Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms
Abstract
We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. 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 H{\'e}non-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 C^2 with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.
Cite
@article{arxiv.math/0412162,
title = {Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms},
author = {Romain Dujardin},
journal= {arXiv preprint arXiv:math/0412162},
year = {2007}
}