  Consider the standard family of complex H\'enon maps $H(x,y) = (p(x) - ay,
x)$, where $p$ is a quadratic polynomial and $a$ is a complex parameter. 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
$(\mathbb{C}-\overline{\mathbb{D}}) \times\mathbb{C}$ by a discrete group of
automorphisms $\Gamma$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. 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 $\mathbb{S}^1\times \mathbb{C}/\,\Gamma$ 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.
