  Given a polynomial diffeomorphism f: C^2 -> C^2 there is a set
$J_f\subset{\bf C}^2$ which we call the Julia set of f. The set $J_f\subset
C^2$ plays the role of the Julia set $J\subset{\bf C}$ for a polynomial map of
C. In the study of polynomial maps of C a great deal of attention has been paid
to the connectivity of the Julia set. The focus of this paper is to investigate
the J-connected/J-disconnected dichotomy in the case of polynomial
diffeomorphisms of C^2.
  The Jacobian determinant of f is constant. We make the standing assumption
that $|det\ Df|\le 1$ (this can always be achieved by replacing f by $f^{-1}$
if necessary). The set $J^-$ is the set of points with bounded backward orbits.
The set $U^+$ is the set of points with unbounded forward orbits. Let p be a
periodic saddle point and let $W^u(p)$ be its unstable manifold. The set
$W^u(p)$ will be a Riemann surface conformally equivalent to C.
  Theorem 1. The following are equivalent:
  1. For some periodic saddle point p, some component of $W^u(p)\cap U^+$ is
simply connected.
  2. The set $J^-\cap U^+$ has a lamination by simply connected leaves so that
for any periodic saddle point p each component of $W^u(p)\cap U^+$ is a leaf of
this lamination.
  3. For any periodic saddle point p, each component of $W^u(p)\cap U^+$ is
simply connected.
  If f satisfies one of these conditions we say that f is unstably connected.
  Theorem 2. The set J is connected if and only if f is unstably connected.
  These results imply that we can determine the connectivity of J by
considering the forward orbits of points in a single unstable manifold. These
results open the door to computer exploration of the topology of two
dimensional Julia sets and the connectivity locus in the parameter space.
