Holomorphic motions for unicritical correspondences

TL;DR

This paper investigates the deformation and mixing properties of hyperbolic Julia sets in holomorphic correspondences of the form z^r + c, revealing holomorphic motions and parameterizations of these complex fractal sets.

Contribution

It introduces a framework for understanding holomorphic motions of Julia sets in unicritical correspondences, including cases with non-integer r and near-zero c.

Findings

Hyperbolic Julia sets move holomorphically in the parameter space.

Projection of Julia sets yields branched holomorphic motions.

Parameterizations of Julia sets by quasiconformal curves are established.

Abstract

We study quasiconformal deformations and mixing properties of hyperbolic sets in the family of holomorphic correspondences z^r +c, where r >1 is rational. Julia sets in this family are projections of Julia sets of holomorphic maps on C^2, which are skew-products when r is integer, and solenoids when r is non-integer and c is close to zero. Every hyperbolic Julia set in C^2 moves holomorphically. The projection determines a branched holomorphic motion with local (and sometimes global) parameterisations of the plane Julia set by quasiconformal curves.