We consider perturbations of integrable area preserving non twist maps of the annulus thoseare maps in which the twist condition changes sign These maps appear in a variety of applications notablytransport in atmospheric Rossby wavesWe show in suitable parameter families the persistence of critical circles invariant circles whoserotation number is the maximum of all the rotation numbers of points in the map with Diophantine rotationnumber The parameter values with critical circles of frequency lie on a one dimensional analytic curveFurthermore we show a partial justication of Greenes criterion If analytic critical curves with Diophantine rotation number exist the residue of periodic orbits that is one fourth of the trace of thederivative of the return map minus with rotation number converging to converges to zero exponentially fast We also show that if analytic curves exist there should be periodic orbits approximating themand indicate how to compute themThese results justify in particular conjectures put forward on the basis of numerical evidence in D delCastillo et al Phys D The proof of both results relies on the successive application of aniterative lemma which is valid also for d dimensional exact symplectic di eomorphisms The proof of thisiterative lemma is based on the deformation method of singularity theory