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Bifurcations de points fixes elliptiques. I: Courbes invariantes. (French) Zbl 0566.58025
This is the first part of a work whose aim is to show that one-parameter families of diffeomorphisms of \({\mathbb{R}}^ 2\) undergoing a bifurcation where two invariant curves neatly cancel each other (as in the vector- field case) play, in the dissipative realm, the role of rotations of the circle in the K.A.M. theory of area-preserving plane diffeomorphisms.
In this paper, it is shown that in certain generic 2-parameter families of local plane diffeomorphisms unfolding a local diffeomorphism with an elliptic fixed point, there are always ”many” 1-parameter subfamilies undergoing such a neat bifurcation, in the same way as a generic area- preserving plane diffeomorphism always has, in the neighborhood of an elliptic fixed point, ”many” smooth invariant curves on which it is smoothly conjugated to a rotation. The proof uses normal forms and Russmann’s translated curve theorem in a version derived by Herman from Hamilton’s implicit function theorem.
The second part appeared in Invent. Math. 80, 81-106 (1985). The third part will appear shortly.

37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] A. Chenciner,Courbes invariantes non normalement hyperboliques au voisinage d’une bifurcation de Hopf dégénérée de difféomorphismes de (R 2 , O), Paris VII, Preprint Université, déc. 1980.
[2] A. Chenciner,ibid.,C.R.A.S. 292 (mars 1981), série I, 507–510.
[3] A. Chenciner, Points homoclines au voisinage d’une bifurcation de Hopf dégénérée de difféomorphismes deR 2 ,C.R.A.S. 294 (février 1982), série I, 269–272.
[4] A. Chenciner, Sur un énoncé dissipatif du théorème géométrique de Poincaré-Birkhoff,C.R.A.S. 294 (février 1982), série I, 243–245.
[5] A. Chenginer, Points périodiques de longues périodes au voisinage d’une bifurcation de Hopf dégénérée de difféomorphismes deR 2 ,C.R.A.S. 294 (mai 1982), série I, 661–663.
[6] A. Chenginer, Orbites périodiques et ensembles de Cantor invariants d’Aubry-Mather au voisinage d’une bifurcation de Hopf dégénérée de difféomorphismes deR 2 ,C.R.A.S. 297 (novembre 1983), série I, 465–467.
[7] A. Chenciner, Hamiltonian-like phenomena in saddle-node bifurcations of invariant curves for plane diffeomorphisms,Proceedings of the conference Singularities and dynamical systems (Heraklion, 1983), to appear at North-Holland in 1984. · Zbl 0561.58036
[8] A. Chenciner,Bifurcations de difféomorphismes de R 2 au voisinage d’un point fixe elliptique, Cours à l’École des Houches, juillet 1981, North Holland, 1983.
[9] A. Chenciner, Bifurcations de points fixes elliptiques, II: Orbites périodiques et ensembles de Cantor invariants,Inventiones Math., à paraître. · Zbl 0578.58031
[10] A. Chenciner,Bifurcations de points fixes elliptiques, III :Orbites homoclines (en préparation).
[11] A. Chenciner etG. Iooss, Bifurcations de tores invariants,Archives for Rational Mechanics and Analysis,69 (2) (1979), 109–198, et71, no 4, 1979, 301–306. · Zbl 0405.58033 · doi:10.1007/BF00281175
[12] G. R. Hall,Bifurcation of an attracting invariant circle: a Denjoy attractor, Preprint University of Minnesota. · Zbl 0523.58027
[13] R. S. Hamilton,The inverse function theorem of Nash and Moser, Preprint Cornell University, 1974.
[14] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,Publications Mathématiques de l’I.H.E.S.,49 (1979), 5–234. · Zbl 0448.58019 · doi:10.1007/BF02684798
[15] M. Herman, Sur les courbes invariantes par les difféomorphismes de l’anneau, chapitre VIII,Astérisque, à paraître.
[16] M. W. Hirsch, C. C. Pugh andM. Shub, Invariant manifolds,Lecture Notes in Mathematics,583, Springer, 1977.
[17] R. Mañe, Persistent Manifolds are normally hyperbolic,Bulletin A.M.S. 80 (1974), 90–91. · Zbl 0276.58009 · doi:10.1090/S0002-9904-1974-13366-5
[18] J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus,Topology, 21, no 4 (1982), 457–467. · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4
[19] D. Ruelle andF. Takens, On the nature of turbulence,Communications in Mathematical Physics,20 (1971), 167–192. · Zbl 0223.76041 · doi:10.1007/BF01646553
[20] H. Rüssmann, Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes,Nachr. Wiss. Göttingen, Math. Phys. Kl. (1970), 67–105. · Zbl 0201.11202
[21] C. L. Siegel andJ. Moser,Lectures on Celestial Mechanics, Springer, 1971. · Zbl 0312.70017
[22] J. C. Yoccoz, Conjugaison des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne,Annales de l’E.N.S., 4e série,17 (1984), 333–359. · Zbl 0595.57027
[23] E. Zendher, Homoclinic points near elliptic fixed points,C.P.A.M. 26 (1973), 131–182. · Zbl 0261.58002
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