Polynomial Diffeomorphisms of C^2: VI. Connectivity of J
  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. 
Eric Bedford and John Smillie