TITLE:
KAM theory and a partial justification of Greene's criterion 
for non-twist maps 

AUTHORS:
Amadeu Delshams(1), Rafael de la Llave(2)

(1) Departament de Matem\`atica Aplicada I,
    Universitat Polit\`ecnica de Catalunya,
    Diagonal 647, 08028 Barcelona, Spain
    E-mail: amadeu@ma1.upc.es

(2) Department of Mathematics,
    University of Texas at Austin,
    Austin, TX, 78712, USA
    E-mail: llave@math.utexas.edu

ABSTRACT:
We consider perturbations of integrable, area preserving
non-twist maps of the annulus (those are maps in which
the twist condition 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 $\omega_0$ lie on a one-dimensional analytic curve.

Furthermore, we show a partial justification of Greene's criterion:
If analytic critical curves 
with Diophantine rotation number $\omega_0$
exist, the residue of periodic orbits
(that is, one fourth of the trace of the derivative of
the return map minus 2) with rotation
number converging to $\omega_0$ 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, in particular, conjectures put forward on the basis of
numerical evidence in D. del Castillo et al.,
{\sl Phys. D.} {\bf 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 diffeomorphisms.
The proof of this iterative lemma is based on 
the deformation method of singularity theory.


KEYWORDS:
KAM theory, non-twist maps, critical circles,
Greene's criterion, deformation method.

1991 MSC numbers:
53C15, 58F05, 58F14