Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → w + r2n+1 is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.