Complex H\'enon maps and discrete groups

Consider the standard family of complex H'enon maps $H(x,y) = (p(x) - ay, x)$,where$p$isaquadraticpolynomialand$a$isacomplexparameter\mathrm{.}Let$U^{+}$ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set $U^{+}$ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of $(\mathbb{C}-\bar{\mathbb{D}})\times \mathbb{C}$ by a discrete group of automorphisms $\mathrm{\Gamma }$ isomorphic to $\mathbb{Z}[1\mathrm{/}2]\mathrm{/}\mathbb{Z}$. On the other hand, the boundary $J^{+}$ of $U^{+}$ is a complicated fractal object on which the H'enon map behaves chaotically. We show how to extend the group action to ${\mathbb{S}}^1\times \mathbb{C}$, in order to represent the set $J^{+}$ as a quotient of ${\mathbb{S}}^1\times \mathbb{C}\mathrm{/}\mathrm{\Gamma }$ by an equivalence relation. We analyze this extension for H'enon maps that are small perturbations of hyperbolic polynomials with connected Julia sets or polynomials with a parabolic fixed point.