Structure and Properties of Fatou, Julia, Escaping and Fast Escaping Sets of Holomorphic Semigroups

Abstract

We study the dynamical behavior of a semigroup generated by holomorphic functions in the complex plane. In particular, we concentrate on semigroup dynamics, where semigroups are generated by transcendental entire functions. It is a study of the behavior of the compositions of a finite set of holomorphic functions in the complex plane. We study Fatou, Julia, escaping and fast escaping sets of such semigroups. The principal aim of this thesis is to investigate the structure and the properties of these sets in the more general settings of holomorphic semigroups. In this thesis, we see to what extent, the structure and the properties of the Fatou, Julia, escaping and fast escaping sets of classical holomorphic dynamics are preserved and generalized to semigroup dynamics, and what new phenomena can occur. A holomorphic semigroup is not abelian in general; however, a cyclic semigroup is abelian, so differences in the dynamics can occur in the structure and the properties of these sets. If a semigroup is abelian, such types of differences will narrow down, and most of the structure and the properties of these sets of classical holomorphic dynamics are preserved and generalized. In this thesis, we generalize the notion of abelian semigroups to nearly abelian semigroups, and we investigate the identical structure and the properties of these sets in such semigroups. On the basis of the algebraic notion of different indices such as finite index, cofinite index and Rees index, we also investigate subsemigroups whose Fatou, Julia and escaping sets coincide with their corresponding parent semigroup. In the holomorphic semigroup setting, there may be empty Fatou sets and empty escaping sets; hence we also investigate certain holomorphic semigroups whose Fatou sets and escaping sets are non-empty on the basis of (partial)fundamental sets and Carlemen sets. Finally, we define fast escaping sets of transcendental semigroups, and we discuss some fundamental structure and properties of these sets.