Dynamics of hyperbolic correspondences

TL;DR

This paper investigates the geometric rigidity and stability of certain holomorphic correspondences defined by -cycles, establishing their hyperbolic nature and describing their Julia sets through conformal iterated function systems.

Contribution

It introduces a novel approach linking conformal iterated function systems to the dynamics of these correspondences, demonstrating their rigidity, stability, and hyperbolicity.

Findings

Holomorphic correspondences are geometrically rigid on the sphere.

Such correspondences are -stable via holomorphic motions in ^2.

Julia sets are characterized as  limit sets of typical points.

Abstract

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.