Polynomial diffeomorphisms of C 2 VI: Connectivity of J

§0. Introduction Polynomial maps g : C → C are the simplest holomorphic maps with interesting dynamical behavior. The study of such maps has had an important influence on the field of dynamical systems. On the other hand the traditional focus of the field of dynamical systems has been in a different direction: invertible maps or diffeomorphisms. Thus we are led to study polynomial diffeomorphisms f : C → C, which are the simplest holomorphic diffeomorphisms with interesting dynamical behavior. Two features are apparent in much of the contemporary work on polynomial maps of C (cf. [DH]). The first is a focus on the connectivity of the Julia set. The second is the use of computer pictures as a guide to research. Computer pictures do not substitute for proofs but they have provided a tool that has been used to guide research. In this paper we consider these ideas in the context of polynomial diffeomorphisms of C. This approach to the study of polynomial diffeomorphisms originates with Hubbard. For a polynomial map of C the “filled Julia set” K ⊂ C is the set of points with bounded orbits, and the Julia set J is defined to be the boundary of K. The Julia set has several analogs for diffeomorphisms of C. Since f : C → C is invertible, we can distinguish properties of points based on both forward and backward iteration. The sets K (resp. K) consist of points with bounded forward (resp. backward) orbits under f . We write U for the complementary sets U = C −K±. The set of points whose orbits are bounded in both forward and backward time is K = K ∩K−. The sets J := ∂K are analogues of the Julia set, as is the set J = J ∩ J. We use the notation J + for J ∩ U. We will see that in some cases the set J + plays the role of the Fatou set for polynomial maps of C. The focus of this paper is to investigate the J-connected/ Jdisconnected dichotomy in the case of polynomial diffeomorphisms of C and relate it to the structure of the sets J ∩ U and J ∩ U. One of the attractive features of the study of polynomial maps of C is that the Julia sets can be drawn by computer. Thus the connectivity properties of the Julia set can often be demonstrated (visually if not rigorously) by means of computer pictures. One of the daunting features of the study of polynomial diffeomorphisms of C is that the sets of fundamental importance are complicated subsets of C. As a consequence of our investigations we will show that it is possible to “see” the connectivity of the Julia set J for polynomial diffeomorphisms of C. Hubbard has suggested the following computer experiment. Let f be a polynomial diffeomorphisms of C and p be a periodic saddle point of f . The unstable manifold of p, W(p), is an immersed submanifold conformally equivalent to C. We have a partition of W(p) into subsets W(p) ∩K+ and W(p) ∩ U. (We can view this as a partition of C.) This partition is easily drawn by computer, and the sets Wu(p)∩K+ have many local features of Julia sets. While the unstable manifolds are easily drawn and yield complicated pictures, these pictures depend on choice of the point p. In order to make these computer pictures a useful