Bifurcations de points fixes elliptiques. II: Orbites periodiques et ensembles de Cantor invariants. (Bifurcations of elliptic fixed points. II: Periodic orbits and invariant Cantor sets). (French) Zbl 0578.58031

[For part I see Publ. Math., Inst. Hautes Etud. Sci. 61, 67-127 (1985; Zbl 0566.58025).]
The first paper of this series was concerned with the existence, in certain generic 2-parameter families of plane diffeomorphisms, of non- normally hyperbolic invariant curves obtained by K.A.M. methods. In this second paper, we address more generally to the problem of the existence of Aubry-Mather invariant sets for such families, that is invariant sets similar to those encountered in the theory of circle diffeomorphisms. After reducing the problem to the case of periodic orbits via uniform Lipschitz estimates (Birkhoff theory), we use the approach of Aubry [see my Bourbaki lecture, Exp. No.622, Astérisque 121/122, 147-170 (1985)]: the Aubry functional does not exist any more in our dissipative setting, but its derivative does, which is enough for our purposes. The existence of zeroes of this derivative is proved by a degree argument which is made possible by the existence of geometric a priori estimates on the orbits.


37G99 Local and nonlocal bifurcation theory for dynamical systems


Zbl 0566.58025
Full Text: DOI EuDML


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