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 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 Dynamics of slowly growing entire functions

2001 ◽  
Vol 63 (3) ◽  
pp. 367-377 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Normal Family ◽  
Entire Functions ◽  
Julia Set ◽  
Maximum Modulus ◽  
Stable Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Simply Connected

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

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 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.

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.

Growth of transcendental entire functions on algebraic varieties

1999 ◽  
Vol 109 (1) ◽  
pp. 253-271
Author(s):  
Stanley M. Einstein-Matthews ◽  
Clement H. Lutterodt
Keyword(s):  
Entire Functions ◽  
Algebraic Varieties ◽  
Transcendental Entire Functions
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