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Hénon mappings in the complex domain. I: The global topology of dynamical space. (English) Zbl 0839.54029
In this important paper the authors discuss the Hénon mapping \(F\) defined by the formula \(F(x,y) : = (x^2 + c - ay,x)\) where \(a \neq 0\), in the complex domain. Let \(F^{on}\) denote the \(n\)-fold composition of \(F\) or \(F^{-1}\) depending on whether \(n\) is positive or negative. The following sets defined for \(F\) are investigated: \[ \begin{aligned} K_+ : & = \biggl \{ (x,y) : \bigl |F^{on} (x,y) \bigr |\text{ does not tend to zero} \biggr\} \\ K_- : & = \biggl\{ (x,y) : \bigl |F^{o - n} (x,y) \bigr |\text{ does not tend to zero} \biggr\} \\ U_+ & = \mathbb{C}^2 - K_+,\;U_- = \mathbb{C}^2 - K_-,\;J_\pm = \partial K_\pm,\;K = K_+ \cap K_-,\;J = J_+ \cap J_-. \end{aligned} \] The set \(J_\pm\) is a generalization of the Julia set. It is shown that \(K\) and \(J\) are compact (and invariant under \(F)\).
There are three main results in the paper: Theorem 6.1 says that the set \(\mathbb{C}^2 - K_\pm\) is homeomorphic to a fibration over the reals with fiber a 3-sphere with a solenoid removed. The analytic structure of \(U_+\) is described completely in Sec. 8 (Theorems 8.1, 8.7 and 8.9). A compactification of \(\mathbb{C}^2\) to which the Hénon mappings extend canonically (analogous to compactifying \(\mathbb{C}\) by adding a circle at infinity) is presented in the last section (Th. 9.1). An algebraic characterization of Hénon mappings is discussed in Section 2; in Section 3 solenoidal mappings, essentially needed in the paper, are investigated.

MSC:
54H20 Topological dynamics (MSC2010)
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
30C10 Polynomials and rational functions of one complex variable
57N35 Embeddings and immersions in topological manifolds
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