Correspondences in complex dynamics

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 $$\mathbf{f}_c(z)=z^β +c, \mbox{ where } 1<β=p/q \in \mathbb{Q},$$ 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 Böttcher 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 ${\mathbf{f}}_c$.

22 pages, 8 figures