- 10-72 Massimiliano Berti, Luca Biasco
- Branching of Cantor manifolds of elliptic tori and applications to PDEs
May 6, 10
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Abstract. We consider in nite dimensional Hamiltonian systems. First we prove the existence of
\Cantor manifolds" of elliptic tori -of any nite higher dimension- accumulating on a given elliptic
KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of
elliptic tori which are ranching" points of other Cantor manifolds of higher dimensional tori. We
also provide a positive answer to a conjecture of Bourgain  proving the existence of invariant elliptic
KAM tori with tangential frequency constrained to a xed Diophantine direction. These results are
obtained under the natural nonresonance and nondegeneracy conditions. As applications we prove
the existence of new kinds of quasi periodic solutions of the one dimensional nonlinear wave equation.
The proofs are based on averaging normal forms and a sharp KAM theorem, whose advantages are
an explicit characterisation of the Cantor set of parameters, quite convenient for measure estimates,
and weaker smallness conditions on the perturbation.