Dynamics of hyperbolic correspondences

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p,$ whose post-critical set is finite in any bounded domain of $\mathbb{C}.$ In spite of being rigid on the sphere, such correspondences are $J$-stable by means of holomorphic motions when viewed as maps of $\mathbb{C}^2.$ The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set. We also show that Julia sets of any perturbation of such correspondences are obtained as $\alpha$ limit sets of typical points, establishing the hyperbolicity of these correspondences.