Kam theory and a partial justification of Greene's criterion for nontwist maps

We consider perturbations of integrable, area preserving nontwist maps of t he annulus (those are maps in which the twist condition changes sign). Thes e maps appear in a variety of applications, notably transport in atmospheri c Rossby waves. We show in suitable two-parameter families the persistence of critical circ les (invariant circles whose rotation number is the maximum of all the rota tion numbers of points in the map) with Diophantine rotation number. The pa rameter values with critical circles of frequency omega(0) lie on a one-dim ensional analytic curve. Furthermore, we show a partial justification of Greene's criterion: If anal ytic critical curves with Diophantine rotation number omega(0) exist, the r esidue of periodic orbits (that is, one fourth of the trace of the derivati ve of the return map minus 2) with rotation number converging to omega(0) c onverges to zero exponentially fast. We also show that if analytic curves e xist, there should be periodic orbits approximating them and indicate how t o compute them. These results justify, in particular, conjectures put forward on the basis of numerical evidence in [D. del Castillo-Negrete, J.M. Greene, and P.J. Mo rrison, Phys. D., 91 (1996), pp. 1-23]. The proof of both results relies on the successive application of an iterative lemma which is valid also for 2 d-dimensional exact symplectic diffeomorphisms. The proof of this iterative lemma is based on the deformation method of singularity theory.