Dynamical properties of some classes of entire functions

doi:10.5802/AIF.1318

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Dynamical properties of some classes of entire functions

Journal Article•DOI•

Dynamical properties of some classes of entire functions

01 Jan 1992-Annales de l'Institut Fourier (Association des annales de l'institut Fourier)-Vol. 42, Iss: 4, pp 989-1020

About: This article is published in Annales de l'Institut Fourier.The article was published on 1992-01-01 and is currently open access. It has received 530 citations till now. The article focuses on the topics: Entire function.

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Journal Article•DOI•

Iteration of meromorphic functions

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Walter Bergweiler

01 Jan 1993-Bulletin of the American Mathematical Society

TL;DR: In this paper, the authors describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions, and some aspects where the transcendental case is analogous to the rational case are treated rather briefly here.

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Abstract: This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

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737 citations

Journal Article•DOI•

Dynamic rays of bounded-type entire functions

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Günter Rottenfußer, Johannes Rückert¹, Lasse Rempe², Dierk Schleicher¹•Institutions (2)

Jacobs University Bremen¹, University of Liverpool²

04 Jan 2011-Annals of Mathematics

TL;DR: In this paper, an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components was constructed, which gave a partial positive answer to the aforementioned question.

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Abstract: We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in B, in particular those of nite order, every escaping point can be connected to 1 by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

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160 citations

Journal Article•DOI•

Escaping Points of Exponential Maps

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Dierk Schleicher¹, Johannes Zimmer²•Institutions (2)

International University, Cambodia¹, California Institute of Technology²

01 Apr 2003-Journal of The London Mathematical Society-second Series

TL;DR: In this article, a complete classification of escaping points is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays.

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Abstract: The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpinska for specific choices of λ.

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148 citations

Cites background or methods from "Dynamical properties of some classe..."

...Tails of rays lie entirely in the Julia set of Eλ: this follows from the classification of Fatou components by Eremenko and Lyubich [6, 7, 8]; see also [1, Theorems 6 and 7]....
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...A. Eremenko and M. Lyubich, ‘Iterates of entire functions’, Soviet Math....
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...This project was inspired by discussions with Bogusia Karpińska and Misha Lyubich at a Euroconference in Crete organized by Shaun Bullett, Adrien Douady and Christos Kourouniotis....
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...M. Lyubich, ‘Measurable dynamics of the exponential’, Sibirsk....
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...It follows that all points in U must escape to infinity within HR by the minimum principle, and this is impossible by the classification of Fatou components (see [6, 7, 8] or [1])....
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Journal Article•DOI•

Iteration of a class of hyperbolic meromorphic functions

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P. J. Rippon¹, Gwyneth M. Stallard¹•Institutions (1)

Open University¹

27 Apr 1999

TL;DR: In this article, the authors consider the class Bn which contains functions f in Bl for which the forward orbits of the singularities of f-1 stay away from the Julia set.

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Abstract: We look at the class Bn which contains those transcendental meromorphic functions f for which the finite singularities of f-n are in a bounded set and prove that, if f belongs to Bn, then there are no components of the set of normality in which fmn(z) oo as m -* oo. We then consider the class B which contains those functions f in Bl for which the forward orbits of the singularities of f-1 stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions f-n and (b) that, for points in the Julia set of f, the derivatives (fn)' have exponential-type growth. This justifies the assertion that B is a class of hyperbolic functions.

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131 citations

Cites background from "Dynamical properties of some classe..."

...Primary 30D05. c©1999 American Mathematical Society 3251 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Eremenko and Lyubich [6] investigated the properties of entire functions in the class B = {f : f is a transcendental meromorphic function with S(f) bounded}....
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...Eremenko and Lyubich [6] investigated the properties of entire functions in the class B = {f : f is a transcendental meromorphic function with S(f) bounded}....
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...Our proof is based on ideas of Eremenko and Lyubich [6, Theorem 1] who proved this result in the case when f is entire and n = 1....
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...Lyubich, Dynamical properties of some classes of entire functions, Ann....
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Monograph•DOI•

Quasiconformal surgery in holomorphic dynamics

[...]

Bodil Branner¹, Núria Fagella²•Institutions (2)

Technical University of Denmark¹, University of Barcelona²

01 Jan 2013

TL;DR: This book discusses surgery and its applications in dynamical systems and actions of Kleinian groups, as well as some of the principles of surgery as applied to extensions and interpolations.

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Abstract: Preface Introduction 1. Quasiconformal geometry 2. Extensions and interpolations 3. Preliminaries on dynamical systems and actions of Kleinian groups 4. Introduction to surgery and first occurrences 5. General principles of surgery 6. Soft surgeries with a contribution by X. Buff and C. Henriksen 7. Cut and paste surgeries with contributions by K. M. Pilgrim, Tan Lei and S. Bullett 8. Cut and paste surgeries with sectors with a contribution by A. L. Epstein and M. Yampolsky 9. Trans-quasiconformal surgery with contributions by C. L. Petersen and P. Haissinsky Bibliography Symbol index Index.

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122 citations

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References

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Book•

Quasiconformal mappings in the plane

[...]

Olli Lehto, K. I. Virtanen

01 Jan 1973

TL;DR: In this article, the authors define a geometric definition of a quasiconformal mapping and apply it to the Hilbert transformation to a set of dimensions of a circle domain and a ring domain.

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Abstract: I. Geometric Definition of a Quasiconformal Mapping.- to Chapter I.- 1. Topological Properties of Plane Sets.- 2. Conformal Mappings of Plane Domains.- 3. Definition of a Quasiconformal Mapping.- 4. Conformal Module and Extremal Length.- 5. Two Basic Properties of Quasiconformal Mappings.- 6. Module of a Ring Domain.- 7. Characterization of Quasiconformality with the Help of Ring Domains.- 8. Extension Theorems for Quasiconformal Mappings.- 9. Local Characterization of Quasiconformality.- II. Distortion Theorems for Quasiconformal Mappings.- to Chapter II.- 1. Ring Domains with Extremal Module.- 2. Module of Grotzsch's Extremal Domain.- 3. Distortion under a Bounded Quasiconformal Mapping of a Disc.- 4. Order of Continuity of Quasiconformal Mappings.- 5. Convergence Theorems for Quasiconformal Mappings.- 6. Boundary Values of a Quasiconformal Mapping.- 7. Quasisymmetric Functions.- 8. Quasiconformal Continuation.- 9. Circular Dilatation.- III. Auxiliary Results from Real Analysis.- to Chapter III.- 1. Measure and Integral.- 2. Absolute Continuity.- 3. Differentiability of Mappings of Plane Domains.- 4. Module of a Family of Arcs or Curves.- 5. Approximation of Measurable Functions.- 6. Functions with Lp-derivatives.- 7. Hubert Transformation.- IV. Analytic Characterization of a Quasiconformal Mapping.- to Chapter IV.- 1. Analytic Properties of a Quasiconformal Mapping.- 2. Analytic Definition of Quasiconformality.- 3. Variants of the Geometric Definition.- 4. Characterization of Quasiconformality with the Help of the Circular Dilatation.- 5. Complex Dilatationl.- V. Quasiconformal Mappings with Prescribed Complex Dilatation.- to Chapter V.- l. Existence Theorem.- 2. Local Dilatation Measures.- 3. Removable Point Sets.- 4. Approximation of a Quasiconformal Mapping.- 5. Application of the Hilbert Transformation to Quasiconformal Mappings21l.- 6. Conformality at a Point.- 7. Regularity of a Mapping with Prescribed Complex Dilatation.- VI. Quasiconformal Functions.- to Chapter VI.- 1. Geometric Characterization of a Quasiconformal Function.- 2. Analytic Characterization of a Quasiconformal Function.

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1,359 citations

"Dynamical properties of some classe..." refers background in this paper

...Proof. — This is a well-known property of quasiconforma l homeomorphisms (see for example [ LV ])....
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Conformal Invariants

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Joseph Miller Thomas

01 Jun 1926

1,047 citations

Journal Article•DOI•

On the dynamics of rational maps

[...]

Ricardo Mañé, Paulo Sad, Dennis Sullivan

01 Jan 1983-Annales Scientifiques De L Ecole Normale Superieure

TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).

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Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1983, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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775 citations

"Dynamical properties of some classe..." refers background or result in this paper

...Remarks. — 1. As in [L2], [ MSS ] Theorems 9 and 10 may be proved for any analytic subfamily M. C M....
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...analog of the theorem obtained in [L2], [ MSS ] for rational maps....
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...b) The map (pf : A —> C is quasiconformal for any f e W [ MSS ]....
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...A-LEMMA. — a) A holomorphic motion (p of a set A may be extended to a holomorphic motion of the closure A [L2], [ MSS ];...
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...THEOREM 10. — The set of structurally stable endomorphisms is open and dense in M. The conjugating homeomorphisms can be chosen to be quasiconformal. Proof (Compare [ MSS ])....
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Journal Article•DOI•

Riemann's mapping theorem for variable metrics*

[...]

Lars V. Ahlfors, Lipman Bers

01 Sep 1960-Annals of Mathematics

630 citations

Journal Article•DOI•

Sur les équations fonctionnelles

[...]

P. Fatou

01 Jan 1920-Bulletin de la Société Mathématique de France

TL;DR: The Bulletin de la S. M. F. as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.html).

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Abstract: © Bulletin de la S. M. F., 1920, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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582 citations