Horseshoes, solenoids, and holomorphic motions for Hénonmaps

Abstract

In this dissertation we show that the McMullen-Sullivan holomorphic motion for topologically conjugate, complex polynomials with connected Julia set follows level sets of the Böttcher coordinate. Analogously, we prove that the Buzzard-Verma motion for hyperbolic, unstably connected, polynomial diffeomorphisms of [special characters omitted] follows level sets of the Bedford-Smillie solenoid map. It follows that this solenoid map is a homeomorphism for those Hénon maps that are perturbations of one-dimensional hyperbolic maps with connected Julia set. Also, we show that there is a natural holomorphic motion of the Julia set in the horseshoe setting as well as more generally; this motion lends itself nicely to a computer algorithm