Manjula Samarasinghe: Quasi-Fuchsian Correspondences
Abstract:
We consider the iterative behaviour of holomorphic correspondences or algebraic functions acting on the Riemann sphere $\overline{\mathbb{C}}$ and their limit sets. A holomorphic correspondence is a polynomial relation $P(z,w)=0$ from $\overline{\mathbb{C}}\times\overline{\mathbb{C}}$ to $\overline{\mathbb{C}}.$ We say that $P$ is an $(n:m)$ holomorphic correspondence if the degrees of $z$ and $w$ are $n$ and $m$ respectively; the limit set of a holomorphic correspondence in this scheme is taken to be the smallest completely invariant closed set with cardinality at least three. We identify a class of $(2:2)$ holomorphic correspondences\\ $P(z,w)=0$ whose limit set is a quasicircle (ie the image of the unit circle under a quasiconformal map) for which the dynamical system $z\mapsto w$ is a perturbation of the action of the Modular group $\mathbf{PSL}(2,\mathbb{C})$ on $\overline{\mathbb{C}}.$