scholarly journals Entire functions for which the escaping set is a spider's web

2011 ◽  
Vol 151 (3) ◽  
pp. 551-571 ◽  
Author(s):  
D. J. SIXSMITH
Keyword(s):  
Entire Functions ◽  
Degree Of Stability ◽  
Transcendental Entire Functions ◽  
Fatou Components

AbstractWe construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.

scholarly journals Bounded Fatou Components of Composite Transcendental Entire Functions with Gaps

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Cunji Yang ◽  
Shaoming Wang
Keyword(s):  
Entire Functions ◽  
Fatou Component ◽  
Transcendental Entire Functions ◽  
Fatou Components

We prove that composite transcendental entire functions with certain gaps have no unbounded Fatou component.

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.

scholarly journals Classifying simply connected wandering domains

2021 ◽  
Author(s):  
Anna Miriam Benini ◽  
Vasiliki Evdoridou ◽  
Núria Fagella ◽  
Philip J. Rippon ◽  
Gwyneth M. Stallard
Keyword(s):  
Entire Functions ◽  
General Technique ◽  
Jordan Curves ◽  
Wandering Domain ◽  
Detailed Classification ◽  
Transcendental Entire Functions ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

AbstractWhile the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

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.

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