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.

Quasiconformal surgery in holomorphic dynamics

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 $f$.

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Deformation of Entire Functions with Baker Domains

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.

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The role of the Ahlfors five islands theorem in complex dynamics

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]).

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Transcendental Dynamics and Complex Analysis: Residual Julia sets of rational and transcendental functions

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.

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The Teichmuller Space of an Entire Function

Let f be a nonconstant meromorphic function. The sequence of the iterates of f is denoted by 

$${f^0}=id,{f^1}=f,\cdots,{f^{n + 1}} = {f^n}(f), \cdots$$

Xiaoling Wang

Papers

Dynamics of Transcendental Functions