A finite dimensional approach to Bramham’s approximation theorem
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 2169-2202.

Using pseudoholomorphic curve techniques from symplectic geometry, Barney Bramham proved recently that every smooth irrational pseudo-rotation of the unit disk is the limit, for the C 0 topology, of a sequence of smooth periodic diffeomorphisms. We give here a finite dimensional proof of this result that works in the case where the pseudo-rotation is smoothly conjugate to a rotation on the boundary circle. The proof extends to C 1 pseudo rotations and is based on the dynamical study of the gradient flow associated to a generating family of functions given by Chaperon’s broken geodesics method.

À l’aide de la théorie des courbes pseudo-holomorphes de la géométrie symplectique, Barney Bramham a récemment montré que toute pseudo-rotation irrationnelle lisse du disque unité est limite, pour la topologie C 0 , d’une suite de difféomorphismes lisses périodiques. Nous donnons ici une preuve du résultat dans un cadre de dimension finie, valable quand la pseudo-rotation est différentiablement conjuguée à une rotation sur le bord du disque. La preuve, qui s’étend aux pseudo-rotations de classe C 1 , est basée sur l’étude dynamique du flot de gradient associé à une famille génératrice de fonctions, obtenue par la méthode des géodésiques brisées de Chaperon.

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DOI: 10.5802/aif.3061
Classification: 37D30,  37E30,  37E45,  37J10
Keywords: Irrational pseudo-rotation, generating function, rotation number, dominated decomposition
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Le Calvez, Patrice. A finite dimensional approach to Bramham’s approximation theorem. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 2169-2202. doi : 10.5802/aif.3061. https://aif.centre-mersenne.org/articles/10.5802/aif.3061/

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