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The Lipschitzian core of some invariant manifold theorems

Published online by Cambridge University Press:  01 October 2008

MARC CHAPERON
MARC CHAPERON*
Affiliation:
Institut de mathématiques de Jussieu and Université Paris 7, UFR de mathématiques, Site Chevaleret, CASE 7012, 75205 Paris Cedex 13, France (email: chaperon@math.jussieu.fr)
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Abstract

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In the Michael Herman Memorial Volume, we stated and proved some rather simple invariant-manifold theorems, having many old and new applications. This paper presents the final state of the Lipschitzian part of the theory: the results are put into a more general framework and new properties are established. Both the hypotheses and the proofs of all our statements are quite simple. Smoothness will be treated as in Chaperon [Stable manifolds and the Perron–Irwin method. Ergod. Th. & Dynam. Sys.24 (2004), 1359–1394] in a forthcoming book.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008, pp. 1419 - 1441
Copyright
Copyright © 2008 Cambridge University Press

References

[1]Chaperon, M.. Géométrie différentielle et singularités de systèmes dynamiques. Astérisque 138–139 (1986).Google Scholar
[2]Chaperon, M.. C k-conjugation of holomorphic flows near a singularity. Publ. Math. Inst. Hautes Études Sci. 64 (1987), 143–183.Google Scholar
[3]Chaperon, M.. Variétés stables et formes normales. C. R. Acad. Sci. Paris, Série 1 317 (1993), 87–92.Google Scholar
[4]Chaperon, M.. Invariant manifolds revisited. Proc. Steklov Inst. 236 (2002), 415–433.Google Scholar
[5]Chaperon, M.. Stable manifolds and the Perron–Irwin method. Ergod. Th. & Dynam. Sys. 24 (2004), 1359–1394; and A. Fathi and J.-C. Yoccoz (Eds.). Dynamical Systems: Michael Herman Memorial Volume. Cambridge University Press, Cambridge, 2006, pp. 89–124.CrossRefGoogle Scholar
[6]Chaperon, M.. A remark on Stable manifolds and the Perron–Irwin method. Dynamical Systems: Michael Herman Memorial Volume. Eds. A. Fathi and J.-C. Yoccoz. Cambridge University Press, Cambridge, 2006, pp. 595–596.Google Scholar
[7]Chaperon, M. and Coudray, F.. Invariant manifolds, conjugations and blow-up. Ergod. Th. & Dynam. Sys. 17 (1997), 783–791.CrossRefGoogle Scholar
[8]de la Llave, R. and Wayne, C. E.. On Irwin’s proof of the pseudostable manifold theorem. Math. Z. 219 (1995), 301–321.CrossRefGoogle Scholar
[9]Fenichel, N.. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971), 193–225.CrossRefGoogle Scholar
[10]McGehee, R. and Sander, E. A.. A new proof of the stable manifold theorem. Z. Angew. Math. Phys. 47(4) (1996), 497–513.CrossRefGoogle Scholar
[11]Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Springer Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[12]Irwin, M. C.. Smooth Dynamical Systems. Academic Press, London, 1980.Google Scholar
[13]Irwin, M. C.. A new proof of the pseudo-stable manifold theorem. J. London Math. Soc. 21 (1980), 557–566.CrossRefGoogle Scholar
[14]Kammerer-Colin de Verdière, M.. Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions. C. R. Acad. Sci. Paris, Série I 335 (2004), 625–629.CrossRefGoogle Scholar
[15]Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.CrossRefGoogle Scholar
[16]Sternberg, S.. On the structure of local homeomorphisms of euclidean n-space II. Amer. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar