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    The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter  , except for a set of ɛ of measure exponentially small with ɛ  0 . In the case that ɛ is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori.Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models.

On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations