Complex H\'enon maps and discrete groups

Abstract

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.