Estimates on invariant tori near an elliptic equilibrium point of a Hamiltonian system
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hdl:2117/832
Tipus de documentArticle
Data publicació1996
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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.
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