Dynamics of Transcendental Functions
22 Jan 2019-pp 311-316
TL;DR: In this article, the sequence of the iterates of a nonconstant meromorphic function is denoted by a sequence of numbers, where the number of iterates is the length of the sequence.
Abstract: Let f be a nonconstant meromorphic function. The sequence of the iterates of f is denoted by
$${f^0}=id,{f^1}=f,\cdots,{f^{n + 1}} = {f^n}(f), \cdots$$
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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.
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.
122 citations
TL;DR: In this paper, the authors studied the space of quasiconformal deformations of an entire map with an invariant Baker domain by studying its Teichmuller space and showed that the dimension of this set is infinite if the Bakerdomain is hyperbolic or simply parabolic.
Abstract: We consider entire transcendental functions $f$ with an invariant
(or periodic) Baker domain $U$. First, we classify these domains
into three types (hyperbolic, simply parabolic and doubly
parabolic) according to the surface they induce when we take the
quotient by the dynamics. Second, we study the space of
quasiconformal deformations of an entire map with such a Baker
domain by studying its Teichmuller space. More precisely, we
show that the dimension of this set is infinite if the Baker
domain is hyperbolic or simply parabolic, and from this we deduce
that the quasiconformal deformation space of $f$ is infinite
dimensional. Finally, we prove that the function $f(z)=z+e^{-z}$,
which possesses infinitely many invariant Baker domains, is rigid,
i.e., any quasiconformal deformation of $f$ is affinely conjugate
to $f$.
36 citations
TL;DR: The Ahlfors five islands theorem has become an important tool in complex dynamics as discussed by the authors, and it can be used to deal with a variety of problems, such as the Hausdorff dimension of Julia sets, the existence of singleton components, and repelling periodic points.
Abstract: The Ahlfors five islands theorem has become an important tool in complex dynamics. We discuss its role there, describing how it can be used to deal with a variety of problems. This includes questions concerning the Hausdorff dimension of Julia sets, the existence of singleton components of Julia sets, and the existence of repelling periodic points. We point out that for many applications a simplified version of the Ahlfors five islands theorem suffices, and we give an elementary proof of this version.
29 citations
01 Jun 2008
TL;DR: The residual Julia set as mentioned in this paper is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set, and it is defined as a subset of the points of a topological topological space.
Abstract: A bstract . The residual Julia set , denoted by J r (f) , is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set. The points of J r (f) are called buried points of J(f) and a component of J(f) which is contained in J r (f) is called a buried component . In this paper we survey the most important results related to the residual Julia set for several classes of functions. We also give a new criterion to deduce the existence of buried points and, in some cases, of unbounded curves in the residual Julia set (the so-called Devaney hairs ). Some examples are the sine family, certain meromorphic maps constructed by surgery and the exponential family. INTRODUCTION Given a map f : X → X , where X is a topological space, the sequence formed by its iterates will be denoted by f 0 ≔ Id, f n ≔ f ∘ f n−1 , n ∈ ℕ. When f is a holomorphic map and X is a Riemann surface the study makes sense and is non-trivial when X is either the Riemann sphere, the complex plane ℂ or the complex plane minus one point ℂ \ {0}. All other interesting cases can be reduced to one of these three. All other interesting cases can be reduced to one of these three. In this paper we deal with the following classes of maps (partially following [12]).
19 citations
01 Jan 2008
TL;DR: In this paper, the authors consider the Teichmuller space of a general entire transcendental function f : C → C regardless of the nature of the set of singular values of f (critical values and asymptotic values).
Abstract: We consider the Teichmuller space of a general entire transcendental function f : C → C regardless of the nature of the set of singular values of f (critical values and asymptotic values). We prove that, as in the known case of periodic points and critical values, asymptotic values are also fixed points of any quasiconformal automorphism that commutes with f and which is homotopic to the identity, rel. the ideal boundary of the domain. As a consequence, the general framework of McMullen and Sullivan [McMullen & Sullivan 1998] for rational functions applies also to entire functions and we can apply it to study the Teichmuller space of f , analyzing each type of Fatou component separately. Baker domains were already considered in citefh, but wandering domains are new. We provide different examples of wandering domains, each of them adding a different quantity to the dimension of the Teichmuller space. In particular we give examples of rigid wandering domains.
18 citations
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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
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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