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

Received : 2013-11-07
Revised : 2015-01-29
Accepted : 2016-02-19
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3061
Classification:  37D30,  37E30,  37E45,  37J10
Keywords: Irrational pseudo-rotation, generating function, rotation number, dominated decomposition
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     author = {Le Calvez, Patrice},
     title = {A finite dimensional approach to Bramham's approximation theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     pages = {2169-2202},
     doi = {10.5802/aif.3061},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_5_2169_0}
}
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. http://www.numdam.org/item/AIF_2016__66_5_2169_0/

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