Several results on complex dynamics are presented in this thesis. Firstly, we study the dynamics of two permutable transcendental entire functions. In 1922, Julia proved that for two given permutable rational functions f and g, their Fatou sets are identical. However, for the case of permutable transcendenal entire functions, it remains open for many decades. By assuming that both the singularities of f-1 and g-l are relatively rare, we have confirmed that their Fatou sets are the same. In addition, we study special forms of permutable functions and show by an example that in general the dynamical properties of two permutable transcendental entire functions are quite different from each other, which is opposite to the case of rational functions. Secondly, we study the dynamics of compos...[ Read more ]

Several results on complex dynamics are presented in this thesis. Firstly, we study the dynamics of two permutable transcendental entire functions. In 1922, Julia proved that for two given permutable rational functions f and g, their Fatou sets are identical. However, for the case of permutable transcendenal entire functions, it remains open for many decades. By assuming that both the singularities of f-1 and g-l are relatively rare, we have confirmed that their Fatou sets are the same. In addition, we study special forms of permutable functions and show by an example that in general the dynamical properties of two permutable transcendental entire functions are quite different from each other, which is opposite to the case of rational functions. Secondly, we study the dynamics of composite functions. By means of Iversen's theorem on parabolic Riemann surfaces and the boundary behavior of univalent functions, we show that for a given pair of transcendental entire functions f and g, f o g and g o f have the same dynamical properties. Thirdly, we give a characterization on the Fatou components of a given transcendental entire function f so that the Fatou exceptional value will always belongs to the Julia set of f.