The Embedding Flow of 3-Dimensional Locally Hyperbolic C ∞ Diffeomorphisms
The Embedding Flow of 3-Dimensional Locally Hyperbolic C ∞ Diffeomorphisms
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发表时间：
2015
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通讯作者：
Xiang Zhang
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作者：
Xiang Zhang
对于有限维局部双曲C∞光滑实微分同态，Li et al. （Am J Math 124:107-127, 2002）在弱非谐振情况下解决了其嵌入流的存在性和光滑性问题。而在弱共振情况下，除了一些例子和一些不存在嵌入流的例子外，没有关于嵌入流存在的一般结果。具有弱共振现象的最简单的双曲微分同态应该是三维的。本文将重点研究这类微分同态的嵌入流问题。在一种情况下，我们获得了具有不嵌入流的微分同态的完整特征，在另一种情况下，我们获得了具有嵌入流的微分同态的一些必要条件。在这种情况下，给出了C∞嵌入流存在的充分条件。我们知道，这是关于弱共振局部双曲C∞微分同态嵌入流存在性的第一个一般结果。这些结果表明，弱共振情况下的嵌入流动问题更为复杂。
For finite dimensional locally hyperbolic C ∞ smooth real diffeomorphisms, the problem on the existence and smoothness of their embedding flows was solved by Li et al. (Am J Math 124:107–127, 2002) in the weakly non-resonant case. Whereas there are no general results on the existence of embedding flows in weakly resonant cases, except for some examples and some ones on the non-existence of embedding flows. The simplest hyperbolic diffeomorphisms having weakly resonant phenomena should be the 3-dimensional ones. This paper will concentrate on the embedding flow problem for this kind of diffeomorphisms. In onecaseweobtaincompletecharacterizationofthediffeomorphismswhichadmitembeddingflows,andintheothercasesweobtainsomenecessaryconditionsforthediffeomorphisms tohaveembeddingflows.Alsointhislattercaseweprovideasufficientconditionforthe existenceof C ∞ embedding flow. As we know it is the first general result on the existence of embedding flows for locally hyperbolic C ∞ diffeomorphisms with weak resonance. These results show that the embedding flow problem in the weakly resonant case is much more involved.