Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1. Complété par un appendice au chapitre 1 de Albert Fathi. (French) Zbl 0532.58011

Herman, Michael R.

Astérisque, 103-104. Publié avec le Concours du Centre National de la Recherche Scientifique. Paris: Sociéte Mathématique de France. 221 p. (1983).

This publication, constituted by four chapters, deals with invariant closed curves, and rings of instability in the sense of Birkhoff, for some conservative surface transformations (twist map of an annulus A here). Chapter 1 gives in a different language an exposition of a part of the results of G. D. Birkhoff [Acta Math. 43, 1-119 (1920); Ann. Inst. Henri Poincaré 2, 369-386 (1932; Zbl 0005.22103)] for the analytical case. In particular a more topological proof of Birkhoff’s theorem about curves being graphs is laid. The problem of instability rings is exposed from more recent results of Robinson and Moser. Chapter 2 shows that Moser’s invariant curve theorem is optimal up to \(\epsilon>0\), and that usually for \(C^{3-\epsilon}\)-perturbations of a \(C^\infty\)-complete integrable monotone twist map, there exists no invariant curve of a fixed rotation number (another proof is also given in chapter 3). In this chapter counterexamples to stability for \(C^{2-\epsilon}\)-perturbations can be found. Chapter 3 is devoted in part to some insight of the geometry of the set of ”standard-diffeomorphisms” of A that leave invariant a curve, after another proof of a chapter 2 result. This proof is based on a counterexample of Denjoy. Chapter 4 proves Rüssmann’s translated curve theorem in class \(C^{3+\epsilon}\), \(\epsilon>0\), if one restricts the rotation number on the translated curve to be a number of constant type. This theorem is a nonconservative generalization of Moser’s invariant curve theorem.

Reviewer: C.Mira

MSC:

37C75	Stability theory for smooth dynamical systems
57R50	Differential topological aspects of diffeomorphisms
39A11	Stability of difference equations (MSC2000)
58C25	Differentiable maps on manifolds
58-02	Research exposition (monographs, survey articles) pertaining to global analysis
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes

Keywords:

surface mapping; invariant closed curves; rotation numbers; rings of instability; Denjoy example; twist map

Citations:

Zbl 0005.22103