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Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms

Romain Dujardin
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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{é}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.

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Romain Dujardin. 2004-12-08. Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms. https://arxiv.org/abs/math/0412162

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