Herman’s last geometric theorem

Bassam Fayad; Raphaël Krikorian

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 2, page 193-219
  • ISSN: 0012-9593

Abstract

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We present a proof of Herman’s Last Geometric Theorem asserting that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F -invariant smooth curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable feature of this theorem is that it does not require any twist assumption.

How to cite

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Fayad, Bassam, and Krikorian, Raphaël. "Herman’s last geometric theorem." Annales scientifiques de l'École Normale Supérieure 42.2 (2009): 193-219. <http://eudml.org/doc/272121>.

@article{Fayad2009,
abstract = {We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable feature of this theorem is that it does not require any twist assumption.},
author = {Fayad, Bassam, Krikorian, Raphaël},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Birkhoff normal forms; KAM theory; invariant curves; Whitney dependence; stability of elliptic fixed; disk diffeomorphisms},
language = {eng},
number = {2},
pages = {193-219},
publisher = {Société mathématique de France},
title = {Herman’s last geometric theorem},
url = {http://eudml.org/doc/272121},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Fayad, Bassam
AU - Krikorian, Raphaël
TI - Herman’s last geometric theorem
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 2
SP - 193
EP - 219
AB - We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable feature of this theorem is that it does not require any twist assumption.
LA - eng
KW - Birkhoff normal forms; KAM theory; invariant curves; Whitney dependence; stability of elliptic fixed; disk diffeomorphisms
UR - http://eudml.org/doc/272121
ER -

References

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  2. [2] J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolmogorov, Arnolʼd, Moser, Rüssmann, Zehnder, Herman, Pöschel,...), Seminaire Bourbaki, vol. 1984/85, Astérisque 133-134 (1986), 113–157. Zbl0602.58021MR837218
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  9. [9] R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque 259 (1999). Zbl0957.37016
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  11. [11] H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems22 (2002), 1551–1573. Zbl1030.37040MR1934150
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  13. [13] Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynam. Systems12 (1992), 621–631. Zbl0768.58042MR1182665
  14. [14] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup.17 (1984), 333–359. Zbl0595.57027MR777374
  15. [15] J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki, vol. 1991/92, exp. no 754, Astérisque 206 (1992), 311–344. Zbl0791.58044MR1206072

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