Polynomial Diffeomorphisms of C^2: VI. Connectivity of J

Abstract

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.