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**Sur les courbes invariantes par les difféomorphismes de l’anneau. Volume 2.**
*(French)*
Zbl 0613.58021

Astérisque, 144. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 248 p. FF 155.00; $ 22.00 (1986).

This publication is the second volume of a work (chapters 5 to 8), the first part of which was published in 1983 (Zbl 0532.58011) (chapters 1 to 4). It is devoted to the study and the proof of new existence theorems of invariant curves when the Hölder conditions on the third derivatives are weakened by \(L^ p\) conditions. It is noteworthy that since Moser’s work Hölder spaces classically are used for perturbation theorems related to small divisors. Chapter 5 concerns the proof of persistence of invariant curves whose rotation number is of constant type for \(C^ 3\)- area preserving, globally canonical, being \(C^ 3\)-perturbations of a completely integrable monotone twist map of the annulus. Chapter 6 is the proof of the translated curve theorem for Besov space. Chapter 7 gives a proof using only \(L^ 2\) Sobolev spaces of the translated curve theorem, for \(C^ 4\) topology, when the rotation numbers are once again of constant type. This permits computation of a series of general constants giving a good precision of results. In chapter 8 the main theorem says that a \(C^ 1\)-diffeomorphism of the circle, having a second derivative of bounded variation (not necessarily continuous) and a rotation number of constant type, is \(C^{2-\delta}\) conjugated to a rotation for every \(\delta >0\).

Reviewer: C.Mira

### MSC:

37C75 | Stability theory for smooth dynamical systems |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

58C25 | Differentiable maps on manifolds |

57R50 | Differential topological aspects of diffeomorphisms |

57R55 | Differentiable structures in differential topology |

39A11 | Stability of difference equations (MSC2000) |