JournalscmhVol. 81, No. 4pp. 827–857

Topological model for a class of complex Hénon mappings

  • Sylvain Bonnot

    Stony Brook University, United States
Topological model for a class of complex Hénon mappings cover
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Abstract

In order to describe the dynamics of the complex Hénon map Ha,c ⁣:(xy)(Pc(x)ayx)H_{a,c}\colon \begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix}P_c(x)-ay\\x\end{pmatrix}, where Pc ⁣:zz2+cP_c\colon z \mapsto z^2+c has an attractive fixed point, we build a global topological model (g,Y)(g,Y). In this model YY is the complement in R4\mathbb{R}^4 of a cone over a solenoid lying in the unit 3-sphere, and g ⁣:YYg\colon Y\rightarrow Y is a map given in spherical coordinates by g(r,θ)=(r2,σ(θ))g(r,\theta)=(r^2,\sigma(\theta)), where σ\sigma is a solenoidal map of degree two. Then we prove the existence of a constant ε>0\varepsilon>0 such that any Hénon map Ha,cH_{a,c} with 0<a<ε0<|a|<\varepsilon is conjugate to our model (g,Y)(g,Y).

Cite this article

Sylvain Bonnot, Topological model for a class of complex Hénon mappings. Comment. Math. Helv. 81 (2006), no. 4, pp. 827–857

DOI 10.4171/CMH/76