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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)