Topological model for a class of complex Hénon mappings

Abstract

In order to describe the dynamics of the complex Hénon map $H_{a,c}:\left(\begin{array}{c}x\\ y\end{array}\right)\mapsto \left(\begin{array}{c}{P}_c(x)-ay\\ x\end{array}\right)$, where $P_c:z\mapsto z^2+c$ has an attractive fixed point, we build a global topological model $(g,Y)$. In this model $Y$ is the complement in ${\mathbb{R}}^4$ of a cone over a solenoid lying in the unit 3-sphere, and $g:Y\to Y$ is a map given in spherical coordinates by $g(r,\theta )=(r^2,\sigma (\theta ))$, where $\sigma$ is a solenoidal map of degree two. Then we prove the existence of a constant $\epsilon >0$ such that any Hénon map $H_{a,c}$ with $0<|a|<\epsilon$ is conjugate to our model $(g,Y)$.