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Iteration of entire functions

01 Jan 2014-
TL;DR: In this article, the results discussed in the lectures at the CIMPA school in Kathmandu in November 2014 were discussed and the proofs of the results were given. But some references to the literature where proofs can be found are given.
Abstract: These notes contain the results discussed in the lectures at the CIMPA school in Kathmandu in November 2014. They contain only some of the proofs, but some references to the literature where proofs can be found are given.

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Citations
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Journal ArticleDOI
TL;DR: In this article, the authors study functions for which each of their levels, and each of its levels, has the structure of an infinite spider's web and show that there are many such functions and that they have strong dynamical properties.
Abstract: Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

92 citations


Cites background from "Iteration of entire functions"

  • ...It was shown in [14] that Eremenko’s construction in [18] of points in I(f) actually gives points that are in A(f)....

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  • ...We begin by showing that A(f) has properties corresponding to the properties of I(f) proved by Eremenko [18] that we listed in (1....

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  • ...For a transcendental entire function f , the escaping set was first studied by Eremenko [18] who proved that...

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  • ...Much of the work on I(f) has been motivated by Eremenko’s conjecture [18] that every component of I(f) is unbounded....

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Journal ArticleDOI
TL;DR: In this article, the authors studied the dynamical behavior of a transcendental entire function in any multiply connected wandering domain of the Fatou set, and showed that the union of these annuli acts as an absorbing set for the iterates of the function in the domain.
Abstract: The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as an absorbing set for the iterates of $f$ in $U$. Moreover, $f$ behaves like a monomial within each of these annuli and the orbits of points in $U$ settle in the long term at particular `levels' within the annuli, determined by the function $h$. We also discuss the proximity of $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n \setminus \bar{C_n}$. These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus.

51 citations


Cites methods from "Iteration of entire functions"

  • ...Eremenko and Lyubich [23] as well as Goldberg and Keen [27] extended Sullivan’s theorem to the Speiser class S of entire functions for which the set of critical and asymptotic values is finite....

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  • ...We thank Alex Eremenko for a useful discussion about the Riemann– Hurwitz formula, Núria Fagella for a conversation that led us to include Theorem 7.1, Jian-Hua Zheng for drawing our attention to the question answered by Theorem 1.9, and Dave Sixsmith and the referee for helpful comments....

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  • ...A. E. Eremenko and M. Yu....

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  • ...To prove several of our theorems, we need the following result which is based on a technique due to Eremenko [22] using Wiman–Valiron theory....

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  • ...L. Rempe, ‘On a question of Eremenko concerning escaping components of entire functions’, Bull....

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Journal ArticleDOI
TL;DR: In this article, the authors studied the connectedness properties of a set of points BU (B) whose iterates under the Fatou component of a function neither escape to infinity nor are bounded.
Abstract: For a transcendental entire function ƒ, we study the set of points BU (ƒ) whose iterates under ƒ neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU (ƒ), then most boundary points of U (in the sense of harmonic measure) lie in BU (ƒ). We prove this using a new result concerning the set of limit points of the iterates of ƒ on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.

47 citations

Journal ArticleDOI
TL;DR: For a survey of the dynamics of functions in the Eremenko-Lyubich class, see, e.g., as mentioned in this paper, for instance, the work of as mentioned in this paper.
Abstract: The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials -- in other words transcendental entire functions -- is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics. In this paper we survey the dynamics of functions in the Eremenko-Lyubich class, $\mathcal{B}$. Among transcendental entire functions, those in this class have properties that make their dynamics most "similar" to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of class $\mathcal{B}$ functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases. We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.

40 citations

References
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Book
01 Jan 1979

2,694 citations

Book
01 Aug 2000
TL;DR: In this article, the dynamics of iterated holomorphic mappings from a Riemann surface to itself are studied, focusing on the classical case of rational maps of the RiemANN sphere.
Abstract: This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattes map has been made more inclusive, and the ecalle-Voronin theory of parabolic points is described. The residu iteratif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

1,620 citations


"Iteration of entire functions" refers background in this paper

  • ...2 can be found in textbooks on complex dynamics [9, 30, 36]....

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  • ...The analogous result for rational functions can be found in standard textbooks on complex dynamics [9, 30, 36]....

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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

DOI
01 Jan 1935
TL;DR: In this article, the applications of homogeneous differential polynomials to the Nevanlinna theory of meromorphic functions in the finite complex plane have been given, and some generalizations by these polynomial coefficients have been shown.
Abstract: In this paper, we have given the applications of homogeneous differential polynomials to the Nevanlinna's theory of meromorphic functions in the finite complex plane and given some generalizations by these polynomials. ÖZET : Bu çal›flmada, homojen diferansiyel polinomlar Nevanlinna kuram›na uyguland› ve bu homojen polinomlarla baz› genellefltirmeler verildi. Anahtar kelimeler: Meromorfik fonksiyon, homojen diferansiyel polinom ve sonlu karmafl›k düzlem.

563 citations


"Iteration of entire functions" refers result in this paper

  • ...These results were proved by Nevanlinna in 1924; see [24, 26, 31]....

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