Complex Hénon maps and discrete groups

Consider the standard family of complex Hénon maps H(x,y)=(p(x)−ay,x)H(x,y)=(p(x)−ay,x), where p is a quadratic polynomial and a   is a complex parameter. Let U+U+ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set U+U+ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of (C−D‾)×C by a discrete group of automorphisms Γ isomorphic to Z[1/2]/ZZ[1/2]/Z. On the other hand, the boundary J+J+ of U+U+ is a complicated fractal object on which the Hénon map behaves chaotically. We show how to extend the group action to S1×CS1×C, in order to represent the set J+J+ as a quotient of S1×C/Γ by an equivalence relation. We analyze this extension for Hénon maps that are perturbations of hyperbolic polynomials with connected Julia set or polynomials with a parabolic fixed point.