An approximate KAM-renormalization-group scheme for Hamiltonian systems

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov-Arn old-Moser (KAM) theory and renormalization-group techniques. It makes the c onnection between the approximate renormalization procedure derived by Esca nde and Doveil and a systematic expansion of the transformation. In particu lar, we show that the two main approximations, consisting in keeping only t he quadratic terms in the actions and the two main resonances, keep the ess ential information on the threshold of the breakup of invariant tori.