Scaling near resonances and complex rotation numbers for the standard map

We consider the radius of convergence rho(omega) of the Lindstedt series for the standard map and study its scaling behaviour as the rotation number tends to a rational value p/q both through real, diophantine numbers and through complex values. We compute numerically rho(omega) by means of Pade approximants, and therefore are able to plunge deeply into the asymptotic regime by computing rho(omega) very close to resonances. The scaling law rho(omega) approximately |omega-p/q|beta/q, with beta=2, is observed; this is consistent with the conjecture that rho(omega) approximately e-2B(omega), where B(omega) is a purely arithmetical function called Brjuno's function. In the case of the first two resonances (p/q=0/1 and p/q=1/2) we prove that the conjugating function to rotations (Lindstedt series) u(theta) tends to a limit u(p/q) (theta) as omega tends to the resonance and epsilon is scaled in such a way to keep the radius of convergence fixed; this limit is analytically computed and its singularities in the complex theta and epsilon planes are found to agree with the results obtained by Pade approximants. The relevance of these results for a perturbative approach to renormalization theory is discussed.