Spinning deformations of rational maps

Abstract

We analyze a real one-parameter family of quasiconformal deformationsof a hyperbolic rational map known as spinning. We show that underfairly general hypotheses, the limit of spinning either exists and is unique, orelse converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodiccritical point in the Julia set, depending on the combinatorics of the datadefining the deformation. The proofs are soft and rely on two ingredients: theconstruction of a Riemann surface containing the closure of the family, andan analysis of the geometric limits of some simple dynamical systems. Aninterpretation in terms of Teichmüller theory is presented as well. © 2004 American Mathematical Society.