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BookDOI

Iteration of Rational Functions

01 Jan 1991-American Mathematical Monthly (Springer New York)-Vol. 100, Iss: 1, pp 90
About: This article is published in American Mathematical Monthly.The article was published on 1991-01-01. It has received 972 citations till now. The article focuses on the topics: Elliptic rational functions & Rational function.
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Book
29 Nov 1994
TL;DR: In this article, the authors present a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics, including geometric function theory, quasiconformal mappings, and hyperbolic geometry.
Abstract: Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set.The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

748 citations

Journal ArticleDOI
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.
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).

737 citations


Cites background or methods from "Iteration of Rational Functions"

  • ...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
19 Nov 2010
TL;DR: In this paper, the authors provide an introduction to the discipline of arithmetic dynamics, the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function.
Abstract: * Provides an entry for graduate students into an active field of research * Each chapter includes exercises, examples, and figures * Will become a standard reference for researchers in the field * Contains descriptions of many known results and conjectures, together with an extensive bibliography This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures. Key features: - Provides an entry for graduate students into an active field of research - Provides a standard reference source for researchers - Includes numerous exercises and examples - Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

679 citations

MonographDOI
14 Oct 2002
TL;DR: In this paper, the authors propose a measure-theoretic entropy Bibliography Index for the literature on topological and complex dynamics, including topological dynamics, symbolic dynamics, hyperbolic dynamics, and low-dimensional dynamics.
Abstract: Introduction 1. Examples and basic concepts 2. Topological dynamics 3. Symbolic dynamics 4. Ergodic theory 5. Hyperbolic dynamics 6. Ergodicity of Anosov diffeomorphisms 7. Low-dimensional dynamics 8. Complex dynamics 9. Measure-theoretic entropy Bibliography Index.

558 citations

Journal ArticleDOI
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.
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

389 citations


Cites background from "Iteration of Rational Functions"

  • ...(In [ 1 ] as well as in [3, 24] only the case of rational functions is discussed, in which case only critical values need to be considered, but the proof extends to the transcendental case, if we also take asymptotic values into account.) Since f has innitely many multiple zeros and since Leau domains related to distinct xed points of g are disjoint, we deduce that the set of critical and asymptotic values of g is innite....

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  • ...Theorem 1 and its corollaries may be useful in many questions involving meromorphic functions of nite order, in particular in the iteration theory of rational [ 1 , 3, 24] and transcendental meromorphic [2] functions....

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