scholarly journals A landing theorem for entire functions with bounded post-singular sets

2020 ◽  
Vol 30 (6) ◽  
pp. 1465-1530
Author(s):  
Anna Miriam Benini ◽  
Lasse Rempe
Keyword(s):  
Entire Function ◽  
Periodic Point ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Polynomial ◽  
Landing Point ◽  
External Rays ◽  
Singular Sets ◽  
External Ray ◽  
Polynomial Dynamics

AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

scholarly journals Periodic points and smooth rays

2021 ◽  
Vol 25 (8) ◽  
pp. 170-178
Author(s):  
Carsten Petersen ◽  
Saeed Zakeri
Keyword(s):  
Periodic Point ◽  
Julia Set ◽  
Periodic Points ◽  
Connected Component ◽  
Landing Point ◽  
Content Type ◽  
Polynomial Map ◽  
External Ray

Let P : C → C P: \mathbb {C} \to \mathbb {C} be a polynomial map with disconnected filled Julia set K P K_P and let z 0 z_0 be a repelling or parabolic periodic point of P P . We show that if the connected component of K P K_P containing z 0 z_0 is non-degenerate, then z 0 z_0 is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one cycle of rays landing at z 0 z_0 may be broken.

scholarly journals Lebesgue measure of escaping sets of entire functions

2018 ◽  
Vol 40 (1) ◽  
pp. 89-116 ◽  
Author(s):  
WEIWEI CUI
Keyword(s):  
Entire Function ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Finite Order ◽  
Infinite Order ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Positive Lebesgue Measure

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

Describing quadratic Cremer point polynomials by parabolic perturbations

1998 ◽  
Vol 18 (3) ◽  
pp. 739-758 ◽  
Author(s):  
DAN ERIK KRARUP SØRENSEN
Keyword(s):  
Infinite Order ◽  
Julia Set ◽  
Main Idea ◽  
Mandelbrot Set ◽  
Basic Tool ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
External Rays ◽  
External Ray ◽  
Single Comb

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

scholarly journals Growth of solutions of second order complex linear differential equations with entire coefficients

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 275-284 ◽  
Author(s):  
Jianren Long
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Nontrivial Solution ◽  
Infinite Order ◽  
Entire Functions ◽  
Order Complex ◽  
Complex Linear ◽  
Linear Differential ◽  
The One ◽  
Growth Of Solutions

Some new conditions on the entire coefficients A(z) and B(z), which guarantee every nontrivial solution of f''+A(z) f'+B(z) f = 0 is of infinite order, are given in this paper. Two classes of entire functions are involved in these conditions, the one is entire functions having Fabry gaps, the another is function extremal for Yang?s inequality. Moreover, a kind of entire function having finite Borel exception value is considered.

scholarly journals Uniqueness on entire functions and their nth order exact differences with two shared values

2020 ◽  
Vol 18 (1) ◽  
pp. 211-215
Author(s):  
Shengjiang Chen ◽  
Aizhu Xu
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Shared Values ◽  
Simple Method ◽  
Exact Difference ◽  
Hyper Order

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

On entire functions of infinite order

1963 ◽  
Vol 14 (1) ◽  
pp. 323-327 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Infinite Order ◽  
Entire Functions

scholarly journals A question of Gross and the uniqueness of entire functions

1995 ◽  
Vol 138 ◽  
pp. 169-177 ◽  
Author(s):  
Hong-Xun yi
Keyword(s):  
Entire Function ◽  
Nevanlinna Theory ◽  
Entire Functions ◽  
Linear Measure ◽  
Finite Linear

For any set S and any entire function f letwhere each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

The distribution of the values of an entire function whose coefficients are independent random variables (II)

1995 ◽  
Vol 118 (3) ◽  
pp. 527-542 ◽  
Author(s):  
A. C. Offord
Keyword(s):  
Entire Function ◽  
Uniform Distribution ◽  
Entire Functions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Almost All

SummaryThis is a study of entire functions whose coefficients are independent random variables. When the space of such functions is symmetric it is shown that independence of the coefficients alone is sufficient to ensure that almost all such functions will, for large z, be large except in certain small neighbourhoods of the zeros called pits. In each pit the function takes every not too large value and these pits have a certain uniform distribution.

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