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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 fromprevious work of J. Hubbard and R. Oberste-Vorth as a quotient of$(\mathbb{C}-\overline{\mathbb{D}}) \times\mathbb{C}$ by a discrete group ofautomorphisms $\Gamma$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. On the otherhand, the boundary $J^{+}$ of $U^{+}$ is a complicated fractal object on whichthe 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 aquotient of $\mathbb{S}^1\times \mathbb{C}/\,\Gamma$ by an equivalencerelation. We analyze this extension for H\'enon maps that are smallperturbations of hyperbolic polynomials with connected Julia sets orpolynomials with a parabolic fixed point.

Complex H\'enon maps and discrete groups