  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.
