scholarly journals Periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions

2020 ◽  
Vol 3 (1) ◽  
pp. 37-46
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Infinite Number ◽  
Entire Functions ◽  
Periodic Component ◽  
Fatou Set ◽  
Transcendental Entire Functions ◽  
Periodic Components

We prove that there exist three different transcendental entire functions that can have infinite number of domains which lie in the different periodic component of each of these functions and their compositions.

scholarly journals Dynamics on the Pre-periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions

2020 ◽  
Vol 19 (1) ◽  
pp. 161-166
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Infinite Number ◽  
Entire Functions ◽  
Periodic Component ◽  
Fatou Set ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Periodic Components

We prove that there exist three entire transcendental functions that can have an infinite number of domains which lie in the pre-periodic component of the Fatou set each of these functions and their compositions.

scholarly journals Orbifold expansion and entire functions with bounded Fatou components

2021 ◽  
pp. 1-40
Author(s):  
LETICIA PARDO-SIMÓN
Keyword(s):  
General Class ◽  
Entire Functions ◽  
Julia Set ◽  
Holomorphic Dynamics ◽  
Fatou Set ◽  
Local Connectivity ◽  
Hyperbolic Functions ◽  
Hyperbolic Orbifold ◽  
Transcendental Entire Functions ◽  
Fatou Components

Abstract Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

scholarly journals Fatou components of entire functions of small growth

1999 ◽  
Vol 19 (5) ◽  
pp. 1281-1293 ◽  
Author(s):  
XINHOU HUA ◽  
CHUNG-CHUN YANG
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Transcendental Entire Functions ◽  
Small Growth ◽  
Fatou Components

This paper is concerned with the dynamics of transcendental entire functions. Let $f(z)$ be a transcendental entire function. We shall study the boundedness of the components of the Fatou set $F(f)$ under some restrictions on the growth of the function. This relates to a problem due to Baker in 1981.

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

scholarly journals The iteration of polynomials and transcendental entire functions

1981 ◽  
Vol 30 (4) ◽  
pp. 483-495 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Functions ◽  
Unbounded Domains ◽  
Transcendental Entire Functions ◽  
Transcendental Functions

AbstractThe iterative behaviour of polynomials is contrasted with that of small transcendental functions as regards the existence of unbounded domains of normality for the sequence of iterates.

scholarly journals Entire Functions with Separated Zeros and 1-Points

2020 ◽  
Vol 20 (3-4) ◽  
pp. 729-746
Author(s):  
Walter Bergweiler ◽  
Alexandre Eremenko
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Specific Form ◽  
Transcendental Entire Functions

AbstractWe consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.

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