Laminations in holomorphic dynamics

Abstract

There is an intriguing dictionary between two branches of conformal dynamics: the theory of Kleinian groups and dynamics of rational maps. This dictionary was introduced by Sullivan, and led him in the early 80's to the no wandering domains theorem, deformation theory and geometric measure theory for holomorphic maps. Thurston's rigidity and realization theory, developed at the same time, was also motivated by this analogy. More recently, McMullen has made important contributions to the renormalization theory motivated by the analogy with 3-manifolds which fiber over the circle