Complex Hénon maps and discrete groups

Abstract

Consider the standard family of complex Hénon 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 Hénon 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 Hénon maps that are small perturbations of hyperbolic polynomials with connected Julia sets or polynomials with a parabolic fixed point.

33 pages, 12 figures

References in corpus (1)

• A structure theorem for semi-parabolic Hénon maps