A compactification of Hénon mappings inC2 as dynamical systems
A compactification of Hénon mappings inC2 as dynamical systems
复制标题
C2 中 Hénon 映射作为动力系统的紧凑化
DOI：
10.1007/bf02392629
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发表时间：
1997
期刊：
影响因子：
3.7
通讯作者：
V. Veselov
中科院分区：
文献类型：
--
作者：
J. Hubbard;P. Papadopol;V. Veselov
在{HO 1}中，证明了在S^3上存在同胚于4-球的拓扑，使得H 'enon映射连续扩张.那篇论文使用了一些渐近展开式的精细分析，例如，理解无限远附近直线的前向图像的结构。计算是相当困难的，并且不清楚如何将它们推广到其他有理映射。
在本文中，我们将提出一种替代方法，涉及爆破，而不是渐近。我们在这里只将其应用于H\'enon映射及其组合，但该方法应该相当普遍，并有助于理解有理映射$f：\Proj^2\ratto\Proj^2$的动力学与不确定点。应用到合成的H\'enon映射证明了一个结果建议Milnor，涉及嵌入在$S^3$这是拓扑不同的那些从H\' enon映射。
In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball such that the H\'enon mapping extends continuously. That paper used a delicate analysis of some asymptotic expansions, for instance, to understand the structure of forward images of lines near infinity. The computations were quite difficult, and it is not clear how to generalize them to other rational maps.
In this paper we will present an alternative approach, involving blow-ups rather than asymptotics. We apply it here only to H\'enon mappings and their compositions, but the method should work quite generally, and help to understand the dynamics of rational maps $f:\Proj^2\ratto\Proj^2$ with points of indeterminacy. The application to compositions of H\'enon maps proves a result suggested by Milnor, involving embeddings of solenoids in $S^3$ which are topologically different from those obtained from H\'enon mappings.