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Stable Manifolds of a Map and a Flow for a Compact Manifold

Published online by Cambridge University Press:  22 January 2016

Masaharu Kato
Masaharu Kato*
Affiliation:
Mathematical Institute Nagoya University
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The purpose of this paper is to generalize the notion of the stable manifolds in Smale [5] and [6], in which the stable manifolds of flows or diifeomorphisms for a singular point or a closed orbit are defined in certain conditions. This generalization is concerned with Fenichel [1], He considers the stable manifolds of flows and diifeomorphisms for a torus. Here, we consider the case of a compact manifold. But our argument does not exactly imply Fenichel’s result.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 44 , December 1971 , pp. 119 - 132
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Fenichel, N., Linearization of maps and flows near an invariant torus, Notices of the Amer. Math. Soc. 17 (1970), p. 406, 673–50.Google Scholar
[2] Lang, S., Introduction to differentiable manifolds, John Wiley and Sons, Inc., New York, 1962.Google Scholar
[3] Hartmann, P., Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1964.Google Scholar
[4] Rosenberg, H., A generalization of Morse-Smale inequalities, Bull. Amer. Math. Soc. 70 (1964), pp. 422–427.CrossRefGoogle Scholar
[5] Smale, S., Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), pp. 43–49.CrossRefGoogle Scholar
[6] Smale, S., Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), pp. 97–116.Google Scholar