Misiurewicz points and subhyperbolicity in unicritical algebraic correspondences

Abstract

We provide the first definition of \emph{Misiurewicz parameter} for the unicritical family of algebraic correspondences $ z^r + c$, with $ r > 1$ rational, and prove that, at every Misiurewicz parameter, the correspondence uniformly expands the canonical orbifold metric on a neighborhood of the Julia set. This is achieved using Thurston's ideas on postcritically finite rational maps, regular branched coverings, and orbifolds, viewing the correspondence as a global analytic multifunction. This result provides the necessary tools for further investigations into the fine structure of the parameter space near Misiurewicz points, particularly in exploring similarities between the local geometry of the parameter space and the Julia sets at such parameters. Finally, we present both rigorous examples and empirical evidence suggesting that Misiurewicz parameters are abundant and may be detected by identifying increasingly small copies of the Multibrot set nested within itself: the smaller the copy, the closer it is likely to be to a Misiurewicz parameter.