Parametrizing Unstable and Very Unstable Manifolds
Parametrizing Unstable and Very Unstable Manifolds
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参数化不稳定和非常不稳定的流形
DOI：
10.17323/1609-4514-2005-5-1-105-124
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发表时间：
2005
影响因子：
0.8
通讯作者：
J. Hubbard
中科院分区：
文献类型：
--
作者：
J. Hubbard
不稳定流形的存在性和唯一性定理是众所周知的。在这里，我们证明某些改进。设f：（C，0）→ C是一个解析双同态的芽，其导数Df（0）的特征值为λ1，λ 2. . .，λn，使得|λ1| ≥ · · · ≥ |λk| > |λk+1| ≥ · · · ≥ |λn|得|λk|> 1.则存在唯一的k维不变子流形，其切空间由与特征值λ1，. . .，λk，它解析地依赖于f。此外，这个“非常不稳定的流形”有一个自然的参数化，当f定义在所有C上时，它可以扩展到解析映射C → C，并且如果f是整体的非同态，它是一个内射浸入。对于稳定流形，我们也给出了相应的陈述，它们在局部上是相似的，但在全局上是完全不同的。2000年数学科目班。小学37 D10;中学37 F15，37 G 05。
Existence and uniqueness theorems for unstable manifolds are well-known. Here we prove certain refinements. Let f : (C, 0) → C be a germ of an analytic diffeomorphism, whose derivative Df(0) has eigenvalues λ1, . . . , λn such that |λ1| ≥ · · · ≥ |λk| > |λk+1| ≥ · · · ≥ |λn|, with |λk| > 1. Then there is a unique k-dimensional invariant submanifold whose tangent space is spanned by the generalized eigenvectors associated to the eigenvalues λ1, . . . , λk, and it depends analytically on f . Further, there is a natural parametrization of this “very unstable manifold,” which can be extended to an analytic map C → C when f is defined on all of C, and is an injective immersion if f is a global diffeomorphism. We also give the corresponding statements for stable manifolds, which are analogous locally but quite different globally. 2000 Math. Subj. Class. Primary 37D10; Secondary 37F15, 37G05.