Iteration of meromorphic functions

doi:10.1090/S0273-0979-1993-00432-4

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Iteration of meromorphic functions

Journal Article•DOI•

Iteration of meromorphic functions

01 Jan 1993-Bulletin of the American Mathematical Society (American Mathematical Society (AMS))-Vol. 29, Iss: 2, pp 151-188

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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Journal Article•DOI•

On the singularities of the inverse to a meromorphic function of finite order

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Walter Bergweiler, Alexandre Eremenko

31 Aug 1995-Revista Matematica Iberoamericana

TL;DR: In this paper, it was shown that if f is a transcendental meromorphic function, then f with n = 1 takes every finite non-zero value infinitely often, which is a conjecture of Hayman.

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Abstract: Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of them, is a limit point of critical values of f. We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n = 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions

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389 citations

Cites background from "Iteration of meromorphic functions"

...Then A := ϕ({w : |w| Rn}) is a component of f−1({w : |w| Rn}), containing the point zn ∈ U(Rn)....
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Book Chapter•DOI•

Complex dynamics in higher dimensions

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John Erik Fornæss¹, Nessim Sibony²•Institutions (2)

University of Michigan¹, University of Paris²

01 Jan 1994

TL;DR: The field of complex dynamics in higher dimensions was initiated in the 1920s by Fa-tou as discussed by the authors, who was motivated by studies in Newton's method, celestial mechanics and functional equations.

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Abstract: The field of complex dynamics in higher dimension was initiated in the 1920’s by Fa-tou. It was motivated by studies in Newton’s method, celestial mechanics and functional equations. Recently, new methods from pluripotential theory have been introduced to the subject. These techniques have produced many new interesting results. We give an introduction to this subject and a summary of the most relevant developments.

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360 citations

Book•

Value Distribution of Meromorphic Functions

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A. A. Gol'dberg¹, Iossif Ostrovskii², Iossif Ostrovskii³•Institutions (3)

Bar-Ilan University¹, National Academy of Sciences of Ukraine², Bilkent University³

13 Aug 2008

TL;DR: A survey of results after 1970 can be found in this paper, where the authors present a survey of meromorphic functions of finite-order functions with respect to Riemann surfaces.

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Abstract: Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties of meromorphic functions and deficiencies Value distribution with respect to the arguments Applications of Riemann surfaces to value distribution On the magnitude of an entire function Notes A survey of some results after 1970 Bibliography References added to the English edition Author index Subject index Notation index.

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242 citations

Journal Article•DOI•

Graphic and numerical comparison between iterative methods

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Juan L. Varona¹•Institutions (1)

University of La Rioja¹

01 Dec 2002-The Mathematical Intelligencer

TL;DR: A simplified version of the Newton-Raphson method was proposed by Raphson in 1690 as mentioned in this paper, which is also known as the Newman-Rodriguez method.

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Abstract: generates a sequence {xn}n=0 that converges to ζ. In fact, Newton’s original ideas on the subject, around 1669, were considerably more complicated. A systematic study and a simplified version of the method are due to Raphson in 1690, so this iteration scheme is also known as the Newton-Raphson method. (Also as the tangent method, from its geometric interpretation.) In 1879, Cayley tried to use the method to find complex roots of complex functions f : C → C. If we take z0 ∈ C and we iterate

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192 citations

Cites background from "Iteration of meromorphic functions"

...(In the internal behavior of Mathematica, when a function is going to be pictured with DensityPlot ,i t is scaled to [0, 1]. However, iterColorAlgorithm has a range of [0, 4 ]; this is the reason for using 4*p in some places in colorLevel and...
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Journal Article•DOI•

On semiconjugation of entire functions

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Walter Bergweiler¹, Aimo Hinkkanen²•Institutions (2)

University of Kiel¹, University of Illinois at Urbana–Champaign²

01 May 1999

TL;DR: In this paper, it was shown that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f).

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Abstract: Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.

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167 citations

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References

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Journal Article•DOI•

Simple mathematical models with very complicated dynamics

[...]

Robert M. May¹, Robert M. May²•Institutions (2)

Princeton University¹, University of Cambridge²

10 Jun 1976-Nature

TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.

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Abstract: First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

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6,118 citations

"Iteration of meromorphic functions" refers background in this paper

...5 per page 1 2 WALTER BERGWEILER 7. Miscellaneous topics References 1. Introduction Mathematical models for phenomena in the natural sciences often lead to iteration. An often-quoted example (compare [105]) comes from population biology. Assuming that the size of a generation of a population depends solely on the size of the previous generation and may thus be expressed as a function of it, questions c...
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Book•

An introduction to chaotic dynamical systems

[...]

Robert L. Devaney

01 Jan 1986

TL;DR: In this article, the quadratic family has been used to define hyperbolicity in linear algebra and advanced calculus, including the Julia set and the Mandelbrot set.

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Abstract: Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

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3,589 citations

Book•

Dynamics and Bifurcations

[...]

Jack K. Hale, Hüseyin Koçak

01 Jan 1992

TL;DR: In this paper, the authors present ideas and examples about the geometry of dynamics and bifurcations of ordinary differential equations, and demonstrate that the basic notion of stability and stability of vector fields can be easily explained for scalar autonomous equations.

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Abstract: This study presents ideas and examples about the geometry of dynamics and bifurcations of ordinary differential equations. The subject of differential and difference equations is an old and much-honoured chapter in science. In recent years, due primarily to the proliferation of computers, dynamical systems have returned to their roots in applications. It is the aim of this book to provide a modest foundation of knowledge for undergraduate and beginning graduate students in mathematics or science and engineering. Equations in dimension one and two constitute the majority of the text. It is demonstrated in particular that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Proceeding, the authors investigate the dynamics of planar autonomous equations where new dynamical behaviour, such as periodic and homoclinic orbits, appears.

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1,252 citations

Book•DOI•

Iteration of Rational Functions

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Robert L. Devaney, Alan F. Beardon

01 Jan 1991-American Mathematical Monthly

972 citations

"Iteration of meromorphic functions" refers background or methods in this paper

...component [82], essentially parabolic domain [24], and domains at ∞ [59] are also used. The term “Baker domain” seems to have been used ﬁrst in [69, 70]. Besides the papers cited already, we refer to [27, 108, 127] for a proof of the classiﬁcation theorem. Here only the case that f is rational is considered, but the changes necessary to handle the case that f is transcendental are minor. We note that if f is en...
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...osely speaking—a contradiction is obtained from the fact that there are many quasiconformal homeomorphisms of U n0 and hence many functions µ but not so many functions fΦ. For the details we refer to [16, 24, 27]. In case (ii) it is not diﬃcult to obtain a contradiction to Theorem 10 if f is entire and contained in S, F, or N. A result similar to Theorem 10 can still be obtained if f has ﬁnitely many poles, a...
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...cent years work on the iteration of transcendental meromorphic functions has also begun. There exist a number of introductions to and surveys of the iteration theory of rational functions. We mention [27, 37, 42, 53, 63, 69, 93, 100, 108, 128] among the more recent ones but also some older ones [40, 46, 110]. There are comparatively few expositions of the iteration theory of transcendental functions. We refer to [18] for the iteration of e...
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Book•

The Beauty of Fractals

[...]

Heinz-Otto Peitgen, Peter H. Richter

01 Jan 1986

TL;DR: A can is a can made of a steel sheet the surface of which is coated with a three-layered chromium coating, consisting of a metallic chromium coated, a crystalline chromium oxide coating and a non-crystalline hydrated chromiumoxide coating in this order.

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Abstract: A can made of a steel sheet the surface of which is coated with a three-layered chromium coating, consisting of a metallic chromium coating, a crystalline chromium oxide coating and a non-crystalline hydrated chromium oxide coating in this order. A layer of an organic enamel or fused film may be provided further on top of the non-crystalline hydrated chromium oxide coating.

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790 citations

"Iteration of meromorphic functions" refers background or methods in this paper

...who work in the ﬁeld, but they do not seem to have been stated explicitly before. As already mentioned, there are beautiful computer graphics related to the theory, and there are many places (besides [112]) where such pictures can be found for rational functions. Although Julia sets (and bifurcation diagrams) of transcendental functions can compete in their beauty and complexity very well with those of...
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...t in the iteration theory of analytic functions, partially due to the beautiful computer graphics related to it (see, for ITERATION OF MEROMORPHIC FUNCTIONS 3 example, the book by Peitgen and Richter [112]), partially due to new and powerful mathematical methods introduced into it (notably those introduced by Douady and Hubbard [62] and by Sullivan [129]). Most of the work has centered around the itera...
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