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 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 Density Conditions for Interpolation in the Ball

2003 ◽  
Vol 46 (4) ◽  
pp. 559-574 ◽  
Author(s):  
Nicolas Marco ◽  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Sufficient Condition ◽  
Density Condition ◽  
Interpolating Sequences ◽  
Class A ◽  
Spaces Of Holomorphic Functions

AbstractIn this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of Cn, n > 1. We first give density conditions for a sequence to be interpolating for the class A−∞ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for n = 1 coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

scholarly journals Some entire functions with multiply-connected wandering domains

1985 ◽  
Vol 5 (2) ◽  
pp. 163-169 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Julia Set ◽  
Multiply Connected ◽  
Wandering Domain ◽  
Wandering Domains

AbstractA component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.

scholarly journals A–∞-interpolation in the ball

1998 ◽  
Vol 41 (2) ◽  
pp. 359-367 ◽  
Author(s):  
Xavier Massaneda
Keyword(s):  
Unit Ball ◽  
Polynomial Growth ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Derivatives Of

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.

scholarly journals Fatou components and singularities of meromorphic functions

2019 ◽  
Vol 150 (2) ◽  
pp. 633-654 ◽  
Author(s):  
Krzysztof Barański ◽  
Núria Fagella ◽  
Xavier Jarque ◽  
Bogusława Karpińska
Keyword(s):  
Relative Position ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Singular Values ◽  
Wandering Domain ◽  
Meromorphic Maps ◽  
Fatou Components ◽  
Meromorphic Map ◽  
Wandering Domains ◽  
Uniformly Bounded

AbstractWe prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

ON WANDERING AND BAKER DOMAINS OF TRANSCENDENTAL ENTIRE FUNCTIONS

2004 ◽  
Vol 14 (01) ◽  
pp. 321-327 ◽  
Author(s):  
XIAOLING WANG ◽  
CHUNG-CHUN YANG
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Multiply Connected ◽  
Wandering Domain ◽  
Transcendental Entire Functions ◽  
Simply Connected

Let f denote a transcendental entire function, and I(f), I0(f), T(f) and A(f) be denoted as follows: [Formula: see text][Formula: see text] Let D denote a Fatou component of F(f). We have established the relationships between D and I(f), I0(f), T(f) or A(f), when D is a Baker domain or a multiply-connected wandering domain or a simply-connected infinitely wandering domain.

An area approach to wandering domains for smooth surface endomorphisms

1991 ◽  
Vol 11 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Alec Norton
Keyword(s):  
Smooth Surface ◽  
Main Lemma ◽  
Rational Map ◽  
Wandering Domain ◽  
Direct Corollary ◽  
Simply Connected ◽  
Simply Connected Domains ◽  
Wandering Domains

AbstractWe prove an infinite-area lemma for maps which are area non-contracting on the boundaries of certain domains; the maps are required to be smooth to the extent that their Jacobians are twice differentiable (see Main Lemma below). It will follow that a hyperbolic rational map has no wandering simply connected domains. As a more direct corollary, a C3 diffeomorphism f of a compact smooth 2-manifold cannot have a wandering domain Δ if f is area non-contracting on the boundary of each forward image of Δ.

Existence of wandering and periodic domain in given angular region

2020 ◽  
Vol 70 (4) ◽  
pp. 839-848
Author(s):  
Vishnu Narayan Mishra ◽  
Garima Tomar
Keyword(s):  
Entire Functions ◽  
Periodic Domain ◽  
Angular Region ◽  
Wandering Domain ◽  
Structures And Properties ◽  
Fatou Components

AbstractDynamics of composition of entire functions is well related to it's factors, as it is known that for entire functions f and g, fog has wandering domain if and only if gof has wandering domain. However the Fatou components may have different structures and properties. In this paper we have shown the existence of domains with all possibilities of wandering and periodic in given angular region θ.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

Separating Singularities of Holomorphic Functions

1998 ◽  
Vol 41 (4) ◽  
pp. 473-477 ◽  
Author(s):  
Jürgen Müller ◽  
Jochen Wengenroth
Keyword(s):  
Approximation Theory ◽  
Classical Result ◽  
Short Proof ◽  
Holomorphic Functions ◽  
Open Mapping ◽  
Complex Approximation ◽  
Basic Tool ◽  
Open Mapping Theorem ◽  
Separation Results

AbstractWe present a short proof for a classical result on separating singularities of holomorphic functions. The proof is based on the open mapping theorem and the fusion lemma of Roth, which is a basic tool in complex approximation theory. The same method yields similar separation results for other classes of functions.

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