On extensions of holomorphic motions—a survey

Summary

Abstract

In this survey article, we discuss some of the main results on extensions of holomorphic motions. Our emphasis is on holomorphic motions over infinite dimensional parameter spaces. The Teichmüller space of a closed set E in the Riemann sphere is a universal parameter space for holomorphic motions of that set. This universal property is exploited throughout the article.

Notation. In this paper we will use the following notations: C for the complex plane, Ĉ for the Riemann sphere, and ∆ for the open unit disk {z ∈ C : |z| < 1}.

Introduction

Motivated by the study of dynamics of rational maps, Mañé, Sad, and Sullivan (in [MSS]) defined a holomorphic motion of a set E in Ĉ, to be acurve φ_t(z) defined for every z in E and for every t in ∆, such that:

1. (i) ϕ₀ (z) = z for all z in E,

2. (ii) z ↦ ϕ_t (z) is injective as a function from E to Ĉ, for each fixed t in ∆, and

3. (iii) t ↦ ϕ_t(z) is holomorphic for |t| < 1, and for each fixed z in E.

As t moves in the unit disk, the set E_t = ϕ_t(E) moves in Ĉ. We think of t as the complex time-parameter for the motion. As Gardiner and Keen remark in the introduction in [GK], although E may start out as smooth as a circle and although the points of E move holomorphically, for every t ≠ 0, E_t = ϕ_t (E) can be an interesting fractal with fractional Hausdorff dimension.