A FINITE DIMENSIONAL APPROACH TO BRAMHAM'S APPROXIMATION THEOREM

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.