Herman's last geometric theorem
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, p. 193-219

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.

Nous présentons une preuve du dernier théorème géométrique d’Herman qui affirme que, si un difféomorphisme F de l’anneau possède la propriété d’intersection, alors toute courbe C F-invariante, sur laquelle le nombre de rotation de F est diophantien, est accumulée par un ensemble de mesure positive de courbes invariantes C sur lesquelles F est C -conjuguée à une rotation. Ceci implique en particulier la stabilité des points fixes elliptiques diophantiens des difféomorphismes du plan qui préservent l’aire. Le caractère remarquable de ce théorème est qu’il ne requiert aucune condition de torsion.

DOI : https://doi.org/10.24033/asens.2093
Classification:  37J40,  37J10,  37E30,  70H14,  70H08
Keywords: Birkhoff normal forms, KAM theory, invariant curves, Whitney dependence, stability of elliptic fixed, disk diffeomorphisms
@article{ASENS_2009_4_42_2_193_0,
     author = {Fayad, Bassam and Krikorian, Rapha\"el},
     title = {Herman's last geometric theorem},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {2},
     year = {2009},
     pages = {193-219},
     doi = {10.24033/asens.2093},
     zbl = {1175.37062},
     mrnumber = {2518076},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_2_193_0}
}
Fayad, Bassam; Krikorian, Raphaël. Herman's last geometric theorem. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, pp. 193-219. doi : 10.24033/asens.2093. http://www.numdam.org/item/ASENS_2009_4_42_2_193_0/

[1] D. V. Anosov & A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč. 23 (1970), 3-36. | MR 370662 | Zbl 0255.58007

[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. | Numdam | MR 837218 | Zbl 0602.58021

[3] C. Q. Cheng & Y. S. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom. 47 (1989/90), 275-292. | MR 1056793 | Zbl 0705.70013

[4] B. Fayad & M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. École Norm. Sup. 38 (2005), 339-364. | Numdam | MR 2166337 | Zbl 1090.37001

[5] J. Féjoz, Démonstration du ‘théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dynam. Systems 24 (2004), 1521-1582. | MR 2104595 | Zbl 1087.37506

[6] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222. | MR 656198 | Zbl 0499.58003

[7] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.É.S. 49 (1979), 5-233. | Numdam | Zbl 0448.58019

[8] M. Herman, Some open problems in dynamical systems, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., 1998, 797-808. | MR 1648127 | Zbl 0910.58036

[9] R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque 259 (1999). | Zbl 0957.37016

[10] J. Moser, Stable and random motions in dynamical systems. With special emphasis on celestial mechanics, Annals of Mathematics studies 77, Princeton University Press, Princeton, 1973. | MR 442980 | Zbl 0271.70009

[11] H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems 22 (2002), 1551-1573. | MR 1934150 | Zbl 1030.37040

[12] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, 1970. | MR 290095 | Zbl 0207.13501

[13] Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynam. Systems 12 (1992), 621-631. | MR 1182665 | Zbl 0768.58042

[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. | Numdam | MR 777374 | Zbl 0595.57027

[15] J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki, vol. 1991/92, exp. no 754, Astérisque 206 (1992), 311-344. | Numdam | MR 1206072 | Zbl 0791.58044