1996/12/18 by Eric Bedford, Bedford, Eric, John Smillie +1
Mathematics · Physics and Astronomy · #32 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.CV #math.DS #msc:32
paper · pdf · doi:10.48550/arxiv.math/9612203
arxiv created 1996/12/18 · openalex publication_date 1996/12/18 · arxiv updated 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a polynomial diffeomorphism f: C2 -> C2 there is a set Jf⊂\bf C2 which we call the Julia set of f. The set Jf⊂ C2 plays the role of the Julia set J⊂\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 C2. The Jacobian determinant of f is constant. We make the standing assumption that |det Df|≤ 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 Wu(p) be its unstable manifold. The set Wu(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 Wu(p)∩ U+ is simply connected. 2. The set J-∩ U+ has a lamination by simply connected leaves so that for any periodic saddle point p each component of Wu(p)∩ U+ is a leaf of this lamination. 3. For any periodic saddle point p, each component of Wu(p)∩ 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.