scholarly journals Wandering domains for entire functions of finite order in the Eremenko–Lyubich class

2019 ◽  
Vol 120 (2) ◽  
pp. 155-191 ◽  
Author(s):  
David Martí‐Pete ◽  
Mitsuhiro Shishikura
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Wandering Domains

Newton's method and a class of meromorphic functions without wandering domains

1993 ◽  
Vol 13 (2) ◽  
pp. 231-247 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Fixed Points ◽  
Finite Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Newton Iteration ◽  
Fatou Set ◽  
Iteration Function ◽  
Wandering Domains

AbstractLetNbe the class of meromorphic functionsfwith the following properties:fhas finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points offare zeros off′ and vice versa, with finitely many exceptions;fhas finite order. It is proved that iff∈N, thenfdoes not have wandering domains. Moreover, iff∈Nand if ∞ is among the limit functions offnin a cycle of periodic domains, then this cycle contains a singularity off−1. (Herefndenotes thenth iterate off) These results are applied to study Newton's method for entire functionsgof the formwherepandqare polynomials and wherecis a constant. In this case, the Newton iteration functionf(z) =z−g(z)/g′(z) is inN. It follows thatfn(z) converges to zeros ofgfor allzin the Fatou set off, if this is the case for all zeroszofg″. Some of the results can be extended to the relaxed Newton method.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

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.

Hankel Convolution Operators on Spaces of Entire Functions of Finite Order

2005 ◽  
Vol 48 (2) ◽  
pp. 161-174 ◽  
Author(s):  
Jorge J. Betancor
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Convolution Operators ◽  
Hankel Transforms ◽  
Hankel Convolution

AbstractIn this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals.

A theorem on divisors of entire functions of finite order

1990 ◽  
Vol 48 (2) ◽  
pp. 799-804
Author(s):  
G. N. Shilova
Keyword(s):  
Finite Order ◽  
Entire Functions

scholarly journals Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

2011 ◽  
Vol 33 (1) ◽  
pp. 284-302 ◽  
Author(s):  
JÖRN PETER
Keyword(s):  
Finite Order ◽  
Hausdorff Measure ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Gauge Function ◽  
Exponential Map ◽  
Hausdorff Measures ◽  
Transcendental Entire Functions ◽  
Bounded Type Functions

AbstractWe show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become ‘smaller’ as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

Sign in / Sign up

Export Citation Format

Share Document