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

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

As t moves in the unit disk, the set Et = ϕt(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, Et = ϕt (E) can be an interesting fractal with fractional Hausdorff dimension.