Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. (Smooth conjugacy of diffeomorphisms of the circle whose rotation number is of diophantine type). (French) Zbl 0595.57027

Yoccoz, Jean-Christophe

Ann. Sci. Éc. Norm. Supér. (4) 17, 333-359 (1984).

Improving a theorem of M. R. Herman, this paper establishes the following result: Let A be the set of those irrational numbers a such that every smooth diffeomorphism of the circle f admitting a for its rotation number is smoothly conjugated to the corresponding rotation; then A is exactly the set of diophantine numbers; there is a more precise statement relating the class of differentiability to the order of the diophantine condition.
The proof follows the same lines as Herman’s. It relies on certain estimates of Schwarzian derivatives leading to an improvement of Denjoy inequality, and then to a crucial estimate of \(f^ n-Id\). Though the details of the proof are highly technical, the exposition is very clear and precise.

Reviewer: J.Pradines

MSC:

57R50	Differential topological aspects of diffeomorphisms
37E10	Dynamical systems involving maps of the circle
28D99	Measure-theoretic ergodic theory

Keywords:

smooth conjugation; smooth diffeomorphism of the circle; rotation number; diophantine numbers; Schwarzian derivatives

References:

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