Dynamics of Hyperbolic Schwarz Reflections

Abstract

We give a conformal mating description for the class ${\Sigma }_d$ of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of ${\Sigma }_d$ can be characterized independently by the class of \emph{annular Schwarz reflections} we introduce. Our main result is that each $\sigma \in {\Sigma }_d$ decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.