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Massimiliano Berti, Philippe Bolle
A Nash-Moser approach to KAM theory
ABSTRACT. Any finite dimensional embedded invariant torus of an Hamiltonian system,
densely filled by quasi-periodic solutions, is isotropic. This property allows us
to construct a set of symplectic coordinates in a neighborhood of the torus in which
the Hamiltonian is in a generalized KAM normal form with angle-dependent coefficients.
Based on this observation we develop an approach to KAM theory via
a Nash-Moser implicit function iterative theorem. The key point is to construct an
approximate right inverse of the differential operator associated to the linearized
Hamiltonian system at each approximate quasi-periodic solution. In the above symplectic
coordinates the linearized dynamics on the tangential and normal directions
to the approximate torus are approximately decoupled. The construction of an approximate
inverse is thus reduced to solving a quasi-periodically forced linear differential
equation in the normal variables. Applications of this procedure allow to
prove the existence of finite dimensional Diophantine invariant tori of autonomous