Bryuno Function and the Standard Map

Abstract:

For the standard map the homotopically non-trivial invariant curves of rotation number ω satisfying the Bryuno condition are shown to be analytic in the perturbative parameter ε, provided |ε| is small enough. The radius of convergence ρ(ω) of the Lindstedt series – sometimes called critical function of the standard map – is studied and the relation with the Bryuno function B(ω) is derived: the quantity |log ρ(ω) + 2 B (ω)| is proved to be bounded uniformly in ω.