Commentarii Mathematici Helvetici

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Volume 81, Issue 4, 2006, pp. 827–857
DOI: 10.4171/CMH/76

Published online: 2006-12-31

Topological model for a class of complex Hénon mappings

Sylvain Bonnot[1]

(1) Stony Brook University, United States

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 $(g,Y)$.

Keywords: Complex Hénon mappings, solenoidal map, topological model

Bonnot Sylvain: Topological model for a class of complex Hénon mappings. Comment. Math. Helv. 81 (2006), 827-857. doi: 10.4171/CMH/76