Normally Hyperbolic Invariant Manifolds: The Noncompact Case
Normally Hyperbolic Invariant Manifolds: The Noncompact Case
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DOI：
10.2991/978-94-6239-003-4
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发表时间：
2013-08
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影响因子：
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通讯作者：
J. Eldering
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文献类型：
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作者：
J. Eldering
在这项工作中，我们证明了正常双曲不变流形的持久性。当不变流形是紧的时，这个结果是众所周知的；我们将其扩展到允许不变流形以及环境空间是非紧流形的设置。环境空间被假定为有界几何的黎曼流形。通常双曲不变流形（NHIM）是双曲不动点的推广。双曲不动点的许多概念、结果和证明都适用于 NHIM。推广到 NHIM 的两个重要性质是不变流形的持久性以及稳定和不稳定流形的存在。
In this work, we prove the persistence of normally hyperbolic invariant manifolds. This result is well known when the invariant manifold is compact; we extend this to a setting where the invariant manifold as well as the ambient space are allowed to be noncompact manifolds. The ambient space is assumed to be a Riemannian manifold of bounded geometry. Normally hyperbolic invariant manifolds (NHIMs) are a generalization of hyperbolic fixed points. Many of the concepts, results, and proofs for hyperbolic fixed points carry over to NHIMs. Two important properties that generalize to NHIMs are persistence of the invariant manifold and existence of stable and unstable manifolds.