Chain Recurrence, Chain Transitivity, Lyapunov Functions and Rigidity of Lagrangian Submanifolds of Optical Hypersurfaces
Chain Recurrence, Chain Transitivity, Lyapunov Functions and Rigidity of Lagrangian Submanifolds of Optical Hypersurfaces
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DOI：
10.1007/s10884-016-9543-5
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发表时间：
2018-03-01
影响因子：
1.3
通讯作者：
Cardin, Franco
中科院分区：
文献类型：
--
作者：
Abbondandolo, Alberto;Bernardi, Olga;Cardin, Franco
该文致力于研究两个问题.一方面，我们讨论了度量空间上流的强链递归性和强链传递性的概念，并利用Lipschitz Lyapunov函数的刚性性质对它们进行了刻画.这一部分将Fathi和Pageault的同胚的一些最新结果推广到流。另一方面，在上的动力学是强链递归的假设下，我们利用这些特征重新讨论了Paternain，Polterovich和Siburg关于余切丛的光学超曲面中的Lagrange子流形的内刚性的一个定理的证明.在上的动力学是强链传递的较强假设下，我们还证明了这样的Lagrange子流形的一个外刚性结果.
The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold , under the stronger assumption that the dynamics on is strongly chain transitive.