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Escaping set and Bounded Orbit Set of Holomorphic Semigroup on C
Abstract
In this paper, we study the structure and properties of escaping sets of holomorphic semigroups. In particular, we study the relationship between escaping set of holomorphic semigroup and escaping set of each function that lies in that semigroup. We also study about the invariantness of escaping sets. Also, in this paper, we define the term bounded orbit set K(H) and the set K'(H) of holomorphic semigroup H. Then we study their invariantness and their relations with escaping sets. We also construct a particular class of holomorphic semigroups generated by two holomorphic functions such that bounded orbit set of holomorphic semigroup is equal to bounded orbit set of its generators.
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We discuss the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory and provide some condition for the complete
invariance of escaping set and Julia set of transcendental semigroups. Some
results on limit functions and postsingular set have been discussed. A class of
hyperbolic transcendental semigroups and semigroups having no wandering domains
have also been provided.
We study the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory. Several results of the dynamics associated
with iteration of a transcendental entire function have been extended to
transcendental semigroups. We provide some conditions for connectivity of the
Julia set of the transcendental semigroups. We also study finitely generated
transcendental semigroups, abelian transcendental semigroups and limit
functions of transcendental semigroups on its invariant Fatou components.
This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for Kleinian groups are essentially the stable basins of infinity for certain polynomial semigroups. Here we discuss the structure of the Fatou and Julia sets and their basic properties. We investigate to what extent Sullivan's 'no wandering domains' theorem remains valid. We obtain a complete generalisation of the classical results concerning classification of basins and their associated dynamics under an algebraic hypothesis analogous to the group-theoretical notion of 'virtually abelian'. We show that, in general, polynomial semigroups can have wandering domains. We put forward some conjectures regarding what we believe might be true. We also prove a theorem about the existence of filled in Julia sets for certain polynomial semigroups with specific applications to the theory of Kleinian groups in mind.
In this paper, we shall study the dynamics on transcendental semigroups. Several properties of Fatou and Julia sets of transcendental semigroups will be explored. Moreover, we shall investigate some properties of Abelian transcendental semigroups and wandering domains of transcendental semigroups.
We show that for a transcendental entire function the set of points whose
orbit under iteration is bounded can have arbitrarily small positive Hausdorff
dimension.
Structure and properties of Fatou, Julia, escaping and fast escaping sets of holomorphic semigroups
- B H Subedi
B. H. Subedi, Structure and properties of Fatou, Julia, escaping and fast escaping sets of holomorphic
semigroups. Ph.D. Thesis, Tribhuvan University, Kathmandu, Nepal, 2020.
Eremenko's conjecture, wandering lakes of wada, and maverick points
- D Marti-Pete
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- J Waterman
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On the set where iterates of an entire functions are neither escaping nor bounded
- J W Osborne
- D J Sixsmith
J. W. Osborne, D. J. Sixsmith, On the set where iterates of an entire functions are neither escaping nor
bounded, 2015. arXiv: 1503.08077 v1 [math. DS]..