Resumen de Topological model for a class of complex Hénon mappings

Sylvain Bonnot

• In order to describe the dynamics of the complex Hénon map $H_{a,c}\colon \begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}P_c(x)-ay\\x\end{pmatrix}$, where $P_c\colon z \mapsto z^2+c$ 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 $g\colon Y\rightarrow Y$ 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 $\varepsilon>0$ such that any Hénon map $H_{a,c}$ with $0<|a|<\varepsilon$ is conjugate to our model $(g,Y)$.