Giovanni Gallavotti, Guido Gentile, and Alessandro Giuliani
Fractional Lindstedt series
(703K, postscript)

ABSTRACT.  The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter $\varepsilon$, i.e. they are not analytic functions of $\varepsilon$. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation (``resonances of order $1$') admit formal perturbation expansions in terms of a fractional power of $\varepsilon$ depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.