International Solvay Institutes for Physics and Chemistry

In the classical perturbation theory of conditionally periodic motions series occur that, due to resonances, diverge on a dense set. In the complement of the resonances, small divisors make convergence problematic. Nonetheless, convergence of the series can be established in a nowhere dense set of positive Hausdorff measure in a suitable dimension. In the product of phase space and parameter space this gives rise to quasi-periodic invariant tori with Diophantine frequency vectors. This kind of result belongs to kam theory, as this developed from Kolmogorov’s 1954 paper [77]. We sketch elements of this development, both in the dissipative and the conservative setting. ∗ Parially supported by grant MB-G-b of the Dutch FOM program Mathematical Physics and by grant HPRN-CT-2000-00113 of the European Community funding for the Research and Training