A proof of existence of whiskered tori with quasi at homoclinic intersections in a class of almost integrable hamiltonian systems

Rotators interacting with a pendulum via small, velocity independent, potentials are considered: the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds (\whiskers"), whose intersections deene a set of homoclinic points. The homoclinic splitting can be introduced as a measure of the splitting of the stable and unstable manifolds near to any homoclinic point. In a previous paper, G1], cancellation mechanisms in the perturbative series of the homoclinic splitting have been investigated. This led to the result that, under suitable conditions, if the frequencies of the quasi periodic motion on the tori are large, the homoclinic splitting is smaller than any power in the frequency of the forcing (\quasi at homoclinic intersections"). In the case l = 2 the result was uniform in the twist size: for l > 2 the discussion relied on a recursive proof, of KAM type, of the whiskers existence, (so loosing the uniformity in the twist size). Here we extend the non recursive proof of existence of whiskered tori to the more than two dimensional cases, by developing some ideas illustrated in the quoted reference.