The embedding flows of $C^\infty$ hyperbolic diffeomorphisms
The embedding flows of $C^\infty$ hyperbolic diffeomorphisms
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DOI：
10.1016/j.jde.2010.12.022
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发表时间：
2014-07
期刊：
影响因子：
--
通讯作者：
Zhang Xiang
中科院分区：
文献类型：
--
作者：
Zhang Xiang
在[{\it American J.}, 124(2002)， 107—127]中，我们证明了一类双曲微分同态，如果其特征值为弱非共振的实对数，则可以嵌入到自治微分系统中。它的证明非常复杂，涉及到嵌入周期向量场的存在性和Floquet理论在非线性周期微分系统中的推广。在本文中，我们将对最后这个结果提供一个简单而直接的证明。接下来，我们将证明在实对数的最后结果中的弱非共振条件对于某些微分同态具有嵌入流是必要的。最后，我们证明了一个双曲微分同态的胚当且仅当不具有负特征值或具有两个相等的负特征值时具有嵌入流，并且它是可对角化的。
In [{\it American J. Mathematics}, 124(2002), 107--127] we proved that for a germ ofhyperbolic diffeomorphismsin, ifhas a real logarithm with its eigenvalues weakly nonresonant, thencan be embedded in aautonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field ofand the extension of the Floquet's theory to nonlinearperiodic differential systems. In this paper we shall provide a simple and direct proof to this last result. Next we shall show that the weakly nonresonant condition in the last result on the real logarithm ofis necessary for somediffeomorphismsto haveembedding flows. Finally we shall prove that a germ ofhyperbolic diffeomorphismswithinhas aembedding flow if and only if eitherhas no negative eigenvalues orhas two equal negative eigenvalues and it can be diagonalizable.