  We construct an approximate renormalization scheme for Hamiltonian systems
with two degrees of freedom. This scheme is a combination of
Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. It
makes the connection between the approximate renormalization procedure derived
by Escande and Doveil, and a systematic expansion of the transformation. In
particular, we show that the two main approximations, consisting in keeping
only the quadratic terms in the actions and the two main resonances, keep the
essential information on the threshold of the breakup of invariant tori.
