Correspondences in complex dynamics

TL;DR

This paper surveys recent results on the dynamics of two families of holomorphic correspondences, generalizing quadratic maps, including properties like Böttcher maps, periodic geodesics, and holomorphic motions.

Contribution

It introduces new dynamical properties for these correspondences, extending classical quadratic polynomial dynamics to more complex holomorphic correspondences.

Findings

Describes Böttcher maps for the family 

Establishes periodic geodesics and Yoccoz inequality

Analyzes holomorphic motions for hyperbolic multifunctions

Abstract

This paper surveys some recent results concerning the dynamics of two families of holomorphic correspondences, namely ${\mathcal F}_a:z \to w$ defined by the relation $$\left( \frac{aw-1}{w-1} \right)^2 + \left( \frac{aw-1}{w-1} \right) \left( \frac{az +1}{z+1} \right) + \left( \frac{az+1}{z+1} \right)^2 =3,$$ and $$\mathbf{f}_c(z)=z^{\beta} +c, \mbox{ where } 1<\beta=p/q \in \mathbb{Q},$$ which is the correspondence $\mathbf{f}_c:z \to w$ defined by the relation $$(w-c)^q=z^p.$$ Both can be regarded as generalizations of the family of quadratic maps $f_c(z)=z^2+c$. We describe dynamical properties for the family $\mathcal{F}_a$ which parallel properties enjoyed by quadratic polynomials, in particular a B\"ottcher map, periodic geodesics and Yoccoz inequality, and we give a detailed account of the very recent theory of holomorphic motions for hyperbolic multifunctions in the family ${\bf f}_c$.