Jordi Villanueva
Kolmogorov Theorem Revisited
(451K, Postscript)

ABSTRACT.  Kolmogorov Theorem on the persistence of 
invariant tori of real analytic Hamiltonian systems is revisited. 
In this paper we are mainly concerned with 
the lower bound on the constant of the Diophantine condition 
required by the theorem. From the existing proofs 
in the literature, this lower bound 
turns to be of $\mathcal{O}(\varepsilon^{1/4})$, where 
$\varepsilon$ is the size of the perturbation. 
In this paper, by means of careful (but involved) 
estimates on Kolmogorov's method, 
we show that this lower bound can be weakened to be of 
$\mathcal{O}(\varepsilon^{1/2})$. 
This condition coincides with the optimal one of KAM Theorem. 
Moreover, we also obtain optimal estimates for the distance 
between the actions of the perturbed and unperturbed tori.