Delshams A., Gutierrez P.
Estimates on invariant tori near an elliptic equilibrium point
of a Hamiltonian system
(68K, LaTeX 2.09)
ABSTRACT. We give a precise statement for KAM theorem in a neighbourhood
of an elliptic equilibrium point of a Hamiltonian system.
If the frequencies of the elliptic point are nonresonant up to
a certain order $K\ge4$, and a nondegeneracy condition is fulfilled,
we get an estimate for the measure of the complement of the KAM tori
in a neighbourhood of given radius. Moreover, if the frequencies
satisfy a Diophantine condition, with exponent $\tau$, we show that
in a neighbourhood of radius $r$ the measure of the complement
is exponentially small in $(1/r)^{1/(\tau+1)}$.
We also give a related result for quasi-Diophantine frequencies,
which is more useful for practical purposes.
The results are obtained by putting the system in Birkhoff normal
form up to an appropiate order, and the key point relies on giving
accurate bounds for its terms.