2004/12/08 by Romain Dujardin, Dujardin, Romain
Mathematics · #37Fxx #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:37Fxx
paper · pdf · doi:10.48550/arxiv.math/0412162
arxiv created 2004/12/08 · arxiv updated 2009/12/01
We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the connectivity of the Julia set, that carries over for certain Hénon-like and birational maps. We study the structure of disconnected Julia sets and the associated invariant currents. This provides a simple approach to some results of Bedford-Smillie, as well as some new corollaries --the connectedness locus is closed, construction of external rays in the general case, etc. We also prove the following theorem: a hyperbolic polynomial diffeomorphism of C2 with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.