Matings of Quadratic Polynomials
by
Adam Epstein
Mathematics Institute, University of Warwick

The moduli space of all quadratic rational maps up to Möbius conjugacy is isomorphic to C². It is possible, and also useful, to regard one of the coordinate axes as the moduli space of quadratic polynomials; the Mandelbrot set, parametrizing the quadratic polynomials with connected Julia set, thereby lies in this slice.

Nearly twenty years ago, Douady conjectured that the quadratic rational maps in the central portion of moduli space might be understood as matings of pairs of quadratic polynomials. The proposed construction is purely topological: one glues filled-in Julia sets back-to-back along complex-conjugate prime ends to obtain a branched cover of the sphere.

Work of Tan Lei and Mary Rees shows that under favorable circumstances, the resulting branched cover is topologically conjugate to an essentially unique quadratic rational map.

According to Milnor, mating is an interesting operation because it possesses none of the usual good properties: it is not injective, surjective, continuous, or even everywhere defined.

We will survey recent results concerning these issues.

Date received: July 16, 2001