Structure and Properties of Fatou, Julia, Escaping and Fast Escaping Sets of Holomorphic Semigroups
Date
2020
Authors
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Journal ISSN
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Publisher
Faculty of Mathematics
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.
Description
Keywords
Holomorphic functions, Escaping sets