Abstract

We discuss work done at Boston University during the last few years on the family of rational maps $z\; \to z2+\lambda /z2$, where λ is a nonzero complex parameter. Maps in this family are interesting because they behave like the map $z\; \to z2$ far from the origin but have a pole at the origin. This dual behavior yields unusual Julia sets, both from a topological and a dynamical point of view. We were motivated to study this family by McMullen's examples of rational maps whose Julia sets are Cantor sets of quasicircles.