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Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains

Gustavo R. Ferreira, Sebastian van Strien

TL;DR

This work extends the theory of structural stability for entire functions beyond finite-type maps by constructing a universal Banach-analytic natural family $T_f$ with a holomorphic covering map $\Phi$ onto the quasiconformal equivalence class $M_f$, thereby endowing $M_f$ with a complex structure even when $S(f)$ is infinite. It introduces distortion sequences for simply connected wandering domains and proves their holomorphic movement in holomorphic parameter families where the Julia set moves holomorphically, via a distortion map $A: \Lambda' \times \overline{\Lambda'} \to \ell^\infty$. Additionally, the paper demonstrates how to perturb Herman-type wandering domains to obtain non-constant distortion maps, using a detailed quasiconformal surgery and linearisation framework to parametrize dynamics in infinite-dimensional parameter spaces. Collectively, these results fuse Teichmüller-theoretic descriptions with wandering-domain dynamics, providing new tools to study stability and parametric variation in broad classes of entire functions.

Abstract

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich's result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set $M_f$ of entire functions quasiconformally equivalent to $f$ admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains -- another hot topic of research in complex dynamics. Given an entire function $f$ with a simply connected wandering domain $U$, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as $f$ moves within appropriate parameter families.

Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains

TL;DR

This work extends the theory of structural stability for entire functions beyond finite-type maps by constructing a universal Banach-analytic natural family with a holomorphic covering map onto the quasiconformal equivalence class , thereby endowing with a complex structure even when is infinite. It introduces distortion sequences for simply connected wandering domains and proves their holomorphic movement in holomorphic parameter families where the Julia set moves holomorphically, via a distortion map . Additionally, the paper demonstrates how to perturb Herman-type wandering domains to obtain non-constant distortion maps, using a detailed quasiconformal surgery and linearisation framework to parametrize dynamics in infinite-dimensional parameter spaces. Collectively, these results fuse Teichmüller-theoretic descriptions with wandering-domain dynamics, providing new tools to study stability and parametric variation in broad classes of entire functions.

Abstract

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich's result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set of entire functions quasiconformally equivalent to admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains -- another hot topic of research in complex dynamics. Given an entire function with a simply connected wandering domain , we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as moves within appropriate parameter families.
Paper Structure (14 sections, 26 theorems, 67 equations)

This paper contains 14 sections, 26 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Natural families of entire maps
  3. Multiplier-like objects for wandering domains
  4. Preliminaries
  5. Internal dynamics of simply connected wandering domains
  6. Holomorphic functions on Banach spaces
  7. Banach analytic manifolds and conjugations
  8. Quasiconformal maps and Teichmüller theory
  9. Parameter families of entire functions
  10. Holomorphic and natural families
  11. Universal natural families
  12. Distortion sequences for wandering domains
  13. Perturbing Herman-type wandering domains
  14. Proving Claims \ref{['cl:approx']} and \ref{['cl:calc']}

Key Result

Theorem A

Let $f$ be an entire function, and assume that $f$ has at least two singular values. Let $F = S(f)\cup\{\infty\}$, where $S(f)$ is the set of singular values of $f$, and assume that $\mathbb{C}\setminus S(f)$ is non-empty. Then, there exists a Banach analytic manifold $T_f = T(F)\times(\mathbb{C}^*\ In particular, $M_f$ admits a complex structure that turns $\Phi$ into a holomorphic map. Furthermo

Theorems & Definitions (56)

  • Theorem A: Universal natural family
  • Remark
  • Corollary 1.1
  • Corollary 1.2
  • Theorem B: A holomorphic deformation defines a distortion sequence
  • Corollary 1.3
  • Corollary 1.4
  • Definition 1.5
  • Remark
  • Theorem C: A distortion sequence defines a holomorphic deformation
  • ...and 46 more