A Diophantine duality and applications to perturbation of quasi-periodic motions

Details A Diophantine duality and applications to perturbation of quasi-periodic motions A Diophantine duality and applications to perturbation of quasi-periodic motions

2012-04-07

arxiv.org/abs/1204.1608v1

Mathematics

Dynamical Systems

Scientific paper

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic motions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability in the model case of a perturbation of a constant vector field on the n-dimensional torus. Our first result is the construction of a "partial" normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno-R\"ussmann condition, we construct an "inverted" normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.

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