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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
##### Keywords:
Hénon mapping; solenoidal mappings
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##### References:
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