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Dynamical Systems
An International Journal
Volume 28, 2013 - Issue 4
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Original Articles

A geometrical proof of the persistence of normally hyperbolic submanifolds

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Pages 567-581 | Received 21 Feb 2012, Accepted 06 Aug 2013, Published online: 23 Sep 2013

References

  • Hirsch , M W , Pugh , C C and Shub , M . 1977 . Invariant manifolds , Berlin : Springer .
  • Fenichel , N . 1971 . Persistence and smoothness of invariant manifolds for flows . Math J, Indiana Univ. , 21 : 193 – 226 . doi: 10.1512/iumj.1972.21.21017
  • Mañé , R . 1978 . Persistent manifolds are normally hyperbolic . Trans Am Math Soc. , 246 : 261 – 283 .
  • Hadamard , J . 1901 . Sur l’itération et les solutions asymptotiques des équations différentielles [On iteration and asymptotic solutions of differential equations] . S M F Bull. , 29 : 224 – 228 .
  • Perron , O . 1929 . Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen [On the stability and asymptotic behaviour of the integrals of differential equations] . Math Z. , 29 ( 1 ) : 129 – 160 . doi: 10.1007/BF01180524
  • Irwin , M C . 1970 . On the stable manifold theorem . Bull Lond Math Soc. , 2 : 196 – 198 . doi: 10.1112/blms/2.2.196
  • Berger , P . 2010 . Persistence of laminations . Bull Braz Math Soc. , 41 ( 2 ) : 259 – 319 . (NS) doi: 10.1007/s00574-010-0013-0
  • Berger , P . 2011 . “ Persistence of stratifications of normally expanded laminations ” . In Mémoire de la société mathématique de France
  • Bates , P W , Lu , K and Zeng , C . 1998 . Existence and persistence of invariant manifolds for semiflows in Banach space . Mem Am Math Soc. , 645 : 1 – 129 .
  • Gourmelon , N . 2007 . Adapted metrics for dominated splittings . Ergod Theory Dyn Syst. , 27 ( 6 ) : 1839 – 1849 . doi: 10.1017/S0143385707000272
  • Hirsch , M W . 1976 . Differential topology , New York : Springer Verlag .
  • Greub , W , Halperin , S and Vanstone , R . 1973 . “ Connections, curvature, and cohomology. Vol. II: lie groups, principal bundles, and characteristic classes ” . In Pure and Applied Mathematics, 47-II , Vol. XXI , 541 New York : Academic Press .
  • Capiński , M J and Zgliczyński , P . 2011 . Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds . Discrete Contin Dyn Syst. , 30 ( 3 ) : 641 – 670 . doi: 10.3934/dcds.2011.30.641
  • McGehee , R . 1973 . A stable manifold theorem for degenerate fixed points with applications to celestial mechanics . J Differ Equations , 14 : 70 – 88 . doi: 10.1016/0022-0396(73)90077-6
  • Moser , J . 2001 . “ Stable and random motions in dynamical systems ” . In With special emphasis on celestial mechanics , Princeton , NJ : Princeton University Press .
  • Katok , A and Hasselblatt , B . 1997 . “ Introduction to the modern theory of dynamical systems. With a supplement by Anatole Katok and Leonardo Mendoza ” . In Encyclopedia of Mathematics and Its Applications , Vol. 54 , 802. Cambridge : Cambridge University Press . XVIII
  • Brown , R F . 1971 . The Lefschetz fixed point theorem , Glenview, Ill. London : Scott, Foresman and Co. .
  • Granas , A and Dugundji , J . 2003 . “ Fixed point theory ” . In Springer Monographs in Mathematics , Vol. XV , 690 New York , NY : Springer .
  • Flett , T M . 1980 . Differential analysis: differentiation, differential equations and differential inequalities , Vol. VII , 359 Cambridge : Cambridge University Press .
  • Newhouse , S Brin , M . 2004 . “ Cone-fields, domination, and hyperbolicity ” . In Modern dynamical systems and applications , 419 – 432 . Cambridge : Cambridge University Press . Dedicated to Anatole Katok on his 60th birthday

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