A finite dimensional approach to Bramham's approximation theorem

Abstract

Using pseudoholomorphic curves 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.

arXiv comment27 pages

References in corpus (2)

• Pseudo-rotations with sufficiently Liouvillean rotation number are C^0-rigid
• Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

Cited by in corpus (1)

• Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces