2002/06/06 by Kevin M. Pilgrim, Lei Tan, Pilgrim, Kevin M. +1
Mathematics · #30F60 #37F30 #37F40 #Analytic and geometric function theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0206054
openalex publication_date 2002/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as \em spinning. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else 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-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well.