COMPENSATIONS IN SMALL DIVISOR PROBLEMS

A general direct method, alternative to KAM theory, apt to deal with s mall divisor problems in the real-analytic category, is presented and tested on several small divisor problems including the construction of maximal quasi-periodic solutions for nearly-integrable non-degenerate Hamiltonian or Lagrangian systems and the construction of lower dimen sional resonant tori for nearly-integrable Hamiltonian systems. The me thod is based on an explicit graph theoretical representation of the f ormal power series solutions, which allows to prove compensations amon g the monomials forming such representation.