Kam Theory and a Partial Justiication of Greene's Criterion for Non-twist Maps

We consider perturbations of integrable, area preserving non-twist maps of the annulus (those are maps that violate the twist condition in a very strong sense: @q 0 =@p changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves. We show in suitable 2-parameter families the persistence of critical circles (invariant circles whose rotation number is the maximum of all the rotation numbers of points in the map) with Diophantine rotation number. The parameter values with critical circles of frequency ! 0 lie on a one dimensional analytic curve. In contrast with recent progress in KAM theorem with degeneration in the frequency map, the curves we consider have rotation numbers on the boundary of the range of the frequency. Furthermore, we show a partial justiication of Greene's criterion: If critical analytic critical curves with Diophantine rotation number ! 0 exist, the residue of periodic orbits with rotation number converging to ! 0 converges to zero exponentially fast. We also show that if analytic curves exist, there should be periodic orbits approximating them and indicate how to compute them. These results justify conjectures put forward on the basis of numerical evidence in D. del Castillo et al., Phys. D. 91, 1{23 (1996). The proof of both results relies on the successive application of an iterative lemma which is valid also for 2d-dimensional exact symplectic diieomorphisms. The proof of this iterative lemma is based on the deformation method of singularity theory.