Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-11T06:33:25.898Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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