Dynamics of quadratic correspondences

We investigate the dynamics of maps z -> z' determined by implicit complex quadratic equations g(z, z') = 0. Such maps are 2-valued and have a 2-valued inverse. They may be considered as a generalisation of both quadratic maps and two-generator Kleinian groups. Under iteration they exhibit not only properties typical of these two classes, but also features of Hamiltonian systems. Maps of this type with a reflection symmetry can be restricted to a half plane, where they are homeomorphisms except for 'zip' discontinuities. When additional symmetries are imposed they have hierarchies of elliptic island chains and an isolated invariant semicircle. Restricted to other domains the same maps have 'Julia sets', which appear to vary continuously with domain.