and Fatou set F(f). We will show some important dynamical properties by studying these four sets, and we get the relationships between the above four sets and the Fatou components of f.

In Chapter 3, as an extension of Baker’s result, we will show that if f and g are two permutable transcendental entire functions with q(g) = aq(f) + b for some nonconstant polynomial q(z) and some two numbers a(≠ 0) and b. Then the above four sets, Julia sets and Fatou sets of f and g correspondently are the same.

In Chapter 4, we study the boundedness, connectivity and boundary of a Fatou component. It’s well-known that if F(f) contains multiply connected components, then all components are bounded. Furthermore, in Chapter 5 we prove that under some weak conditions, F(f) contains only bounded components. These subjects have attracted much interests among complex analysts and many related results were obtained earlier by Anderson, I.N. Baker, A. Hinkkanen, X.H. Hua, G.S. stallard, Y.F. Wang, C.C. Yang, J.H. Zheng and etc.

Many results as well as some conjectures of this thesis have been published in, for example, Indian J. Pure Appl. Math., J. Math. Anal. Appl. and Inter. J. Bifur. Chaos.