Topological model for a class of complex Hénon mappings
The Hénon map \(H_{a,c}:\mathbb C^2\to \mathbb C^2\) is defined by \((x,y)\mapsto (x^2+c+ay, ax)\), where \(c\in \mathbb C\) is contained in the main cardioid of the Mandelbrot set for the quadratic polynomial \(z\mapsto z^2+c\) and \(a\in\mathbb C\) is a parameter. The aim of the paper under review is to provide a topological model for such a map, for small parameters \(a\). More precisely, let endow \(\mathbb R^4\) with polar coordinates \((r,\theta)\in \mathbb R^+\times \mathbb S^3\). Write \(\mathbb S^3\) as the union of two solid tori \(\mathbb T_0, \mathbb T_1\) glued along their boundaries and let \((\zeta,z)\in \mathbb S^1\times \Delta=\mathbb T_0\) be coordinates. The solenoid map \((\zeta,z)\mapsto (\zeta^2, \frac{1}{2}\zeta+\varepsilon \frac{z}{\zeta})\) from \(\mathbb T_0\) to \(\mathbb T_0\) is known to extend as a homeomorphism \(\sigma:\mathbb S^3\to \mathbb S^3\). Let \(\Sigma=\cap_{n\leq 0} \sigma^{n}(\mathbb T_1)\) be the invariant solenoid defined by the backward iterates of \(\sigma\). Let \(Y=\mathbb R^4\setminus \{(r,\theta)\in \mathbb R^+\times \mathbb S^3: r\geq 1, \theta\in\Sigma\}\). Finally, let \(g(r,\theta):=(r^2, \sigma(\theta))\). Then \(g(Y)=Y\). The author proves that, fixed \(c\), there exists \(\varepsilon>0\) such that for any \(a\) satisfying \(0<| a| <\varepsilon\), there exists a homeomorphism \(h:\mathbb C^2 \to Y\) such that \(h\circ H_{a,c}\circ h^{-1}=g\). The couple \((Y,g)\) provides a topological model for the Hénon map. The idea of the proof is to find suitable coordinates on the partition of \(\mathbb C^2\) given by the basin of attraction of \(H_{a,c}\), its boundary and the complement [see \textit{J. E. Fornaess} and \textit{N. Sibony}, Duke Math. J. 65, 345--380 (1992; Zbl 0761.32015)]. This task is rather complicated because of the fractal nature of the boundary of the domain of attraction.
scientific article; zbMATH DE number 5071523