Hyperbolic polynomial diffeomorphisms of C2. II: Hubbard trees

This paper is a sequel to Part I [Y. Ishii, Hyperbolic polynomial diffeomorphisms of C2. I: A non-planar map, Adv. Math. 218 (2) (2008) 417–464]. In the current article we construct an object analogous to a Hubbard tree consisting of a pair of trees decorated with loops and a pair of maps between them for a hyperbolic polynomial diffeomorphism f of C2. Key notions in the construction are the pinching disks and the pinching locus which determine how local dynamical pieces are glued together to obtain a global picture. It is proved that the shift map on the orbit space of a Hubbard tree is topologically conjugate to f on its Julia set. Several examples of Hubbard trees are also given.