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

\AUTHOR Amadeu Delshams 
\FROM Departament de Matem\`atica Aplicada I
Universitat Polit\`ecnica de Catalunya 
Diagonal 647,  08028 Barcelona, Spain 
\EMAIL amadeu@ma1.upc.es

\AUTHOR Rafael de la Llave 
\FROM Department of Mathematics
The University of Texas at Austin
Austin, TX 78712-1082, U.S.A. 
\EMAIL llave@math.utexas.edu

\ENDTITLE



\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.

\ENDABSTRACT

\SECTION Introduction 

\SUBSECTIONNONR The motivation

The main goal of this paper is to provide rigorous proofs of several 
phenomena discovered empirically by del Castillo, Greene and Morrison
in \cite{CGM1}.
Even if our results will apply for a more general  class of 
maps---see Definition 1.3, 1.4, etc., for more precise 
definitions---we will start by describing the results of
that paper and the applications 
of the results we present here.

In \cite{CGM1}, the authors consider the two-parameter family of 
area preserving maps, called there the ``quadratic standard map''
$$T_{\omep} (p,q) = \left(p+\ep \sin (2\pi q) ,q -(p+\ep \sin 2\pi q)^2 +  
\omega \Pmod{1}\right).
\EQ(standardmap)$$

One motivation for such study is that qualitatively similar
maps appear naturally in the 
study of geostrophic flows and indeed in many problems in hydrodynamics
and in other applications, mentioned briefly later.

The ``unperturbed'' map $T_{\omega,0}$
$$
T_{\omega,0} (p,q) = \left(p , q + \Gamma(\omega,p)\Pmod{1}\right),
\qquad \Gamma(\omega,p) =\omega - p^2
\EQ(integrablemap)
$$
describes a situation
where  particles in a fluid are moving in a  laminar flow 
whose velocity is  faster in the middle ($p=0$) but slower as we move 
away from the center of the stream. This is 
a very common situation in fluid 
motion, where often the motion slows down as we move closer to edges
of the stream.
In many applications, it is natural to consider  $q$ as an angle.
For example,  in  the  description of the jet stream,
 $q$ corresponds to the longitude and $p$ is a
range of latitudes. 

The map $T_{\omega,0}$ is an integrable map, since all the 
circles with fixed $p$ are invariant under $T_{\omega,0}$
and the motion in them is a rigid rotation with rotation number
$\Gamma(\omega, p)$.
The quantity $\partial \Gamma/\partial p$---usually 
called the twist---measures the anisochronicity---i.e., the rate of
change of of frequencies among different invariant circles---. 
The condition $\partial \Gamma/\partial p$ is called
``twist condition'', and a map which satisfies the twist condition
is called a (monotone) twist map. This twist condition does not
hold in any map $T_{\omega,0}$ given in \equ(integrablemap), since
$\partial \Gamma/\partial p$ changes sign in $p=0$. Accordingly,
$T_{\omega,0}$ is called a non-twist map.  Note that 
changing of sign is stronger than the twist 
vanishing in some circle, but being otherwise positive. 
These are the small twist maps, which appear
also in many applications.
The relevance of the twist condition comes from the celebrated KAM 
theorem which establishes that the invariant circles 
whose frequency satisfies a Diophantine condition, persist 
under an small enough---in a smooth norm---area
preserving perturbation with zero mean flux. 
That is to say, twist mappings under perturbation 
look integrable for a large area. Non-twist maps, 
on the other hand, experience  new phenomena in the area 
where the twist changes sign. (See later in this 
introduction for more references.)

The extra term modified by the small parameter $\ep$ is representative
of the maps that arise when one considers the
physical effect of a small
periodic oscillation transverse to the channel flow.
Such phenomena occur frequently in hydrodynamics when channel flows 
are destabilized through a Hopf bifurcation.
This happens in jet flows in the atmosphere due to 
Rossby waves.
We refer to \cite{C} and \cite{CM} for a detailed description 
of the fluid mechanics motivation of such models.  In particular
for the justification of the use of a two-dimensional approximation.
In this interpretation, it is very important the existence of
invariant circles, since they are
complete barriers for the mixing of the material
in the pole---one of the edges of the latitude $p$---with the material 
near the equator---the other edge of $p$. 
In the particular model for the atmosphere,
these barriers give rise to the creation of
``ozone holes'' since they isolate the ozone 
created in the tropics from the  regions 
near the poles.

For area preserving perturbations of twist maps,
the Twist Theorem (see \cite{He} for a quantitative version, and
\cite{BHS} for an exhaustive description of KAM theory),
ensures the persistence, for $|\ep|$ small enough, of
those invariant curves with a Diophantine rotation number $\omega_0$:
$$
\exists C>0, \theta\ge 0 :
|k \cdot\omega_0 -m|^{-1} \le C |k|^{\theta -1},\forall k \in \integer ,
 m\in \integer \setminus \{0\}.
\EQ(Diophantineeq)
$$
The set of Diophantine numbers has full measure. A paradigmatic example
is $(\sqrt5 -1)/2$, which satisfies the inequalities above for $\theta=0$.

Unfortunately, given a Diophantine rotation number $\omega_0$,
the Twist Theorem cannot be applied to the map $T_{\omega_0,\ep}$
close to the invariant circle $p=0$, since the twist condition
breaks, and moreover the associated rotation number $\omega_0$ lies
on the boundary of the range of the rotation numbers $\Gamma(\omega_0,p)$.
The paper \cite{CGM1} finds
numerically---among other results---numerical evidence for
the following:

\CLAIM Claim(claimone)
Let $\omega_0 = (\sqrt5 -1)/2$.
Then, for $ |\ep|\ll 1$ there is a smooth curve $\omega(\ep)$
with $\omega(0)= \omega_0$ such that
\iitem{a)} If $\omega > \omega (\ep)$, then 
$T_{\omega,\ep}$ admits two invariant circles with rotation number 
$\omega_0$. 
\iitem{b)} If $\omega < \omega (\ep)$, then 
$T_{\omega,\ep}$ admits no invariant circles with rotation number 
$\omega_0$.
\iitem{c)} If $\omega = \omega(\ep)$, then 
$T_{\omega(\ep),\ep}$ admits an invariant circle with rotation number 
$\omega_0$. 

The circle in c), moreover is  ``critical'', that is, there exists a 
change of variables $(p,q) \to (A,\varphi)$
in its neighborhood  in such a way that
$$h^{-1} \circ T_{\omega(\ep),\ep}\circ h(A,\varphi) 
= (A,\varphi +\omega_0 + \kappa A^2) + O(A^3)\qquad \kappa\ne 0$$ 
(in fact, $\kappa < 0$ for the example in \equ(standardmap)).

It is worth noticing that the method used in \cite{CGM1} to assess
the existence of the invariant circles is the Greene's criterion,
introduced in \cite{Gr}. This criterion asserts that 
there exists an invariant circle 
with rotation number $\omega_0$ if and only if
$$\Res (O_{m,n}):= \frac14
\left[\tr \left(D T_{\omega,\ep}^n \left(O_{m,n}\right)\right)-2\right]
\mathop{\longrightarrow}\limits_{m/n\to\omega_0} 0$$
for any sequence of periodic orbits $O_{mn}$ of type $m/n$ converging
to $\omega_0$.

For the maps $T_{\omep}$ as in \equ(standardmap), the 
Greene's criterion can be implemented numerically very efficiently.
These maps are reversible and, for reversible maps,  the 
search for periodic orbits of type $m/n$ (those are $n$-periodic orbits
which make $m$ complete turns in the angle variable $q$)
in some symmetry lines---not all of
of the map---can be reduced to finding zeros of
one-dimensional functions, a tractable numerical task.
In the paper \cite{CGM1} the authors succeed in implementing this criterion,
and therefore they also find numerical evidence for:

\CLAIM Claim(claimtwo)
Greene's criterion applies. 

In this paper, we will prove rigorous results that justify the
experimental results we stated in detail above.
We will state and prove a result that justifies 
\clm(claimone) and another one that justifies one of the implications 
in \clm(claimtwo).
Namely that, if there exists an invariant circle,
the residue goes to zero.

To our knowledge, the converse---that is, if the residue 
goes to zero for any sequence of periodic orbits $O_{mn}$ of type $m/n$
converging to $\omega_0$, one can find an invariant circle 
with rotation number $\omega_0$---remains an open problem even 
for twist maps.
However we call attention to the work of \cite{KO}, 
which proves that if there are periodic orbits
of twist maps  which 
are, in a precise sense, well distributed, one can 
find an invariant circle with rotation number  related to that of
the periodic orbit. We also note that if the 
renormalization  group picture can be justified, 
at least to a certain extent, the Greene's
criterion will be also justified and indeed
several improvements on   that 
give precise asymptotic of the residue 
(see \cite{McK}.)

It is worth remarking that an easy argument, which we will detail later
in Proposition~4.4, shows that if there is a critical invariant circle as
%%fixed label
above, indeed it is approximated by periodic orbits of 
type $m/n$ with $m/n$ converging to $\omega_0$. Hence, this 
criterion is rather effective.


The general theory we will develop will not depend on the
exact form for the map, but on qualitative features that can be verified
in the realistic models. Of course, the map \equ(standardmap) 
is a concrete model
introduced for the purpose of  discovering 
qualitative features through numerical calculations. 

We also point out that other models having 
non-twist maps  have  appeared with other motivations. For example, 
they appear in celestial mechanics in problems such as the 
``critical inclination'' \cite{K} and in 
the study of billiards with a boundary moving periodically in time
\cite{KMOP1,KMOP2} or in the study of the motion of particles in 
magnetic fields \cite{ZZUSC}.
As a matter of fact, since the iterates of a twist map
are not, in general, twist maps, we expect that they appear also 
as descriptions of regions of iterates of twist maps.
(See. e.g. \cite{BST,Si}.)

These non-twist maps exhibit  a very rich phenomenology that is 
only now started to be explored.  The papers cited 
above as well as \cite{VG,HH1,HH2,Si,Ha1,Ha2} contain descriptions 
and  studies of
a wealth of phenomena such as  ``scaling relations'', ``reconnection'',
``meandering curves'' etc., that deserve to be investigated further.
Notably in \cite{Si,Ha1}, there are studies of new phenomena
that happen in higher dimensional non-twist maps.
In a very recent paper \cite{DMS}, it is shown that in a generic
unfolding of the tripling bifurcation of a fixed point of an area
preserving map, give rise to non-twist maps and therefore critical
invariant curves appear.


\SUBSECTIONNONR The methodology

In this paper, we will develop rigorous techniques 
that can produce results on two problems of the ones mentioned above:
The existence of critical invariant circles and the validity of 
Greene's criterion. Needless to say, we hope that the techniques that 
we develop for this purpose (e.g. finding appropriate normal forms 
and  quantitative error estimates of  them in neighborhoods) can eventually 
be used in the study of some of these other phenomena.




About the method of proof we note that there are
two basic methods in KAM theory to prove the persistence
of invariant tori of exact symplectic mappings or Hamiltonian flows. One is
based on applying successive transformations close to the invariant torus
and another one is based on solving
 functional equations that express invariance.
Both methods have complementary advantages. 
The functional equation method leads to very 
crisp proofs and they are more 
natural for numerical implementations. 
On the other hand, the methods based on 
transformation theory yield more 
information about the behavior of 
the map on a neighborhood of the invariant torus. 

Since in this paper we wanted to 
discuss the partial justification 
of the Greene's criterion, we certainly 
needed a method based on the transformation theory
and it was natural to use the same method
for the proof of the persistence of the invariant 
tori. 
In the future, we plan to come back to the 
functional method, specially in connection 
with a numerical implementation.

The proof  we present here will be
based on the deformation method. 
This method was introduced in the 
study of singularities of mappings
\cite{TL,Mat}
and it is very well suited for 
study of equivalence of maps in 
situations  where 
geometric structures are present 
\cite{BLW}, like families of exact symplectic diffeomorphisms.
One can use it also for the regular KAM theorem 
\cite{Ll}.
In our case, the use of the deformation method is 
very natural since the unknown involves a
family of maps.


Note that in this situation we are
trying to study the persistence of 
invariant circles whose frequency is on the 
boundary of the frequencies that are
present on the integrable map. 
This is in contrast with KAM theory, where
the non-degeneracy conditions---the so 
called twist condition or the more 
sophisticated R\"ussmann conditions (see \cite{BHS}, chapter~4)---imply
that the frequency under study is 
in the interior of the frequencies 
of the invariant circles in the integrable case.

Since the frequency we want to study is on the
boundary of the frequencies, 
it is  not difficult to consider a perturbation of 
the integrable case  in which there is no invariant circle
with the frequency we want. (It suffices to consider
an integrable 
perturbation in which we just add---or subtract---an
extra rotation so that all the invariant circles
persist, but their rotation number is 
changed.)

Speaking heuristically, what we will do
is to consider the regular perturbation theory
supplemented with a choice of $\omega(\ep)$.
The regular perturbation theory may 
force the $\omega_0$ out of the
range of frequencies, but we will find 
the extra rotation $\omega(\ep)$
that puts it on the boundary.
Since in this method of proof
one needs to consider families all
the time, the use of the deformation method
seems particularly well justified.

On a more technical level, we note that the proof 
will be based on an iterative lemma (Lemma~3.6), that describes
%%fixed label
how it is possible to obtain  transformations that
reduce the system to integrable. Moreover, we will present 
bounds on the error of this reduction depending on the domain.
This iterative lemma can be applied repeatedly in different 
ways depending on how one 
plays the trade off between domain loss and 
accuracy. One can try to make the error decrease very fast
at the price that the domain decreases very fast 
or one can make the error decrease slowly on a larger domain.
In this way, one can obtain a unified approach
towards KAM theory and towards
exponentially small estimates, which  we will show justify Greene's criterion.
This approach has precedents in  \cite{DG1}.
Since the iterative lemma as well as the 
deformation method are  widely applicable, we have 
developed it in an arbitrary dimension. The geometric considerations that 
lead to the KAM theorem for critical circles and 
to the Greene's criterion seem to be different in higher dimensions, 
so we have postponed the discussion of this part.




\SUBSECTIONNONR The results

Now we turn to making all these ideas more precise.


\CLAIM Definition(critical) 
We say that a circle $S$, invariant under an
area preserving  map $T$ of $\real \times\torus ^1
\equiv M$, is a critical invariant circle if there exists a canonical
transformation
$h:[-\delta,\delta] \times\torus ^1\to M$ in such a way that 
$$h^{-1}\circ T\circ h(A,\varphi) = (A,\varphi +\omega_0 +\kappa A^2)+O(A^3)$$ 
with $\kappa \ne 0$ and $h( \{0\}\times \torus ^1 ) = S$.

\REMARK
The definition of a critical circle includes in its hypothesis that the motion 
on the circle is conjugate to a rotation of $\omega_0$. 
We will not include the $\omega_0$ in the notation since it will be 
understood from the context. 

We also recall---and we will develop it in more detail later 
in Lemma~4.2---that there is an analogue of Birkhoff normal form 
%%fixed label
in a neighborhood of an invariant circle with a Diophantine rotation.
(In the twist map case, this was also considered in 
\cite{OS,FL}.)
Given $N\in\natural$, it is possible to find coefficients 
$\kappa_1,\ldots,\kappa_N$ and a canonical transformation
$h$ such that  
$$
h^{-1}\circ T\circ h(A,\varphi) 
= (A,\varphi +\omega_0 + \kappa_1 A+ \kappa_2 A^2 + \cdots +  \kappa_M A^M)
+ O(A^{M+1})
\EQ(normalform)
$$ 
The coefficients $\kappa_1,\ldots,\kappa_N$ are uniquely defined and are properties 
of the invariant circle.
In this language, critical circles are those for which $\kappa_1=0$, $\kappa_2\ne0$. 

\CLAIM Definition(non-degenerate)
We will call an invariant circle non-degenerate when 
the normal form \equ(normalform) does 
not vanish identically. That is, 
we can find $M \in \natural$ such that 
$\kappa_1 = \cdots = \kappa_{M-1} = 0$,
$\kappa_M \ne 0$.



Our result to justify \clm(claimone) is:

\CLAIM Theorem(saddlenode)
Let $\omega_0$ be a Diophantine number as in \equ(Diophantineeq),
$f_{\omega,\ep}$ be a family of 
mappings from $\real ^1\times \torus ^1$ to itself satisfying 
\smallskip
\iitem{i)} $f_{\omega,\ep} (p,q)$ is analytic in 
$$|\omega-\omega_0| < \rho_0\ ,\quad 
|\ep| <  \rho_0\ ,\quad 
|\Im\  q| < \beta_0\ ,\quad 
|p|  < \rho_0 .$$
and takes real values for $\omega,\ep,p,q$ real
\smallskip
\iitem{ii)} $f_{\omega,\ep}$ is exact symplectic for all $\omega,\ep$
\iitem{iii)} $f_{\omega,0}(p,q) = (p,q+\Gamma(\omega,p))$ with
$$
\displaylines{\Gamma(\omega_0,0)=\omega_0,\qquad
{\partial\over\partial p} \Gamma(\omega_0,0) = 0\cr 
{\partial^2\over\partial p^2} \Gamma (\omega_0,0) = t<0,\qquad 
{\partial\over\partial \omega} \Gamma (\omega_0,0) = s>0\cr}
$$
Then, we can find a $\delta >0$ and an analytic function $\omega$ defined
for $|\ep|$ 
sufficiently small and taking real values for $\ep$ real in such a way that 
\smallskip
\iitem{a)} $f_{\omega(\ep),\ep}$ has exactly one critical invariant circle
in $[-\delta , \delta] \times \torus ^1$
\iitem{b)} if $\omega<\omega(\ep)$, $f_{\omega,\ep}$
has no points
in $[-\delta , \delta] \times \torus ^1$ with 
rotation number $\omega_0$, and if $\omega >\omega(\ep)$ there are two 
invariant circles of $f_{\omega,\ep}$
in $[-\delta , \delta] \times \torus ^1$
which are  not critical. 

\REMARK
It is possible to change hypothesis $iii)$ of \clm(saddlenode),
to be that $t$ is positive.
It suffices to change the inequalities between $\omega$, $\omega(\ep)$
in part~$b)$ of the conclusions and the proof goes through without change.
Similarly, if $s$ is negative in $iii)$.

The precise meaning in which Greene's criterion can be justified is 
the following. We will show 

\CLAIM Theorem(Greene)
Let $f_{\omep}$ be  an  analytic area preserving diffeomorphism of 
the annulus.
Assume that $f_{\omep}$ admits an
analytic invariant circle
on which the motion is analytically conjugate to a rotation with 
Diophantine number $\omega_0$ and 
which is non-degenerate in the sense of 
\clm(non-degenerate).
\vskip 0 em
Then,
we can find $C_1, C_2, \mu  > 0$  
(depending on $\omega_0$, the map and  the torus)  such that
for any sequence of periodic orbits $O_n$ of type $p_n/q_n$ which 
are converging to the analytic invariant circle and such that 
$|\omega_0 - p_n/q_n| \le 1/ q_n$,
we have
$$
\Res (O_n) \le C_1  \exp \left(-C_2 |\omega_0 - p_n/q_n|^{-\mu}\right)
\EQ(Greeneconclusions)
$$

We will also show that there is one such sequence of periodic orbits
converging to the non-degenerate circle. 
Of course, when the circle is critical, depending on the sign of 
$\omega - p_n/q_n$ we will find either  two or four periodic orbits.
For more general non-degenerate circles, 
when $M$ is even we will find 
two or  four periodic orbits of 
type $p_n/q_n$ depending on the sign 
of $\omega - p_n/q_n$
and when $M$ is odd we will find two 
irrespective of the sign or $\omega - p_n/q_n$.

\REMARK
The proof that the residue goes to zero faster than any power is 
significantly easier than the proof  with an explicit rate.

\SECTION The deformation method 

In this section we recall the basis of the deformation 
method for symplectic maps.
This method was introduced in 
singularity theory \cite{TL,Mat}, but it was 
remarked later that it can be used very effectively 
to obtain structure theorems for volume preserving 
maps of a manifold \cite{Mo1}, or for symplectic 
maps \cite{W} giving a very direct proof of Darboux theorem.
More details and other applications can be found in 
\cite{LMM,Ll,BLW} and  in several other places.

In this section, the dimension of the space will not 
play a role, so we will consider $M$ a $2d$-dimensional manifold.

We recall that a 2-form $\twof $  on $M$ is a symplectic form if it is closed and 
has full rank. (Of course, the fact that $\twof$ has full rank implies that the 
dimension of $M$ is even, this is why we chose the notation $2d$ for it.)
We will  be specially interested in  the case when $\twof $ is exact. 
That is, there exist a 1-form $\onef $ such that $\twof = d\onef $. 

A  diffeomorphism $f$ is symplectic when $f_*\twof  = \twof $. 
For $\twof $ exact, this is equivalent to $d(f_* \onef -\onef )=0$. 
We say that a symplectic map $f$ is exact when $f_*\onef -\onef =dS$
for some function $S$, called the primitive function of $f$.

Given a family of diffeomorphisms $f_\ep$, we denote by $\F_\ep$
the vector field defined by 
$${d\over d\ep} f_\ep = \F_\ep \circ f_\ep 
\EQ(generator)$$ 
and refer to $\F_\ep$ as the generator of $f_\ep$. 
Note that a family determines the generator and, conversely, by the 
uniqueness theorem for O.D.E.'s, 
a family is determined by its initial point $f_0$ and its generator, when the 
generator is ${\cal C}^1$. 
(We will always assume that this is the case.) 
 
The main idea of the deformation method is to work always 
with the generators, which, when the families are differentiable enough
so  that the uniqueness theorem for O.D.E.'s applies, is equivalent to 
working with the families. When the
diffeomorphisms  are symplectic, further simplifications are possible.
Using Cartan's formula for Lie derivatives and that $\twof $ is closed we 
obtain 
$$\eqalign{
{d\over d\ep} {f_\ep}_* \twof   
& = {f_\ep}_* (d(i(\F_\ep)\twof ) +i(\F_\ep)\, d\twof ) 
	= {f_\ep}_* (d(i(\F_\ep)\twof ))\cr 
{d\over d\ep} {f_\ep}_* \onef  
& = d({f_\ep}_* (i(\F_\ep))\onef ) + {f_\ep}_* (i(\F_\ep)\twof )\cr} 
\EQ(derivative)$$ 

If $f_\ep$ is symplectic, ${d\over d\ep} {f_\ep}_*\twof =0$, 
and then we see that
$$d(i(\F_\ep)\twof )=0 
\EQ(symplectic)$$ 
If  $f_\ep$ is exact symplectic, 
$ d\left( {d \over d \ep} S_\ep \right) -
{f_\ep}_* d(i(\F_\ep)\twof )= {f_\ep}_* (i(\F_\ep)\onef )$
and, therefore,
$$i(\F_\ep)\twof  = dF_\ep 
\EQ(exact)$$ 
with $F_\ep =  \left({ d \over d \ep} S_\ep\right) \circ {f_\ep}
- i(\F_\ep) \onef $.

Conversely, if $\F_\ep$ satisfies \equ(symplectic) or \equ(exact) and 
$f_0$ is symplectic or exact symplectic, the family $f_\ep$ is symplectic 
or exact  symplectic as can be seen integrating \equ(derivative). 

Along this paper, we will refer to $F_\ep$ as the Hamiltonian
for the family $f_\ep$. 
Note that given $f_\ep$, \equ(exact) determines $F_\ep$ up to a function of 
zero differential hence, constant on each connected component of 
its domain of definition.
This justifies calling $F_\ep$ ``the Hamiltonian'' if we think of 
Hamiltonians as equivalent when they differ in a function with zero 
differential. This identification is 
natural since two Hamiltonian differing by 
a  function with zero differential 
generate the same dynamics.

Conversely, for a ${\cal C}^2$ Hamiltonian $F_\ep$,
given that $\twof $ is full rank,
\equ(exact) determines $\F_\ep$, and it is ${\cal C}^1$. 
This $\F_\ep$ and $f_0$ determine $f_\ep$ by the uniqueness result for
O.D.E.'s.

Hence, for sufficiently smooth families it is equivalent to work with the 
Hamiltonians and the initial points of the families. 

The main  idea of the deformation method 
for  exact symplectic maps is to reformulate all the problems 
in terms of Hamiltonians. 
As it turns out, the equations involving generators are linear. This is to 
be expected since we can heuristically think of generators as   
infinitesimal transformations and all the equations among infinitesimal 
quantities are linear.  
Moreover, using Hamiltonians, the otherwise complicated constraint of the 
transformations being  exact symplectic is implemented automatically,
and the resulting equations only involve functions. Hence, rather than dealing
with non-linear equations among  diffeomorphisms 
satisfying non-linear constraints,
we just have to deal with a linear equation among functions.

We will follow the convention of denoting families in lower case $f_\ep$, 
their generators in calligraphic font $\F_\ep$ and the Hamiltonians in 
upper case $F_\ep$. 

\CLAIM Proposition(calculus) 
Let $f_\ep, g_\ep$ be exact symplectic families and $k$ an exact symplectic 
diffeomorphism. Then, the Hamiltonian of the families formed out of them 
are given in the following table. 
$$\vbox{\offinterlineskip
\halign{\hfil$\strut#$\hfill\quad&\vrule#&\quad$#$\hfill\cr
\hbox{family}&&\hbox{Hamiltonian}\cr
\noalign{\hrule} 
f_\ep\circ g_\ep&&F_\ep + {f_\ep}_* G_\ep = F_\ep + G_\ep\circ f_\ep^{-1}\cr 
f_\ep^{-1}&&-F_\ep \circ f_\ep\cr 
g_\ep^{-1} \circ f_\ep \circ g_\ep&&F_\ep\circ g_\ep -G_\ep\circ g_\ep 
+ G_\ep \circ f_\ep^{-1} \circ g_\ep\cr 
k^{-1} \circ f_\ep\circ k&&F_\ep \circ k\cr 
f_\ep \circ k&&F_\ep\cr}}$$

The computations needed to work out this table can 
be found in \cite{LMM,BLW}. In the latter 
paper one can find similar tables for 
volume preserving or contact families.


Since in perturbation theory one does not always have a family of 
diffeomorphisms
but just two diffeomorphisms that  are close, 
it is worth remarking that given two 
symplectic diffeomorphisms 
that are  close, one can always interpolate them by a family with 
small Hamiltonian. 
If the two maps are exact, the family can be chosen to be exact. 
This is an immediate consequence of the general fact that symplectic 
(or exact symplectic) maps form a Banach manifold (see \cite{W}). 
We just sketch a direct construction whose details
 appear in \cite{BLW}. 
An alternative, old fashioned proof can be obtained using generating 
functions.
(Interpolate the generating functions.) 
Unfortunately, since it is impossible to obtain generating functions that 
are globally defined, one has to also use partitions of unity and 
fragmentation lemmas and the proof becomes cumbersome.

Given $f_0,f_1$  symplectic and 
close enough, we can find a family of diffeomorphisms 
$f_\ep$ interpolating between them (e.g., $f_\ep (x) = \exp_{f_0(x)} \ep 
\exp_{f_0(x)}^{-1} f_1(x)$ where $\exp$ is the Riemannian exponential map). 
The family $f_\ep$ will not be symplectic. 
In general, ${f_{\ep}}_* \twof  = \twof _\ep$ where $\twof _\ep$
is a family of 
symplectic forms. 
Note that, by our assumptions $\twof _0 = \twof _1 = \twof $. 
Using Moser's construction \cite{Mo1}---we refer to 
\cite{LMM,BLW} for the elementary justification of 
the smooth dependence on parameters in Moser's 
construction---we can find $h_\ep$ 
close to the identity in such a way that 
${h_{\ep}}_* \twof _\ep =\twof $. 
Moreover, $h_0 = h_1 = \Id$. 
Then $\tilde f_\ep = h_\ep \circ f_\ep$ satisfies $\tilde f_0 = f_0$, 
$\tilde f_1 = f_1$, ${\tilde {f_{\ep}}}_* \twof  = \twof $. 
If $\twof  = d\onef $ then $\twof _\ep = d\onef _\ep$ with 
$\onef _\ep = {f_{\ep}}_* \onef $. 
Also ${(h_\ep \circ f_\ep)}_* \onef -\onef $ is closed. 
It is then possible to choose $g_\ep$  close to the identity
in such a way that 
${(g_\ep \circ h_\ep\circ f_\ep)}_*\onef -\onef $ is exact (e.g., on the 
annulus choose translations in the radial direction and in another 
manifolds choose a displacement in a neighborhood of paths that 
generate the homology.)

We have, therefore, established 

\CLAIM Lemma(interpolation) 
Let $f_0$ be a ${\cal C}^\infty$ (resp. ${\cal C}^\omega$) symplectic
(resp. exact symplectic) diffeomorphism of a manifold. 
\vskip 0 em
If $f_1$ is a symplectic (resp. exact symplectic) 
diffeomorphism close to $f_0$ we can find a ${\cal C}^\infty$ 
(resp. ${\cal C}^\omega$) family $f_\ep$ of symplectic (resp. exact
symplectic) diffeomorphisms interpolating between $f_0$ and $f_1$. 
\vskip 0 em
Moreover, we can arrange that the generators
and therefore the Hamiltonians of the isotopy are arbitrarily 
small in the ${\cal C}^\infty$ (resp. ${\cal C}^\omega$) topology by assuming
that $f_1$ is arbitrarily close to $f_0$. 


\SECTION Proof of \clm(saddlenode) using the deformation method

\SUBSECTION Heuristic discussion

The  proof we  present here starts with  the observation that the result 
would be obvious if we had a family of the form 
$$i_{\omega,\ep} (p,q) = (p,q+\Omega (\omega,\ep,p))
\EQ(desired)$$
in which the  $p$ is conserved and the $q$ is translated
by $\Omega (\omega,\ep,p)$, which depends on $p$
and on external parameters and is close to the frequency
$\Gamma(\omega,p)$ satisfying hypothesis $iii)$ of \clm(saddlenode).
We will refer to such families as integrable.

If we require that the set $p=p_0$ is an invariant circle with rotation 
$\omega_0$, we obtain  the implicit equation
$$
\Omega ( \omega, \ep, p_0) = \omega_0
\EQ(fold)
$$ 
The possibility of finding solutions of \equ(fold) is described by 
singularity theory and the phenomenon of a critical invariant circle
corresponds to the situation when 
$\Omega (\omep,p_0)-\omega_0$ has a fold:
$$
\Omega (\omep,p_0)-\omega_0=0,
\qquad \partial_p \Omega (\omep,p_0)=0
$$
The equation for $\omega (\ep)$ is precisely the equation for the edge of 
a fold.
We will parameterize the folding surface \equ(fold) as 
the set of points $ (\Upsilon(\ep,p), \ep,p)$ for an
appropriate function $\Upsilon$:
$$
\Omega ( \omega, \ep, p) = \omega_0 \iff \omega = \Upsilon(\ep,p) 
\EQ(parameterization)
$$ 
Then, a critical invariant circle takes place at $p=p_0=p_0(\ep)$
if $\partial_p \Upsilon (\ep,p_0)=0$, and
$\omega(\ep)=\Upsilon (\ep,p_0(\ep))$.

A standard technique in KAM theory is to make changes of variables so that 
in the new system of coordinates, the properties of the map 
are apparent from its expression.
In the present case, we try to find $g_\ep$ in such a way that 
$$\tilde f_{\omega,\ep} = g_\ep^{-1}\circ f_\ep \circ g_\ep
\EQ(changes)$$ 
 has the desired form \equ(desired). 

Unfortunately, in general it is not  possible to obtain 
a change of variables 
reducing to \equ(desired) in the whole phase space. 
We only know how to do it 
approximately in a subset of the domain in $(\omega,\ep,p)$ 
for which $\Omega (\omega, \ep,p)=\omega_0$. 

Hence we will use an iterative scheme in which at step $n$, the system will be 
(described in the notation of the deformation method by the initial point 
of the isotopy and the generating Hamiltonian)
$$
f_{\omega,0}^n (p,q) = (p,q+\Gamma(\omega,p)) ; \quad \quad
F^n_{\omega,\ep}(p,q) = I^n_{\omega,\ep}(p) + E^n_{\omega,\ep}(p,q)
\EQ(nstep)
$$ 
where $E_{\omega,\ep}^n$ is ``small'' in a neighborhood of 
$\{p=0\}$. 

The Hamiltonian $I^n_{\omega,\ep}(p)$ corresponds to a 
deformation  of the form 
$$
i_{\omega,\ep}^n (p,q) = (p,q+\Omega^n (\omega,\ep,p))
\EQ(intpartn)
$$ 
where 
$$\Omega^n (\omega,\ep,p) =
\Gamma(\omega, p) + \int_0^\ep ds {\partial\over\partial p}
I^n _{\omega,s}(p)
\EQ(Omegan)
$$
when we assume that $i_{\omega,0} = f_{\omega,0}$.
Hence, the $I^n_{\omega,\ep}$ should be thought of as the integrable part
of the Hamiltonian $F^n_{\omega,\ep}$.
We will think of $E^n_{\omega,\ep}$ as an error
term that is to  be made smaller and smaller in the iterative process.

\REMARK
We note that the decomposition of a Hamiltonian into 
an integrable part and an small part is not  uniquely defined.
A particularly natural one would be to take the integrable
part to be the average over the $q$. Nevertheless, we 
will not be assuming that this natural decomposition is taken, 
just that such a decomposition exists.

\REMARK
Note that when we consider perturbations of 
an integrable system, we can write the integrable 
part  in $\Omega$ and, hence, assume 
that
$I^0_\omep(p) = 0$.


The main ingredient of the  proof of \clm(saddlenode)
will be an algorithm that, given
a family as in \equ(nstep), finds a transformation
$g_{\omep}^n$ defined in a 
neighborhood of the surface $\Omega^n (\ep,\omega,p)= \omega_0$ such that 
setting $f_{\omega,\ep}^{n+1} = (g_{\omega,\ep}^n)^{-1}\circ 
f_{\omega,\ep}^n\circ g_{\omega,\ep}^n$ we have 
$$
F_{\omega,\ep}^{n+1} (p,q) 
=  I^{n+1}_{\omega,\ep}(p)+ E_{\omega,\ep}^{n+1}(p,q)$$ 
where $E^{n+1}_{\omega,\ep}$ is much smaller
than $E^n_{\omega,\ep}$ and $I^{n+1}_{\omega,\ep}$ differs 
little from $I^n_{\omega,\ep}$ in a domain which will be chosen appropriately
(a smaller neighborhood of the surface $\Omega^{n+1} = \omega_0$.)

Since $\Omega^{n+1}$ is close to $\Omega^n$, the folding surfaces defined by
$\Omega^{n+1} = \omega_0$ and by $\Omega^n=\omega_0$ are very close.
Quantitative estimates will show that the $E^n_{\omega,\ep}$'s
decrease super-exponentially and that the $g^n_{\omega,\ep}$'s
differ from the identity by a super-exponentially small quantity
in neighborhoods of the surfaces $\Omega^{n+1} = \omega_0$. 
As it turns out, we will have to choose these 
neighborhoods to become super-exponentially thinner. 
The  transformations will be defined in these thin
slivers in the $\omega,\ep,p $ coordinates 
and in  domains in $q$ which include complex 
extensions of $\torus ^1$ so that the 
size of the size of the imaginary extension of 
the domain remains bounded from below.

Similarly, the functions $\Omega^n$ converge to a function $\Omega^\infty$. 
Therefore, the surfaces
$\hat\Omega^n \equiv (g^1\circ \cdots \circ g^n)^{-1}\{\Omega^n=\omega_0\}$
converge to a surface $\hat\Omega^\infty$. 
Since each of the surfaces $\{\Omega^n = \omega_0\}$
is foliated by smooth circles
invariant by $F\circ g^1\circ \cdots \circ g^n$ up to super-exponentially 
small errors, it follows that $\hat \Omega^\infty$ is foliated
by smooth circles invariant by $F$. 

For the benefit of
experts, we point out that an alternative method to prove \clm(saddlenode)
could have been to use the non-degeneracy in $\omega$ to prove a
KAM theorem for all small enough $\ep$  and  $p$.
(That is, we fix $\ep$ and  $p$, but 
allow 
ourselves to choose the $\omega$).  Even if 
not all methods to prove KAM theorems would have worked, it 
seems that  methods based on the ``translated curve method'' 
works since one can use the 
$\omega$ to adjust the frequency. 
Then, one needs to prove the analytic dependence of 
the circle on the parameter $\ep$ and to 
prove that  there is indeed a fold. 

The method we develop in this paper seems more appealing since one 
has an understanding of the folding surface at all the 
stages of the iteration and it is certainly not longer to write in 
all detail.

Moreover, we can use much of the technology developed 
along these lines, to prove the partial converse of Greene's theorem.
In particular, Lemma~3.6 is the crux of the iterative 
%%fixed label
step in  the proofs of both problems. 
The difference between the KAM theorem and the proof
of the exponentially small estimates that imply 
Greene's criterion, lies only in different choices
on how  we iterate the method. In the KAM theorem,
we lose domain very fast and drive the errors to 
zero very fast. In the exponentially small estimates, 
we reduce the domains  more slowly and do 
not obtain convergence but the estimates are 
valid in a larger domain.

We also call attention to the fact that 
Lemma~3.6 is valid in any dimension. It is 
%%fixed label
only the geometric considerations about domains
that one uses to conclude \clm(saddlenode)
and \clm(Greene) that require the fact that we 
are working in an annulus. 
We think that this restriction can be lifted with some 
small amount of extra effort.


\SUBSECTION Notation and elementary estimates

Since the iterative step will rely on making transformations on 
functions in such a way that the errors become smaller, we
will need to  define appropriate norms. 
We will also need to  be able to manipulate sets where our transformations
will be defined. (As usual in KAM theory, one has to consider functions 
defined in decreasing sets.)
In this section, we collect the definitions of the norms,
parameterizations of sets  that we will
use later as well as some elementary lemmas and propositions 
dealing with them.

Since Lemma~3.6 is valid in any number of dimensions,
%%fixed label
we will be considering maps  
in $\real^d \times \torus^d$ till the end of Section 3.5. 

We recall  the standard definition that 
$\omega_0 \in \real^d$ is said to be 
Diophantine of exponent $\theta $ if
we can find a $C > 0$ such that
for all 
$ k \in \integer^d, m \in \integer$ we have
$$
| k \cdot \omega_0 - m |^{-1} \le C |k|^{\theta -1}
\EQ(Diophantinevector)
$$
This is the definition of Diophantine vectors that appears naturally
in KAM theory for maps. (The definition that appears naturally in 
KAM theory for flows is slightly different.)

Besides the above standard definition, in this paper
we will use the following notations.



We will denote by $I_{a,b}$ the real interval $[a,b]$, 
by $B_{x,c}$ the closed ball in $\reald $ with center
$x\in\reald $ and radius $c>0$, and by $\torusd $ the
$d$-dimensional torus $\reald /\integerd $.

We will also denote by 
$I_{a,b,\delta} = \{z\in \complex \mid d(z,I_{a,b})\le \delta \}$, 
$B_{x,c,\delta} = \{z\in \complex^{d} \mid d(z,I_{a,b})\le \delta \}$. 
Similarly we will denote by $\torusd _\beta $ the complex extensions on 
the torus $\torusd $ of a distance $\beta $.


Given a set 
$ U= B_{x_1,c_1,\delta} \times I_{a_2,b_2,\delta} \times B_{x_3,b_3,\delta}$
and a function 
$\Omega: U \to \complex^{d}$,
we will denote  for $\alpha, \beta >0$
$$
\eqalign{
\Sgm{U}{\beta} 
&= \{ (\omega,\ep,p,q) \mid (\omega,\ep,p) \in  U,
 | \Im\  q | \le \beta
\}=U \times \torusd _\beta
\cr
\Sigma_{\Omega,\alpha,\beta,U} 
&= \{ (\omega,\ep,p,q) \in \Sgm{U}{\beta} \mid
 |\Omega(\omega,\ep,p) -\omega_0| \le \alpha
\}}
\EQ(Sigma)
$$

The way to think about $\Sigma_{\Omega,\alpha,\beta,U}$ is as
the Cartesian product of 
a thin film---of width $\alpha$, which will 
be extremely small in the proof---around  a portion  of
surface given by the equation $\Omega(\omega,\ep,p) = \omega_0$
and a complex extension of width $\beta$ of the 
torus. The parameter $U$ just limits which portion 
of the surface we are considering and it plays a
somewhat minor role.

Note that, for the sake of notation, we are suppressing some of
the parameters on which $\Sigma_{\Omega,\alpha, \beta,U}$
depends. Notably $\omega_0$. 
We hope that this does not lead to confusion in the proof since the values 
of these parameters will be kept fixed.
The $\omega_0$ will be that appearing in 
\clm(saddlenode) and, 
hence, will not change throughout the proof. 


We will introduce the notation $U_\sigma$ to denote 
a domain  
formed by restricting the domain only in  the 
variable $p$ by an amount $\sigma > 0$,
that is, 
$ U= B_{x_1,c_1,\delta} \times I_{a_2,b_2,\delta}
 \times B_{x_3,b_3,\delta - \sigma}$.

This will be used later since we need to 
reduce the domains in phase space 
(to guarantee that compositions
make sense)  but the domains
in parameters are not affected.

Given a complex domain $\Sigma $, we will denote 
by $\|F\|_\Sigma 
\equiv \sup_{x\in \Sigma} |F(x)|$ and by $\chi^\Sigma$ the Banach 
space of functions 
analytic in $\Sigma$ 
(analytic in the interior and continuous up to the boundary) 
equipped with the norm $\|\cdot\|_\Sigma$. 
In particular, for
$\Sigma = \Sgm{U}{\beta}$, 
$\Sigma = \Sigma_{\Omega,\alpha,\beta,U}$ of the form \equ(Sigma), 
for typographical reasons, we will write
$\|\cdot\|_{\Sgm{U}{\beta}}$ 
as $\NN{\cdot}{U}{\beta}$ and
$\|\cdot\|_{\Sigma_{\Omega,\alpha,\beta,U}}$ 
as $\|\cdot\|_{\Omega,\alpha,\beta,U}$.

For a function $F: U \times \torusd _\beta\to \complex$,
where
$ U= B_{x_1,c_1,\delta} \times I_{a_2,b_2,\delta}
 \times B_{x_3,b_3,\delta}$.
we define the partial Fourier expansion 
$$F_{\omep}(p,q) = \sum_{k\in\integerd } \hat F_{\omep;k} (p) 
e^{2\pi i\,  k\cdot q}$$
The coefficients are unique in the regularity classes we 
will be considering.

For this kind of functions depending on 
parameters, we will use the notation 
$\nabla$ to denote the derivatives 
with respect to the variables, not 
with respect to the parameters.
Hence 
$$
\nabla F_\omep(p,q) =
\left( {\partial \over \partial p} F_\omep(p,q),  
{\partial \over  \partial q} F_\omep(p,q) \right)
$$
In the cases that we will need to 
consider derivatives with respect to 
the parameters, we will write them 
explicitly.



We recall  that  the well known 
Cauchy inequalities allow us to bound derivatives (in a
domain)  and 
Fourier coefficients of a function in terms of its 
size in a (slightly larger) domain.

\CLAIM Lemma(cauchy) 
Let
$ U= B_{x_1,c_1,\delta} \times I_{a_2,b_2,\delta} \times B_{x_3,b_3,\delta}$,
$\tilde U \subset U$  be a domain that 
is at a distance $\sigma > 0 $ from the complement of $U$,
and $F: U \times \torusd _\beta\to \complex$ analytic. Then,
$$\eqalign{
 \NN{\nabla^m F}{\tilde U}{\beta-\sigma}
 & \le K\sigma^{-m} \NN{F}{\tilde U}{\beta}\cr
 \NN{\partial _\omega^m F}{\tilde U}{\beta},
 \NN{\partial _\ep^m F}{\tilde U}{\beta}
 & \le K\sigma^{-m}\NN{F}{\tilde U}{\beta}\cr
|\hat F_{\omep;k}(p)|
& \le Ke^{-2\pi \beta |k|} \NN{F}{\{(\omep,p)\}}{\beta}
}
$$

The well known proof is based on expressing the Fourier coefficients 
or derivatives as integrals over paths and deforming them in the 
complex domain. It can be found in many reference books and we will
not reproduce it here.


\SUBSECTION The iterative step

In this subsection, we will specify  the iterative step of the 
algorithm and we develop quantitative estimates that will later lead 
to the possibility of iterating it and  showing it converges. 
Most of these estimates will be used also in \clm(Greene) on
the partial justification of Greene's criterion. 

We recall that for the purposes of the iterative lemma 
Lemma~3.6, the dimension of the space will be irrelevant, 
%%fixed label
so we will state the results in the  $2d$-annulus $\real^d \times \torus^d$.

At the beginning of the iterative step, we will be given a family of exact 
symplectic maps $f_{\omega,\ep}$ defined on a subset
of $\real^d \times \torus^d$
endowed with the standard symplectic structure.
$$
f_{\omega,0} (p,q) = (p,q+\Gamma (\omega,p)) \quad \quad
F_{\omega,\ep}(p,q) = I_{\omega,\ep}(p) + E_{\omega,\ep}(p,q)
\EQ(nstep2)
$$ 
where $F_{\omega,\ep}$, the Hamiltonian of the deformation $f_{\omega,\ep}$,
is defined in a set
$\Sigma_{\Omega,\alpha,\beta,U}$ 
of the type described  in \equ(Sigma), 
with
$$
U= B_{\omega_0,\gamma,\delta} \times  I_{[-1,1], \delta}
\times  B_{0,\gamma,\delta}
$$
for some $\gamma>0$, $0<\delta < 1$,
where $\omega_0$ is a Diophantine vector
(e.g. it satisfies \equ(Diophantinevector))
and 
$$\Omega (\omega,\ep,p) =
\Gamma(\omega, p) + \int_0^\ep ds {\partial\over\partial p}
I _{\omega,s}(p)
\EQ(Omega)
$$

Since $\Omega (\omega,0,p)=\Gamma(\omega, p)$, from the hypotheses of
\clm(saddlenode) we will also assume that
$\Omega$ is non-degenerate, that is, that we have
$$
\NNN{(\partial_\omega \Omega)^{-1}}{U} \le A,  \qquad
\NNN{(\partial^2_p \Omega)^{-1}}{U}  \le B
\EQ(non-deg)
$$

The goal of the iterative
step is to determine $g_{\omega,\ep}$, $g_{\omega,0}= \Id$, in 
such a way that $\tilde f_\ep = g_{\omega,\ep}^{-1} \circ f_{\omega,\ep} 
\circ g_{\omega,\ep}$ has Hamiltonian 
$$
\tilde F_\omep (p,q) = \tilde I_\omep (p) + \tilde E_\omep(p,q)
\EQ(tildeHam)
$$
where  $\tilde I_{\omega,\ep}$, $\tilde E_{\omega,\ep}$ will be defined in 
an slightly smaller domain than $I_{\omega,\ep}$, $E_{\omega,\ep}$ and
where $\tilde E_{\omega,\ep}$ is much smaller than $E_{\omega,\ep}$ 
and $\tilde I_{\omega,\ep} - I_{\omega,\ep} $ is of the same order
of magnitude than $ E_{\omega,\ep} $ with all these functions  defined in 
an slightly smaller domain than the original ones.


According to \clm(calculus), the Hamiltonian of $g_{\omega,\ep}^{-1}\circ 
f_{\omega,\ep}\circ g_{\omega,\ep}$ is 
$$
F_{\omega,\ep}\circ g_{\omega,\ep} - G_{\omega,\ep} \circ g_{\omega,\ep} 
+ G_{\omega,\ep} \circ f_{\omega,\ep}^{-1} \circ g_{\omega,\ep}
\EQ(transformed)
$$ 
Heuristically, assuming that 
$G_{\omega,\ep}$ and
$E_{\omega,\ep}$ are small  and
of the same order---and therefore that 
$g_{\omega,\ep} - \Id$ and $f_{\omega,\ep} - i_{\omega,\ep}$
are small,
where $i_{\omega,\ep}$ is  
the integrable part of $f_{\omega,\ep}$ 
as in \equ(intpartn)---the main terms in \equ(transformed) are
$$
F_{\omega,\ep} - G_{\omega,\ep} + G_{\omega,\ep} \circ i_{\omega,\ep}^{-1}
$$ 
Hence, to make the new error $\tilde E_{\omega,\ep}$ zero in this 
linear approximation, we need to determine $G_\omep$ 
in such a way that  these main terms give just 
an integrable system (which we will call $\tilde I_\omep$). 
This is formulated as the equation for
$G_\omep$, $\tilde I_\omep$, given $F_\omep$:
$$ \tilde I_\omep(p)  = F_\omep (p,q) - G_{\omega,\ep}(p,q) 
+ G_{\omega,\ep} \circ i_\omega^{-1}(p,q)
$$

Equivalently, we look for an approximate solution of
$$
\Delta_\omep(p) = E_\omep (p,q) - G_{\omega,\ep}(p,q) 
+ G_{\omega,\ep} \circ i_\omega^{-1}(p,q)
\EQ(tosolve)$$
where
$\Delta_\omep(p) := \tilde I_\omep(p) - I_\omep(p)$.

This approximate solution
will be  used to construct a $g_{\omega,\ep}$, which will lead to a 
Hamiltonian which is much closer to integrable.

Indeed, the approximate solution of  \equ(tosolve) will be chosen as
an exact solution of
$$\Delta_\omep(p) = E_{\omega,\ep} (p,q) - G_{\omega,\ep} (p,q) 
+ G_{\omega,\ep} (p,q-\omega_0)
\EQ(tosolvesimple)$$ 
which can be solved by taking Fourier coefficients.
We will show  that, 
if we restrict ourselves to a domain
$\Sigma_{\Omega, \alpha, \tilde \beta, \tilde U}$, 
with $\alpha$ very small, the solutions of \equ(tosolvesimple) solve 
\equ(tosolve) up to errors that can be controlled by $\alpha$. 
Then, the system will be reduced very approximately to a
new integrable one. If the frequency  function 
$\Omega$ is non-degenerate, we 
can apply the implicit function theorem and 
express the domain in terms of the new 
frequency function $\tilde \Omega$. 
We call attention that it is only in this last step 
that the non-degeneracy of the frequency function is used.


To justify the above heuristic argument, we will just find the 
$g_{\omep}$ obtained by the procedure detailed 
above and estimate rigorously  the remainder after we
conjugate the original problem with it. 
This task will take  most of the present section.
We will collect all the estimates systematically 
and, at the end of the section we will formulate 
the final result precisely. 
Once we have these results, we will also need to 
estimate how the integrable part has changed and,
in particular, how much  the  folding surface $\Sigma$ 
and its parameterization $\Upsilon$ introduced in 
\equ(parameterization)  have changed.
This is the task we will undertake in the next section.
Then, in a subsequent section, we will
show that the procedure can be iterated
indefinitely (when 
some of the arbitrary choices are made 
appropriately), 
and that the transformations 
converge to a limiting transformation 
that reduces the system to integrable.



\SUBSECTION The iterative step. Estimates

In this subsection, we present detailed quantitative estimates 
for the iterative step that we described informally in the previous section.

Following standard practice, we denote by $K$ 
sufficiently large positive  constants that depend only 
on the dimension, the number $\omega_0$ 
and other elements that remain constant  during the proof
and denote by $K^{-1}$ all sufficiently small  positive
constants.
We will also need to assume that some quantities related to the integrable
part of the system remain bounded under the iteration. 
We will use $K_1,K_2$ for these constants that 
depend on the integrable part. 
The constants $K$ may depend on these $K_1,K_2$ but not 
viceversa. 
When we discuss the iteration, we will see that these 
$K_1,K_2$ are  chosen in the first step and then, they 
remain unaltered.
In particular, we will need to assume that the 
constants $A$ and $B$ that quantify the 
non-degeneracy assumptions \equ(non-deg) 
satisfy
$$
A \le K_1, \ \ B \le K_2
\EQ(hyp1)
$$


Recall that the goal was, given a Hamiltonian with an error term $E$, 
defined in a set $\Sigma_{\Omega,\alpha, \beta,U}$
of the form defined in \equ(Sigma),  perform a
transformation that  has an error term $ \tilde E$ which is much 
smaller even if defined in a smaller set 
$\Sigma_{\tilde \Omega, \tilde \alpha, \tilde \beta, \tilde U}$.

As it turns out, we will take a  number $\sigma$ and 
take $ \tilde \beta = \beta - \alpha - 4 \sigma$,
$ \tilde U = U_{4\sigma}$. At the $n$ step $\sigma_n$ will
be $\sigma_0 2^{-n}$ but $\alpha$ will have to decrease
super-exponentially.

Our goal will be to show that, under appropriate 
hypotheses, which we will assume inductively, 
we can perform the transformation and  obtain estimates of 
the form
$$
\| \tilde E\|_{  \tilde \Omega, \tilde \alpha, \tilde \beta, \tilde U}
\le K \sigma^{-\tau} \|E\|_{\Omega,\alpha,\beta,U} (
\|E\|_{\Omega,\alpha,\beta,U}  + \tilde \alpha)
\EQ(goodestimates1)
$$
for some  fixed positive number $\tau$ (we will 
show later that it suffices to take $\tau = 2 \nu + 3 $ where 
$\nu = \theta +d -1 $, and $\theta $
is the Diophantine exponent of $\omega_0$).

We will also establish that  
$\Upsilon$ and $\tilde \Upsilon$---the parameterizations
\equ(parameterization) of the
surfaces $\Omega = \omega_0$ and $ \tilde \Omega  = \omega_0$
respectively---, are defined in very similar domains and  differ by an
small amount
$$
\| \Upsilon - \tilde \Upsilon \|_{\tilde U} \le 
K \sigma^{-1} \| E\|_{\Omega, \alpha,\beta,U}
\EQ(goodestimates2)
$$

The proof  will be conveniently divided into 
two parts. In the first one, we obtain 
estimates in terms of the old domains 
parameterized by $\Omega$ and $\alpha$.
In this first part---culminated in  Lemma~3.6---we
%%fixed label
will not need to use any 
non-degeneracy hypothesis in $\Omega$ and 
indeed $\omega$ and $\ep$ will just go along for the ride. 
In a second part
of the inductive step, 
we  adjust the domains to the new frequency map. 
This part will require that we assume 
that $\Omega$ is non-degenerate and 
we will have to lose some domain in $\omega$.
This division is natural since the 
first part is exactly the same as that 
used in the proof of \clm(Greene).

\REMARK 
For the experts in KAM theory, we call attention to the 
fact that the right hand 
side of  \equ(goodestimates1) is  not  quadratic in 
$ \| E\|_{\Omega,\alpha,\beta,U}$---the size of the error.
Nevertheless, the linear term  is multiplied by the 
number $\tilde \alpha$. As we will see in the following subsection,
as $ \tilde \alpha$ goes to zero super-exponentially with the number of 
steps taken, it is possible to recover the super-exponential 
convergence of KAM theory that beats the small divisors.

As is customary in KAM theory, in order to 
be able to carry out the iterative step, we will need to
assume that certain quantities are sufficiently small with respect to
others---so that for example, compositions  have domains that match, 
implicit function theorems can be applied, etc.
As it will turn out all the conditions necessary to perform 
the iterative step will be implied  by smallness conditions 
of $\| E\|_{\Omega,\alpha,\beta,U}$ with respect to other 
quantities. Since the iterative step implies that this 
goes to zero extremely fast, the conditions will be 
recovered from one step to the next.

Hence, for the proof of \clm(saddlenode),
the main result of this subsection will be
Lemma~3.7 below, which states that, under some explicit 
%%fixed label
conditions, the iterative step can be performed and that the 
result satisfies 
\equ(goodestimates1) and \equ(goodestimates2).

Since the proof of  Lemma~3.7 will consist in 
%%fixed label
walking through the steps outlined before and 
just record the conditions needed for them to 
go through, it is natural to start with the proof of
the lemma and postpone its precise statement.


Using \clm(calculus), the Hamiltonian 
of $g_{\omega,\ep}^{-1} \circ f_{\omega,\ep} \circ g_{\omega,\ep}$---if
it is possible to define all the compositions---is
$ I_\omep\circ g_\omep
+ E_\omep \circ g_\omep - G_\omep \circ g_\omep +
G_\omep \circ f^{-1}_\omep \circ g_\omep
$, which adding and subtracting appropriate terms becomes
$$
\eqalign{
& \overline{ I_\omep \circ g_\omep }
	+ ( I_\omep \circ g_\omep - \overline{ I_\omep \circ g_\omep } )  \cr
&  + \overline{ E_\omep}  \cr
& + ( E_\omep - \overline{ E_\omep} ) +
	( E_\omep\circ g_\omep -  E_\omep )  \cr
& - G_\omep + (-G_\omep\circ g_\omep  + G_\omep)  \cr
& + G_\omep \circ T^0 + (G_\omep\circ i^{-1}_\omep - G_\omep \circ T^0)
	+  (G_\omep \circ f^{-1}_\omep \circ g_\omep - G_\omep \circ i^{-1}_\omep)
}
\EQ(breakup)
$$
where we have used the notation
$\overline{\phantom{E_\omep }}$ to indicate 
average over the $q$ variables and $T^0(p,q) = (p,q -\omega_0)$.

The main idea will be to show that it is possible to 
choose $G_\omep$ in such a way that the first terms in the last 
three lines of \equ(breakup) add to zero. That is,
$$
E_\omep - \overline{ E_\omep} - G_\omep + G_\omep \circ T^0 = 0
\EQ(simplelinear)
$$
and that this $G_\omep$ satisfies estimates which will
guarantee that the compositions we used are  indeed defined. 
(We call attention to the fact that  \equ(simplelinear)
is the linearized equation that always appears in KAM theory.)
Then, the transformed system will have an integrable part
$ \tilde I_\omep = \overline{ I_\omep \circ g_\omep} + \overline{ E_\omep}$
and the other terms appearing in \equ(breakup) will be the error
part of the new Hamiltonian. We will estimate them and show that, in 
a precise sense, they will be smaller than the other ones.

\REMARK 
For the  experts in KAM theory,
we note that this procedure has two error terms that are 
linear in $G$---and hence  first order in $E$---, namely 
$ (G_\omep\circ i^{-1}_\omep - G_\omep \circ T^0)$
and 
$( I_\omep \circ g_\omep - \overline{ I_\omep \circ g_\omep } )$---recall
that $I$ will not be converging to zero.

Even if full details will be given later, we advance that
for the first term, in the domains that we are
considering, $i^{-1}_\omep$ and $T^0$ are indeed close and the 
distance is  measured by $ \tilde \alpha$. The mean value theorem 
will give an estimate that contains the factor $\|E\|{\tilde \alpha}$
multiplied by the small divisors. This is the estimate that appears 
in one of the terms in
\equ(goodestimates1).
The second term will turn out to be quadratic because of the 
fact that $g_\omep$ is exact symplectic. This is the 
only place in all the  estimates  where we use that the maps
are exact symplectic.

As usual in KAM theory, we start by obtaining bounds
on $G_\omep$ and we will use them to obtain bounds 
on all the other terms.




\CLAIM Lemma(linearestimates1) 
For any $E_{\omega,\ep}(p,q)$ defined in $\Sigma_{\Omega,\alpha,\beta,U}$,
we can find 
unique $\Delta_\omep(p)$, $G_{\omega,\ep}(p,q)$ satisfying
$$
\displaylines{
\Delta_\omep(p) = E_\omep(p,q) 
- G_\omep(p,q) + G_\omep(p,q-\omega_0)\cr 
 \int_{\torusd } G_\omep(p,q)\,dq = 0}
$$
Moreover, these $\Delta$, $G$ satisfy 
$$
\NN{G}{U}{\beta-\sigma} \le K\sigma^{-\nu} \NN{E}{U}{\beta}, \qquad
\NN{\Delta}{U}{\beta} \le \NN{E}{U}{\beta}
\EQ(estimateslinear)
$$
where $\nu=\theta +d -1$.

\PROOF
The proof is quite standard. 
We note that integrating in $q$ we have 
$$
\Delta_\omep (p) = \overline{E_\omep}(p):=\int_{\torusd } dq \, E_\omep(p,q)
\EQ(Deltagiven)
$$
hence, the 
first estimate in \equ(estimateslinear) follows. 

If we take Fourier transforms in the variable $q$ we obtain: 
$$
\hat G_{\omep;k} (p) = {1\over (e^{- 2\pi ik\cdot \omega_0}-1) } 
\hat E_{\omep;k} (p)
\EQ(coefficientequation)
$$

By the Cauchy estimates of \clm(cauchy), we have
$|\hat E_{\omep;k}(p)|
\le K e^{- 2 \pi \beta |k|} \NN{E}{U}{\beta}$ and,
by the Diophantine assumptions,
$|e^{-2\pi ik\omega_0}-1|^{-1} \le C|k|^{\theta -1}$. 
Hence, 
$$
|\hat G_{\omep;k}(p)| \le K
|k|^{\theta -1} e^{- 2 \pi \beta |k|}  \NN{E}{U}{\beta}
$$
and,  therefore
$$
\eqalign{
\NN{G}{U}{\beta-\sigma}
& \le \sum_{k \in \integer^d} |\hat G_{\omep;k}(p)|
      e^{ 2 \pi (\beta-\sigma)|k|}
\le K \left(\sum_{k \in \integer^d} |k|^{\theta -1}
      e^{- 2 \pi \sigma |k|}\right) \NN{E}{U}{\beta}\cr
&\le K \left(\sum_{ l \in \natural} |l|^{\theta -1 + d - 1}
     e^{- 2 \pi \sigma l}\right) \NN{E}{U}{\beta}
\le K\sigma^{-\nu} \NN{E}{U}{\beta}
}
$$ 
where $\nu=\theta +d -1$.

We refer to \cite{SM} for more details but point out that it is possible 
to obtain better exponents in $\sigma$ (see e.g., \cite{Ru}). 
Of course, since the rest of 
the  proof goes through for any exponent, this does not 
affect the subsequent reasoning. 
\QED

A small generalization of these estimates is:

\CLAIM Proposition(linearestimates2)
With the notation of \clm(linearestimates1)
$$
\N{\nabla^m G}{\sigma}{\beta -\sigma} \le
K \sigma^{-\nu-m} \NN{E}{U}{\beta}
\EQ(coeffestimates)
$$


\PROOF
Using \clm(cauchy) and \equ(coefficientequation)
we obtain that, for $(\omega,\ep,p) \in U_\sigma$,
we have
$$
| \partial_p^i {\hat G}_{\omep;k}(p) | \le 
K \sigma^{-(i + \theta -1)} e^{-2 \pi |k| \beta} \NN{E}{U}{\beta}
\EQ(est1)
$$
Similarly, we have
$$
| \partial_q^j ({\hat G}_{\omep;k}(p) e^{2 \pi i k\cdot q})|
 \le K |k|^j |{\hat G}_{\omep;k}(p)| 
 \le K |k|^{j +\theta -1} e^{-2\pi |k| \beta}
\NN{E}{U}{\beta}
\EQ(est2)
$$

On $\Sg{\sigma}{\beta -\sigma}$
we have $| \Im q | \le \beta - \sigma$
and hence 
$| e^{2 \pi i k\cdot q} | \le e^{2 \pi | k|( \beta -\sigma)}$.
Therefore, using the above  estimates  \equ(est1) and
\equ(est2) in the same way as in \clm(linearestimates1), 
we obtain the desired result.
\QED

Now, we can prove estimates for the flow of $G_{\omep}$

\CLAIM Proposition(flow)
Assume that the conditions of \clm(linearestimates2)
are met and that, furthermore
$$
K \sigma^{-\nu -1} \NN{E}{U}{\beta} \le \sigma/2
\EQ(hyp3a)
$$
Then:
\item{i)}
for $(\omep,p,q)\in \Sg{2\sigma}{\beta-2\sigma}$,
the flow $g_\omep(p,q)$ generated by the Hamiltonian $G_\omep$
is well defined, and
$\left(\omep,g_\omep(p,q)\right) \in \Sg{\sigma}{\beta-\sigma}$
\item{ii)}
$ \N{g - \Id}{2\sigma}{\beta-2\sigma}
\le \N{\nabla G}{\sigma}{\beta-\sigma}
 \le K   \sigma^{-\nu -1} \NN{E}{U}{\beta}
$

\PROOF
It follows from hypothesis \equ(hyp3a), \clm(linearestimates2) and
the local existence theorem for solutions of O.D.E.'s.
\QED

 From now on, we will assume that \equ(hyp3a) holds, and we will proceed
to estimate the terms in \equ(breakup).

By \clm(flow), the compositions $G_\omep \circ g_\omep$,
$E_\omep \circ g_\omep$ are well defined on
$\Sg{2\sigma}{\beta-2\sigma}$. Using the mean value theorem
and Cauchy inequalities from \clm(cauchy),
we can bound
$$
\N{G - G\circ g}{2\sigma}{\beta-2\sigma} 
\le
\N{\nabla G}{\sigma}{\beta-\sigma}\,
\N{g - \Id}{2\sigma}{\beta-2\sigma}
 \le K \sigma^{- 2\nu -2}
\NN{E}{U}{\beta}^{2}
\EQ(term3)
$$
$$
\N{E - E\circ g}{2\sigma}{\beta-2\sigma} 
\le
\N{\nabla E}{\sigma}{\beta-\sigma}\,
\N{g - \Id}{2\sigma}{\beta-2\sigma}
\le K \sigma^{- \nu -2}
\NN{E}{U}{\beta}^{2}
\EQ(term2)
$$

These estimates  show that two of the terms in 
\equ(breakup) are quadratically small in the 
original error.

Now, we turn to estimate the last term in \equ(breakup),
which, as we will show, will also be quadratic in $\| E\|$.
The reason is that $f_\omep $ and $i_\omep $ 
satisfy differential equations whose difference can be controlled by $\|E\|$
and the same initial conditions. Hence,
$\| f^{-1} - i^{-1}  \| \le K \| E\|$
under some mild extra assumptions
that  guarantee that domains match etc., 
and we can now apply the mean value theorem.
The precise details are
a walk through the standard proof of the existence and 
uniqueness for O.D.E.'s,
as we detail below.


First, we recall that $i_{\omega,\ep}$ has the form \equ(intpartn):
$i_{\omega,\ep} (p,q) = (p,q+\Omega (\omega,\ep,p))$, 
with $\Omega(\omep,p)$ given in \equ(Omega), and we note that
$i_{\omega,\ep}^{-1} (p,q) = (p,q-\Omega (\omega,\ep,p))$. Hence, for
$$
\NNN{i-T_0}{U}=\NNN{i^{-1}-T_0}{U}=\NNN{\Omega -\omega_0}{U}\le \alpha
\EQ(alphaDef)
$$
we have
$$
(\omep,p,q)\in \Sgm{U}{\beta -\alpha}
\Longrightarrow
\left(\omep,i_\omep(p,q)\right),
\left(\omep,i_\omep^{-1}(p,q)\right)
\in \Sgm{U}{\beta}
\EQ(iDomain)
$$

Assuming
$$
\NNN{{{\partial \Omega}\over{\partial p}}}{U} \le K_3
\EQ(hyp2)
$$
(where without loss of generality, we assume,
to simplify some formulas that $K_3 > 1$), we can bound
$$
\NNN{\nabla i}{U}=\NNN{\nabla i^{-1}}{U} \le K
\EQ(DiBound)
$$

We recall now that $f_{\omega,\ep} $ is the solution of 
$$
\eqalign{
f_{\omega,\ep}(x) &=
f_{\omega,0}(x)+ \int_0^\ep ds \, \F_{\omega,s} \circ f_{\omega,s}(x) \cr
&= f_{\omega,0}(x)+ \int_0^\ep ds \,
 \left[ 
\I_{\omega,\ep} \circ f_{\omega,s} (x) +
\E_{\omega,\ep} \circ f_{\omega,s} (x)
\right]
}
\EQ(fixedpoint)
$$
while $i_{\omep}$ satisfies 
$ i_{\omep} (x)  = i_{\omega,0}(x)+
\int_0^\ep ds \, \I_{\omega,s} \circ i_{\omega,s} (x)
$, with $ f_{\omega,0}(x)= i_{\omega,0}(x)$.
By hypothesis \equ(hyp3a), using standard arguments
of O.D.E.'s based on the Gronwall inequality, we get that
for $(\omep,p,q)\in \Sg{2\sigma}{\beta-\alpha-2\sigma}$,
the flow $f_\omep(p,q)$ is well defined, and satisfies 
$$
(\omep,p,q)\in \Sg{2\sigma}{\beta-\alpha-2\sigma}
\Longrightarrow
\left(\omep,f_\omep(p,q)\right) \in \Sg{\sigma}{\beta-\sigma}
\EQ(fDomain)
$$
$$
\N{f - i}{2\sigma}{\beta-\alpha-2\sigma}
 \le e^{K_3}  \N{\nabla E}{\sigma}{\beta-\sigma}
 \le K  \sigma^{-1} \NN{E}{U}{\beta}
\EQ(f-iBound)
$$

 From \equ(DiBound), and \clm(cauchy) applied to \equ(f-iBound), we can bound
$\nabla f_\omep$:
$$
\N{\nabla f}{3\sigma}{\beta-\alpha-3\sigma}
\le \NNN{\nabla i}{U}
+ \N{\nabla (f-i)}{3\sigma}{\beta-\alpha-3\sigma}
\le K
\EQ(DfBound)
$$

Applying the Implicit Function Theorem to the estimates above,
it turns out that for $(\omep,p,q)\in \Sg{2\sigma}{\beta-\alpha-2\sigma}$,
$f_\omep ^{-1}(p,q)$ is well defined, satisfies 
$\left(\omep,f_\omep ^{-1}(p,q)\right) \in \Sg{\sigma}{\beta-\sigma}$
and
$$
\N{f^{-1} - i^{-1}}{2\sigma}{\beta-\alpha-2\sigma}
 \le K  \sigma^{-1}\| \NN{E}{U}{\beta}
\EQ(Invf-InviBound)
$$

As before, from \equ(DiBound), and \clm(cauchy) applied to
\equ(Invf-InviBound), we can bound $\nabla f^{-1}_\omep$:
$$
\N{\nabla f^{-1}}{3\sigma}{\beta-\alpha-3\sigma}
\le K
\EQ(DInvfBound)
$$

Using the mean value theorem,
\equ(DInvfBound) and the bounds on $g_\omep - \Id$
established in \clm(flow),
we obtain:
$$
\N{f^{-1} -  f^{-1} \circ g}{3\sigma}{\beta-\alpha-3\sigma}
\le K \sigma^{-\nu - 1} \NN{E}{U}{\beta}
\EQ(fgBound)
$$

Putting together \equ(Invf-InviBound) and \equ(fgBound), 
by the triangle inequality, we obtain
$$
\N{f^{-1} \circ g - i^{-1}}{3\sigma}{\beta-\alpha-3\sigma}
\le K \sigma^{-\nu - 1} \NN{E}{U}{\beta}
\EQ(fgiBound)
$$
Using the mean value theorem, the estimates  in \clm(linearestimates2)
and \equ(fgiBound), we can bound the last term in \equ(breakup) as
$$
\N{G\circ f^{-1} \circ g -G\circ i^{-1}}{3\sigma}{\beta-\alpha-3\sigma}
\le K \sigma^{-2\nu - 2} \NN{E}{U}{\beta}^{2}
\EQ(term4)
$$

Now, we turn our attention to the first term in 
\equ(breakup). It will depend on the approximate 
expression $g^0_\omep = \Id + \int_0^\ep ds \, \G_{\omega,s}$ for $g_\omep$:
$$
g^0_{\omega,\ep}(p,q) =
\left(p - \int_0^\ep ds \,  {\partial \over \partial q} G_{\omega,s}(p,q) 
\ , q +
\int_0^\ep ds \,  {\partial \over \partial p} G_{\omega,s}(p,q) \right)
\EQ(g0)
$$

\CLAIM Proposition(approxg)
Under our standing hypotheses, we have
$$
\N{g-g^0}{2\sigma}{\beta-2\sigma}
\le K \sigma^{-2\nu-3}\NN{E}{U}{\beta}^{2}
$$

\PROOF
Note that our standing assumptions imply
$$
\N{g^0-\Id}{\sigma}{\beta-\sigma}
\le \N{\nabla G}{\sigma}{\beta-\sigma}
\le K \sigma^{-\nu-1} \NN{E}{U}{\beta}
$$
and consequently
$\left(\omep,g^0_\omep (p,q)\right) \in \Sg{\sigma}{\beta-\sigma}$
for $(\omep,p,q)\in \Sg{2\sigma}{\beta-\alpha-2\sigma}$.

We can write $g_{\omega,\ep}$ as the solution of 
a fixed point problem. 
Namely,
$$
g_{\omega,\ep} = 
\Id +  \int_0^\ep ds \, 
\G_{\omega,s}\circ g_{\omega,s} \equiv \Tau(g)_{\omep}
$$
and we have the identity
$$
\Tau(g^0)_{\omep} - g^0_{\omep} = 
\int_0^\ep ds \, [ \G_{\omega,s} \circ g^0_{\omega,s} - \G_{\omega,s}]
$$
If we estimate the integrand of the R.H.S. by the mean value theorem, 
we have
$$
\N{\Tau(g^0)  - g^0}{2\sigma}{\beta-2\sigma}
\le \N{\nabla^2 G}{\sigma}{\beta-\sigma}
\,\N{g^0-\Id}{\sigma}{\beta-\sigma}
\le K \sigma^{-2\nu-3}\NN{E}{U}{\beta}^{2}
\EQ(tauerr)
$$

We also obtain, under \equ(hyp3a), that $\Tau$ is a contraction 
of factor $1/2$. Hence, there is a fixed point of 
$\Tau$ whose distance from $g^0_\omep$ is not
bigger than $ 1/(1 - 1/2) = 2$ times the 
R.H.S. of \equ(tauerr).
\QED

We note that, because $I_\omep(x)$ does not depend on $q$,
denoting by $\Pi_p, \Pi_q$ the projections  on the 
$p$ and $q$ components respectively, we have
for $x = (p,q)$
$$
\eqalign{
I_\omep(g_\omep(x)) &= I_\omep(\Pi_p g_\omep(x))\cr
&=I_\omep(p) + 
\partial_p I_\omep(p) \Pi_p\left[ g_{\omega,\ep}(x) - x\right]
+ R_2\left( \omega,\ep,x,g_{\omega,\ep}(x) \right) \cr
&= I_\omep(p) + 
\partial_p I_\omep(p) \Pi_p\left[ g^0_\omep(x) - x\right]\cr
&\qquad + \partial_p I_\omep(p) \Pi_p\left[ g_\omep(x) - g^0_\omep(x) \right]
+ R_2\left( \omega,\ep,x,g_{\omega,\ep}(x) \right)
}
\EQ(expansion)
$$
where we have denoted by $R_2$ the 
remainder of the second order Taylor expansion in $p$.

Note that  $ \Pi_p\left[ g^0_\omep(x) - x\right] = 
\partial _{q} G_{\omega,\ep}(x) $
(see \equ(g0) ) and that 
$\overline{ \partial _{q} G_{\omega,\ep} } = 0$
since $q$ is a periodic variable.
Hence, observing that $\partial_p I $ is independent of $q$,
we obtain 
$$
\overline{ \partial_p I  \int_0^\ep  ds \, 
\partial _{q} G_{\omega,s} } = 0
\EQ(cancellation)
$$
That is, the second term in the 
R.H.S. of the formula of \equ(expansion) has 
zero average. We call attention to the fact that this is the only part
in the whole proof of the estimates where we use the exact symplectic
character of the deformation, which is equivalent to the fact that $G$ is a
function on the annulus and not just on the universal cover.


Since $I_{\omega,\ep}$ depends only on 
$p$ we have that $\overline{ I_{\omega,\ep}} = I_{\omega,\ep}$.

Under the assumption
$$
\N{\nabla^2 I}{\sigma}{\beta -\sigma} \le K_4
\EQ(hyp5)
$$
we can bound the last two terms in 
\equ(expansion) by terms that are quadratic in $\|E\|$.

Since the last two terms in \equ(expansion) are 
the only ones that contribute to 
$ I_\omep\circ g_\omep  -  \overline{ I_\omep \circ g_\omep} $,
we obtain from \clm(approxg)
$$
\N{I\circ g -  \overline{I \circ g}}{2\sigma}{\beta-2\sigma} 
\le K \sigma^{-2\nu-3}\NN{E}{U}{\beta}^{2}
\EQ(term1)
$$

The only term in 
\equ(breakup) that remains to be estimated is 
$G_{\omega,\ep} \circ i^{-1}_{\omega,\ep} - G_{\omega,\ep} \circ T^0$.
We note that, by \equ(alphaDef),
we have
$$
\NNN{i^{-1} -  T^0}{U} \le \alpha
$$

Therefore, using the estimates 
in \clm(linearestimates2)
$$
\N{G \circ i^{-1} - G \circ T^0}{\beta-\alpha-\sigma}{\sigma}
\le K \sigma^{-\nu-1} \alpha \NN{E}{U}{\beta}
\EQ(term5)
$$

If we add the estimates in 
\equ(term3), \equ(term2), \equ(term4), \equ(term1)  and \equ(term5),
for the terms that has to be bounded in \equ(breakup), and claim them
only  in the domain
$\Sg{4\sigma}{\beta-\alpha-4\sigma}$,
which is smaller than any of 
the domains in which we have bounds, we obtain
$$
\N{\tilde E}{4\sigma}{\beta-\alpha-4\sigma}
\le
K \sigma^{-\tau} \NN{E}{U}{\beta}
\left(\NN{E}{U}{\beta} + \alpha \right)
\EQ(almost1)
$$
where $\tau:=2\nu+3$ and $\NNN{\Omega -\omega_0}{U}\le \alpha$.

We also notice that from \clm(flow) and \equ(fDomain), it follows that
if $(\omep,p,q)\in \Sg{4\sigma}{\beta-\alpha-4\sigma}$,
then
$\left(\omep,g_\omep^{-1}\circ f_\omep \circ g_\omep (p,q)\right)
\in \Sg{\sigma}{\beta-\sigma}$.

On the set $\Sigma_{\Omega,\alpha,\beta\alpha-4\sigma,U_{4\sigma}}$
introduced in \equ(Sigma), equation \equ(almost1) reads as
$$
\| \tilde E\|_{\Omega,\alpha,\beta-\alpha-4\sigma,U_{4\sigma}}
\le K \sigma^{-\tau} \|E\|_{\Omega,\alpha,\beta,U}
(\|E\|_{\Omega,\alpha,\beta,U}  + \alpha)
\EQ(almost2)
$$

This is very similar to the estimates desired in \equ(goodestimates1) and 
it only differs from them in the fact that the  norm in the 
L.H.S. of \equ(almost2) is referred to the domain specified by $\Omega$ 
and not by $\tilde \Omega$.

To remedy that, we will  estimate the change in $\Omega$ and 
the attendant change in the parameterizations $\Upsilon$ of the 
surface and the domain $\Sigma$. Using that the 
frequency function $\Omega$ is non-degenerate, this 
will allow us to transform the expression of the 
domain in which we have improved estimates into 
an expression involving the new frequency function. 

We will find it convenient to 
state formally what we have already accomplished 
without using non-degeneracy conditions in 
$\Omega$. 
We call attention that this lemma 
will also play an important role in 
the proof of \clm(Greene). 
Later, we will prove  Lemma~3.7 
%%fixed label
that takes into account the 
change in the frequency function
and which indeed uses the non-degeneracy
assumptions in $\Omega$.


\CLAIM Lemma(estimates2) 
Given the Hamiltonian $F=I+E$ of $f_\omep$ introduced in \equ(nstep2),
choose $G$, $\Delta$ as given by \clm(linearestimates1),
and consider the new Hamiltonian $\tilde F=\tilde I+\tilde E$
of $g^{-1}_\omep \circ f_\omep \circ g_\omep$ as given in \equ(tildeHam).
Assume that $\sigma$ is such that 
\equ(hyp3a), \equ(hyp2), and \equ(hyp5) are met, and let $\tau=2\nu+3$. 
Then
$$
\| \tilde E\|_{ \Omega,\alpha,\beta-\alpha-4\sigma,U_{4\sigma}}
\le K \sigma^{-\tau} \|E\|_{\Omega,\alpha,\beta,U}
(\|E\|_{\Omega,\alpha,\beta,U}  + \alpha)
\EQ(hestimates)
$$
$$
\|\Delta\|_{\Omega,\alpha, \beta, U} 
\le \|E \|_{\Omega,\alpha, \beta, U},\qquad 
\| \nabla \Delta \|_{\Omega, \alpha, \beta - \sigma, U_{\sigma} }
\le  K \sigma^{-1} \| E\|_{\Omega,\alpha, \beta, U}
\EQ(deltaestimates)
$$

The way of interpreting these estimates is that 
\equ(hestimates) indicates that, after the transformation, the 
resulting Hamiltonian is essentially an integrable one 
(albeit in a smaller domain):
the right hand side of \equ(hestimates) consists on  
two terms, one of which is quadratic in $\|E\|$ and the 
other one contains $\| E\| \alpha$. 
If we choose $\alpha$ sufficiently small, we will be 
able to make the right hand side of \equ(hestimates)
much smaller than the original one. This will  overcome 
the small divisors $\sigma^{-\tau}$.

We call attention to the fact that  \clm(estimates2) 
does not need the non-degeneracy assumption on 
$\Omega$ and that does not lose any domain in the 
parameters. This lemma will a basic 
tool for  the estimates of the inductive steps 
both in the proof of the KAM theorem and in the 
justification of Greene's criterion.
The difference between the two results 
will be that that the inductive steps will have
different domain loses and that 
we will have to apply them repeatedly in different 
ways, losing domain at different rates.

\SUBSECTION  The KAM inductive step. Geometry of domains


To complete the work for the  bounds of 
the inductive step in the KAM theorem, we need to study the 
change in $\Omega$, the surface $\Sigma$ defined by 
$\Omega = \omega_0$ and its natural parameterization 
$\Upsilon$
defined  in \equ(parameterization). 
In particular, we will need to provide
estimates for the changes of the bounds  in 
\equ(non-deg)  that quantify the non-degeneracy assumptions.
Since we are also taking into account the derivative of $\Omega$
with respect to $\omega$, instead of \equ(hyp2), we are going
to assume:
$$
\NNN{{{\partial \Omega}\over{\partial p}}}{U} \le K_3, \qquad
\NNN{{{\partial \Omega}\over{\partial \omega}}}{U} \le K_3
\EQ(hyp2')
$$

Again, we emphasize that most of the results in this section 
are true for arbitrary $d$.
The only exception is $iv)$ in 
Lemma~4.7 below.
%%fixed label




Given the estimates that we have on $\Delta$, it 
will be very easy to estimate the change in 
$\Omega$ and all the other estimates will follow
by an application of the implicit function theorem.
We note that since  $\Delta$ is small, and 
$\Omega$ depends linearly on the integrable part,
the change in $\Omega$ will be of the same order of 
magnitude and hence also small. All the changes
in the surface and in the parameterization will be small
and hence can be estimated by $\|E\|$ possibly multiplied 
by some factors that  come from the fact that we have to 
involve derivatives and control them by Cauchy 
estimates.


More precisely, we have:

\CLAIM Lemma(implicit)
Let $\Omega$ be the frequency 
function  \equ(Omega) for the family $f_{\omega,\ep}$ \equ(nstep2)
defined on $\Sigma_{\Omega, \alpha, \beta, U}$ as in \equ(Sigma).
Let $\Delta$ be given by \equ(Deltagiven)
and let $\sigma$ be a positive number.
Assume that 
\equ(hyp1), \equ(hyp3a), \equ(hyp2'), and \equ(hyp5), hold.
Consider $\tilde \Omega$, the 
new frequency function  defined by
$$
\tilde \Omega(\omega,\ep,p) = \Omega(\omega,\ep,p) 
+\int_0^\ep\, ds\, {\partial \over \partial p }  \Delta(\omega, s,p)
\EQ(tildeomegadef). 
$$
Denote by $\Upsilon$  and $\tilde \Upsilon$ 
the parameterizations \equ(parameterization) corresponding
to $\Omega$ and $ \tilde \Omega$.
\vskip 0 em
 Then, for any $\tilde \alpha \le \alpha$ satisfying
$$
K \sigma^{-1} \| E\|_{\Omega,\alpha,\beta,U} \le \tilde \alpha
\EQ(hyp6)
$$
we have:
\item{$i)$}
$
\left\|\Omega - \tilde \Omega\right\|_{U_\sigma} \le
K \sigma^{-1} \|E\|_{\Omega,\alpha, \beta, U} \le  \tilde \alpha
$
\item{$ii)$}  For $\tilde \alpha$ as before, 
$\tilde  \beta = \beta - \alpha - 4 \sigma$,
$\tilde U \equiv  U_{4 \sigma }$,
we have:
$$
\Sigma_{\tilde \Omega,  \tilde \alpha,  \tilde \beta, \tilde U }
\subset
\Sigma_{\Omega,  2 \alpha , \beta - 4 \sigma, U_{4 \sigma} }
$$
\item{$iii)$} 
$$
\left\|\left({\partial \over \partial \omega}  \tilde \Omega \right)^{-1}
\right\|_{\tilde U}  \le
\left\| \left({\partial \over \partial \omega}  \tilde \Omega \right)^{-1}
\right\|_{U_{4\sigma}}  
 \le \left\| \left({\partial \over \partial \omega}   \Omega \right)^{-1}
\right\|_{U} 
+ K \sigma^{-2} \|E\|_{\Omega,\alpha,\beta,U}
$$
\item{$iv)$}
When $d = 1$,
$$
\left\| \left({\partial^2 \over \partial p^2}  \tilde \Omega \right)^{-1}
\right\|_{\tilde U}  \le
\left\| \left({\partial^2 \over \partial p^2}  \tilde \Omega \right)^{-1}
\right\|_{U_{4\sigma}} 
\le
\left\| \left({\partial^2 \over \partial p^2}   \Omega \right)^{-1} \right\|
_{U} 
+ K \sigma^{-3} \|E\|_{\Omega,\alpha,\beta,U}
$$
\item{$v)$}
$
\| \Upsilon - \tilde \Upsilon \|_{ \tilde U} 
\le K \sigma^{- 1} \| E \|_{\Omega,\alpha,\beta,U}
$
\item{$vi)$} The inequalities  \equ(goodestimates1) hold. 
That is, for $\tau = 2\nu + 3$
$$
\| \tilde E\|_{  \tilde \Omega, \tilde \alpha, \tilde \beta, \tilde U}
\le K \sigma^{- \tau} \|E\|_{\Omega,\alpha,\beta,U} (
\|E\|_{\Omega,\alpha,\beta,U}  + \tilde \alpha)
$$

\PROOF
Part $i)$ follows  immediately from the formula
\equ(tildeomegadef)
for $ \tilde \Omega$  
and the estimates that 
we have for $\Delta$ in 
\clm(linearestimates1).
The last inequality in $i)$ is 
just a re-statement of \equ(hyp3a), which 
is one of the hypotheses of the lemma.

Part $ii)$ follows because of 
\equ(hyp6).


Parts $iii)$ and $iv)$ follow because we can use
Cauchy estimates to estimate the derivatives of $\Delta$.
Then, we can use Cauchy estimates to bound the 
derivatives of $\Omega$.

The existence of $\tilde \Upsilon$ and its estimates are 
a very simple consequence of the implicit function theorem.
Recall  the well known result that if an analytic function $\Phi$ 
satisfies $|\Phi(0) | \le \ep$ and 
$|\Phi'|^{-1} \le a$ on a ball around zero of 
radius $a \ep$
there is one and only one zero in  this ball.
Moreover, if $\Phi$ depends
analytically on parameters, the zero depends analytically 
on parameters.
We can apply this result to 
$\Phi(s) = \Omega( s + \Upsilon(\ep,p), \ep,p) - \omega_0$ and
then, the result follows.

Part $vi)$ is a consequence of
the estimates in \clm(estimates2) and 
part $ii)$  of this Lemma.
\QED

Notice that the only places where we had to consider derivatives with 
respect to $\omega$ are $iii)$ and $v)$. Hence, this will 
be easy to adapt to the situation 
in the justification of the Greene's criterion where there is some degeneracy
in the  frequency function.

\REMARK
Notice also that it is only in these non-degeneracy assumptions that we have
to consider the one-dimensional properties of the map. It seems that 
with some appropriate notion of critical circle in higher dimensions
(one has to consider invariant tori with `degenerate torsion'),
one could develop an analogous converging KAM process, and
a subsequent geometrical interpretation could provide the
structure of invariant objects nearby the critical torus. 


\SUBSECTION Iteration of the KAM inductive step. Convergence 

In this subsection, we verify that if we start with a sufficiently small 
perturbation $E$, the iterative step can be repeated infinitely many times  
and, moreover, converges to a solution.
The estimates are very similar to those in 
the paper \cite{Ru2} on the translated curve method. 
Along the rest of this section, we will assume that $d = 1$.

The main idea is that the loss of domain has to be
fast---say exponentially fast---in the variables 
$q$ so that we have some 
domain left. On the other hand, we have to 
decrease super-exponentially fast the variable 
$\alpha$  which  controls the thickness of 
the approximations to the surface $\Sigma$.
This will achieve that the $\| E \|$
decreases super-exponentially and that,
as a consequence, the process can be iterated 
indefinitely.

We will choose $\alpha_n, \sigma_n$, and 
show that if $\|E^0\|_{\Omega^0,\alpha_0,\beta_0,U^0}$ 
is small enough, the iterative step  
described in the previous 
section can be repeated indefinitely
and the transformations converge to a
solution that indeed solves the problem.

We point out that these smallness conditions 
can always be adjusted 
by switching to another variable 
$\ep' = \ep \lambda$. 
If we choose $\lambda$ small 
enough, the remainder is made 
arbitrarily small while all the 
other parameters in the problem are
left unaltered. (That is, when we have families, we can obtain the 
smallness conditions by considering $\ep$ restricted to 
an small domain.) Of course, when our families are
obtained by interpolating between two diffeomorphisms, 
as in \clm(interpolation), the smallness assumptions
in the family can be accomplished by assuming that the diffeomorphisms
we are interpolating are close.


We will start by picking $\sigma_n = \sigma^* 2^{-n}$,
where we pick $\sigma^* < \beta_0/8$ so 
that $\beta_n$ defined  in 
\clm(implicit)
by $\beta_{n+1} = \beta_n - 4 \sigma_n$ is 
bounded away from zero, and $\sigma^* < \delta/8$ so that
all the domains $U^{n+1}=U^n_{\sigma_n}$ contain the open domain
$U^0_{2\sigma^*}$.
Now, we will show that it is possible to choose
$\alpha_n$ in such a way that if
$\| E^0\|_{\Omega^0,\alpha_0,\beta_0,U^0}$
is small enough, the process can be iterated 
indefinitely and it converges.

Introducing the notation 
$e_n = \| E_n\|_{\Omega^n,\alpha_n,\beta_n,U^n}$,
$a_n = \alpha_{n+1}$, 
$A = 2^{\tau}$, $C =  K/\sigma^*$,
the recursion equation  in $vi)$ of \clm(implicit)
becomes
$$
e_{n+1} \le C A^n\, e_n(e_n + a_n) 
\EQ(recsimple)
$$

We claim that

\CLAIM Lemma(iterability)
If $e_0$ is small enough, 
it is possible to choose $0<\rho < 1$ 
in such a way that 
setting 
$a_n = \rho^{2^n}  (AB)^{-n}$, for $B>1$,
the conditions for \clm(implicit) 
are satisfied for all $n$ 
and 
$$
e_n \le {a_n \over C\, 2^n} = {{\rho^{2^n}}\over{C (2AB)^n}}.
\EQ(finalinequalities)
$$

\PROOF
Assume that  
\equ(finalinequalities) holds for 
a certain $n$ and that we have chosen
$a_n$ as indicated and that the iterative
step can be applied at this step.

Then,  by \equ(recsimple)
we have
$$
\eqalign{
e_{n+1} &\le {{\rho^{2^n}}\over{C (2AB)^n}}
\left({{\rho^{2^n}}\over{C (2AB)^n}}
+{{\rho^{2^n}}\over{ (AB)^n}}\right) C A^n\cr 
&= {{\rho^{2^{n+1}}}\over{C (2AB)^{n+1}}}{2ABC\over B^n}
\left({1\over C\,2^n}+1\right)
\le {{\rho^{2^{n+1}}}\over{C (2AB)^{n+1}}}{4ABC\over B^n}.
}
\EQ(iterationbounds)
$$

If $n > N_0 (A,B,C)$ we have that 
$$
{4ABC\over B^n}\le 1
\EQ(assumption1)
$$ 
so that indeed the formula
\equ(finalinequalities) holds for $n+1$.

We also observe that, if $a_n$ and $e_n$ are
of the form that we claimed, there is 
an $N_1(A,B,C) \ge N_0$ so that all the hypotheses
\equ(hyp1),
\equ(hyp3a),
\equ(hyp2'),
\equ(hyp5),
\equ(hyp6)
are satisfied for $n > N_1$.

Therefore, it suffices to ensure that 
$e_0$ is so small that the iterative 
step can be performed $N_1$ 
times and that the inequalities
\equ(finalinequalities) hold for $n \le N_1$.
Then, the argument in  \equ(iterationbounds)
will show that  \equ(finalinequalities)
continue to hold, and that 
the hypotheses needed to 
perform the iterative step
and \equ(assumption1) hold.
\QED

Clearly, from \equ(finalinequalities), we obtain that the 
error of the solution goes to zero on the surfaces.
Similarly, using the estimates in 
\clm(implicit) we can show that 
the parameterizations $\Upsilon$ of the 
surface converge. (It suffices to check that the 
increments are summable.)

Moreover, defining 
$h^n_{\omega,\ep} = g^0_{\omega,\ep} \circ \, \cdots \, \circ g^n_{\omega,\ep}$
we have that 
$$
\eqalign{
\| h^n -  h^{n-1}\|&_
{\Omega^n,\alpha_n,\beta_n,U^n} = 
\| h^{n-1} \circ g^n -  h^{n-1}\|_
{\Omega^n,\alpha_n,\beta_n,U^n} \cr
&\le \sigma_{n-1}^{-1}K 
\| h^{n-1}\|_
{\Omega^{n-1},\alpha_{n-1},\beta_{n-1},U^{n-1}}
\| g^n - \Id \|_{\Omega^n,\alpha_n,\beta_n,U^n}
}
\EQ(hchange)
$$

 From  \equ(hchange) and 
the estimates in  $ii)$ of \clm(flow), it is immediate to 
show by induction that  
$
\| h^n \|_{\Omega^n,\alpha_n,\beta_n,U^n}
$
remains bounded independently of 
$n$. Then, using $ii)$ of \clm(flow), the 
RHS of \equ(hchange) is summable in $n$. 
Hence $h^n_\omep$ converges in the limiting domain
$\Sigma_{\Omega^\infty,\alpha_\infty,\beta_\infty,U^\infty}$,
with $\alpha_\infty =0$, consisting on the points $(\omep,p,q)$ with 
$(\omep,p)\in U^\infty=U^0_{2\sigma^*}$ such that
$\Omega^\infty(\omep,p)=\omega_0$ and 
$|\Im q|\le \beta_\infty=\beta_0-2\sigma^*\ge \beta_0/2$.

This finishes the proof of \clm(saddlenode).


\SECTION Partial justification of Greene's criterion 

To assess numerically the existence of invariant circles, the most frequently 
used method is the so-called Greene's criterion, formulated in \cite{Gr} 
for two-dimensional maps. 

This criterion asserts that a smooth invariant circle with motion smoothly  
conjugate to a rotation $\omega$ exists if and only if it is possible to 
find a sequence of periodic orbits of type $m/n$ whose ``residue'' 
(that is, the trace of the derivative of
the return map minus 2) converges to zero as the 
$m/n $ converges to $\omega_0$.

As it turns out, this criterion has not been proved to hold, nevertheless, 
parts of it can be established rigorously.

For standard KAM tori, Mather (see \cite{McK} Section 1.3.2.4)
suggested a method 
to prove that if KAM tori existed, the residue should go to zero faster than 
any power of $|\omega- p_n/q_n|$. 
This method was implemented in \cite{FL,McK2} for two-dimensional 
maps to show that the residue is smaller than 
$\exp (-c|\omega- p_n/q_n|^{-\alpha})$ for some $\alpha >0$. 


The main goal of this section is to prove one of the implications of Greene's 
criterion for critical circles. 
We will prove that if a critical circle exists, then any sequence of 
periodic orbits converging to it has residual converging to zero. 
We will also show that, if a critical circle exists, indeed there is at 
least one such sequence. 
Actually, for any $m/n $ such that $m/n <\omega,  |m/n -\omega|\ll 1$, 
we can find at least 2 periodic orbits of type $m/n $ and, under
mild non-degeneracy conditions, at least $4$.

Again, we will assume in this section that
$d =1$. 
We note that for higher dimensional maps, in 
\cite{T1} and \cite{T2} there are versions of 
Greene's criterion for higher dimensional 
twist maps (a rigorous justification of 
one of the implications and numerical evidence respectively).
There are some  differences between the proofs in higher
dimensional cases and the case considered here 
of $d =1$ and we will comment on them after the proof of
our results.





The main part of the proof will consist in showing that, in a neighborhood 
of the invariant circle, it is possible to find changes of variables that 
reduce the system almost to integrable.
Once we have that, 
the result will follow word for word 
the result in \cite{FL}.

Of course, the estimates near the invariant torus are 
a more general result than that of the Greene's 
criterion and they allow to control not only the behavior of 
the periodic orbits, but also other dynamical objects.
Other papers in which similar estimates are
obtained for non-degenerate circles 
are \cite{OS,PW,JV,DG2}.


Most of the work has been done already in Section 3.
The estimates that we will use are the same as those 
of the iterative step and the only difference is 
that we will be in the iterative step that 
makes different choices. 
This unified approach between the KAM theorem and 
exponentially small estimates appears also 
in \cite{DG1}. 



\SUBSECTION Preliminary estimates and notation.

We will be considering 
area preserving maps
$f$ which are 
defined in a neighborhood of 
$[-\delta,\delta]\times \torus ^1$ to
itself. 

These maps will 
have the form
$$
f(p,q) = (p, q +  \omega_0 + \kappa p^M) + O(p^{M+1})
$$
for some $\kappa \ne 0$.

By \clm(interpolation), we can find 
an $f_\ep$ in such a way that the 
$f_0(p,q) = (p,q + \omega_0 + \kappa p^M) $.
The Hamiltonian of this deformation 
will be $F_\ep = O(p^{M+1})$.

We will write for these type of 
families $F_\ep(p,q) = I_\ep(p) + E_\ep(p,q)$,
where again $I_\ep$ will be thought of as the integrable part.
We will denote by $i_\ep$ the deformation 
with initial point $f_0$ and with Hamiltonian $I_\ep$:
$
i_\ep(p,q) =
\left(p, q + \omega_0 + \kappa p^M 
+ \int_0^\ep ds \,  \partial_p I_s(p)\right)
$.


We note that these  families are a particular case of 
the families we have considered in  Section 3.
(In particular, $\kappa <0$ and $M=2$ for the example \equ(standardmap).)
In that section, we allowed a  dependence in another parameter $\omega$.
The families we consider here can be considered as embedded in 
families depending on $\omega$ but such that the dependence 
on $\omega$ is trivial. 
Clearly, all the results of 
Section 3 that do not rely on the dependence 
on $\omega$ being non-trivial 
will  go through as stated using the elementary 
device of writing the extra variable 
$\omega$ and noticing that the 
functions we consider do not depend on 
$\omega$. We will use this completely 
elementary device without too much of 
an explicit mention.

For the purposes of this section, it will 
be sufficient to use 
particular cases of the neighborhoods 
$\Sigma_{\Omega, \alpha, \beta, U}$.
Since all the objects we will consider 
will not depend on $\omega$, we will not need to consider
objects that depend on this, in particular we can suppress
$U$ from the notation. 

We will also  introduce the simplified domains
$$\Sigma _\delta = \{(p,q,\ep)\mid |p|\le\delta,\quad 
|\Im\  q |\le\delta,\quad d(\ep,[0,1)) \le\delta \}$$ 
and, given a family of functions $H_\ep(p,q)$, we 
will denote by 
$$\| H \|_\delta = \sup_{(p,q,\ep) \in \Sigma_\delta} | H_\ep(p,q) |$$


Since we will be working with functions that vanish at the origin to a 
high order, it is worth remarking that Cauchy bounds can be improved 
for them. 

\CLAIM Proposition(Cauchy2) 
Let  $H_\ep(p,q)$ be such that $H_\ep (p,q)=p^n J_\ep (p,q)$. 
Then, provided that the norms are defined,
\smallskip
\iitem{i)} $\|J\|_\delta= \delta^{-n}\|H\|_\delta$
\smallskip\noindent 
and, for $\delta' <\delta$, we have 
\smallskip
\iitem{ii)} $\|H\|_{\delta'} \le (\delta'/\delta)^{n} \|H\|_\delta$ 
\iitem{iii)} $\|\nabla H\|_{\delta'} \le (n/\delta^{\prime }+
(\delta -\delta')^{-1}) (\delta'/\delta)^n 
\|H\|_\delta$

\PROOF 
By the maximum modulus principle 
$$
\|H\|_\delta
= \sup_{\Sigma_\delta} |H_\ep (p,q)| 
= \sup_{{\scriptstyle |p|=\delta\atop\scriptstyle |\Im\ q |\le\delta} 
	\atop \scriptstyle d(\ep,[0,1])\le\delta} 
	|H_\ep (p,q)| 
= \delta^n \sup_{\Sigma_\delta} |J_\ep (p,q)| 
= \delta^n \|J\|_\delta
$$
This proves i). 
Then, 
$$\|H\|_{\delta'} = \delta'^{n} \|J\|_{\delta'} 
\le \delta'^{n} \|J\|_\delta = (\delta'/\delta)^n \|H\|_\delta$$ 
Furthermore, 
$$\eqalign{
\|\nabla (p^n J_\ep) \|_{\delta'} 
& = \|(np^{n-1} J_\ep + p^n \partial_p J_\ep, p^n \partial_q J_\ep) 
\|_{\delta'}  
\le n\delta^{\prime(n-1)}\delta^{-n} \|H\|_\delta 
	+ \delta^{\prime n} \|\nabla J_\ep\|_{\delta'}\cr 
&\le n\delta^{\prime -1} (\delta'/\delta)^n \|H\|_\delta 
        + \delta^{\prime n} (\delta -\delta')^{-1} \|J_\ep\|_\delta\cr 
&\le (n\delta^{\prime -1}
+(\delta -\delta')^{-1}) (\delta'/\delta)^n \|H\|_\delta\cr}$$
\QED

\SUBSECTION Reduction of maps to integrable
in a neighborhood of a Diophantine circle

The key step in the proof of \clm(Greene) is the 
following. Once we prove this result, 
the proof will be the same as in \cite{FL}.

\CLAIM Lemma(reduction) 
Let $\omega_0$ be a Diophantine number, $M$ an integer.
Let $f$ be an analytic area preserving map
of the form 
$$
f(p,q) = \left(p, q + \omega_0 + \kappa p^M \right) + O\left(p^{M+1}\right) 
$$ 
for some $\kappa \ne 0$.
Then, 
\item{$i)$}
For every $N\in \natural$ we can find an analytic 
canonical transformation  such that 
$$
g_N^{-1} \circ T\circ g_N (p,q) = (p, q + \Omega_N(p) )+R_N (p,q)
\EQ(reduction)
$$ 
with $\Omega_N$  analytic,
$\Omega_N(p) =  \omega_0 + \kappa p^M + O(p^{M+1})$,
and $|R_N(p,q)| \le C_N |p|^N$. 
\vskip 0 em
\item{$ii)$}
Moreover,  we can 
find $\mu_1, \mu_2 > 0 $
depending only on $M$ and the Diophantine properties of 
$\omega_0$, such that  for sufficiently small 
$\delta$, choosing  $N = K \delta^{\mu_1}$,
we have
$$
\| R_N\|_\delta \le   K \exp( - K^{-1} \delta^{- \mu_2} )
\EQ(quantitative)
$$


\REMARK
We note that \clm(reduction), besides giving some 
control on the periodic orbits that 
we will use to prove \clm(Greene), also
provides control over other orbits. 
Notably, it shows that critical circles are
 approximated by KAM circles. 
Indeed, the density of KAM circles in a neighborhood of size $\delta$ 
of a critical circle will be bigger than $1- C_1\exp (-C_2\delta^{-\alpha})$ 
for some positive $C_1,C_2,\alpha$. 

\REMARK
We observe that the  first part of the claim, the reduction 
to an integrable form could go through with less differentiability. 
If we only want that $g_N \in {\cal C}^4$ 
(which we will show is  enough to show that the 
residue goes to zero faster than $|\omega_0 - m/n|^{N/M} $)
it would suffice to assume
that $f$ is ${\cal C}^r$ with $r$ depending on $N$ and 
the Diophantine properties of $\omega_0$. Of course, the 
quantitative estimates  \equ(quantitative) 
depend on the analyticity properties.
The first part of the claim is much easier to 
prove, since, as we will see, 
only entails matching powers of 
$p$ in an equation that expresses the desired result. 
We note that this is enough to show 
using the methods that we will develop later
that if  there is a finitely differentiable 
circle, then the residue of a periodic orbit of
type $m/n$ is smaller than 
a power of $|\omega_0 - m/n|$. 
This power can be made as large as 
we want by assuming that the differentiability is 
high enough.

%\PROOF
\medskip\noindent{\bf Proof of $i)$.\ }
If we denote by $f_0 (p,q) = (p, q + \omega_0+\kappa p^M )$, 
by \clm(interpolation) we can find an analytic family $f_\ep$ that interpolates 
between $f_0$ and $f$. 
The Hamiltonian of this family $F_\ep^0$ will be
an analytic function of $(p,q,\ep )$ in a complex 
neighborhood of $\Sigma_\delta$.

To prove that we can find 
$g_N $ so  that  \equ(reduction) holds, we proceed by induction in $N$ 
and assume that for some $N\ge 2$ we can write our Hamiltonian as
$$F_\ep^N (p,q) = I_\ep^N (p) + E_\ep^N (p,q) 
\EQ(decomposition)$$ 
with 
$$E_\ep^N (p,q) = p^N R_\ep^N (p,q)$$ 

We seek Hamiltonians $G_\ep^N(p,q) = p^N S_\ep^{N}(q)$ determined in 
such a way that the family $g_\ep^N$ with this Hamiltonian and starting 
in the identity is such that 
$$j_\ep = (g_\ep^N)^{-1}\circ f_\ep^N \circ g_\ep^N
\EQ(jdefined)$$ 
has a Hamiltonian which is integrable up to a higher order
error in $p$.

We note that
$$
g_\ep^N(p,q) = \left(p+p^N\Delta_p(p,q),  q+p^{N-1}
\Delta_q (p,q) \right)
\EQ(gexpression)
$$
where $\Delta_p,\Delta_q$ are analytic functions.
Therefore, the compositions needed to define $j_\ep$ in \equ(jdefined) 
make sense in a sufficiently small neighborhood of the circle. 

 From \clm (calculus) and \equ(gexpression),
we can compute the Hamiltonian of $j_\ep $
$$
J_\ep = I_\ep^N \circ g_\ep^N + (p+p^N\Delta_p)^N\cdot R_\ep^N \circ g_\ep 
- G_\ep\circ g_\ep^N + G_\ep^N \circ (f_\ep^N)^{-1} \circ g_\ep^N 
\EQ(hamiltonianJ)
$$ 

Expanding the above formula and denoting $R_\ep^N(p,q) = \sum_{i\ge 0} p^i 
R_\ep^{N,i}(q)$---and analogously for other functions---we obtain 
$$
J_\ep(p,q)  = I_\ep^N (p) + p^N \overline {R_\ep^{N,0} }
+ p^N \left\{\left( R_\ep^{N,0} (q) -  
 \overline {R_\ep^{N,0}} \right)
  - S_\ep^{N}(q) 
+ S_\ep^{N} (q-\omega_0)\right\} + O(p^{N+1})
$$

Using \clm(linearestimates1) we now that we can find an analytic  $S^N$ so that the 
term in braces is zero in the domain where the function 
is defined, which includes a strip around the torus.
By the form of the functions, all the compositions needed to 
define $j_\ep$ will be defined in a sufficiently 
small strip around of the torus.

This establishes the first part of the claim,
the fact that we can reduce to any order.


\REMARK 
Rather than using an inductive argument, as we have done, it is possible
to show that \equ(reduction) holds to all orders by matching terms in 
\equ(hamiltonianJ). 
We note that the terms of order $p^{N+m}$ have the form: 
$$R_\ep^{N,m} (q) - S_\ep^{N,m}(q) + S_\ep^{N,m} (q -\omega_0) +
\tilde R_\ep^{N,m-1}(p,q)$$ 
where $\tilde R_\ep^{N,m-1}$ is a polynomial expression in $R_\ep^{N,i}$, 
$S_\ep^{N,i}$, $i\le m-1$ and their derivatives and the derivatives of $I$. 
Again, we can use \clm(linearestimates1) to prove that a solution exists to 
all orders in $p^n$. 

This method clearly shows that the coefficients of 
the expansion in the reduction are uniquely determined 
by the map and the torus, and are independent of 
the procedure. For example, in \cite{OS}, a 
different procedure using generating functions is 
used for twist maps and one can find the remark
that the coefficients of this normal 
form are unique.  (For the situation 
we are considering here, generating functions 
are not so convenient since the mixed variables
are not a good system of coordinates in 
a neighborhood of the invariant torus. 
Nevertheless, the formalism that we 
developed above allows us to reach the 
same uniqueness conclusions.)

To obtain the estimates on the remainders of the reduction, 
we use  an slightly different procedure.
We use \equ(breakup) and  determine $G_\ep$ in exactly the same
way as in section 3.
We can apply \clm(estimates2)---which does not depend
on $\Omega$ being non-degenerate---to obtain, 
with the notation introduced there,
\equ(hestimates) and \equ(deltaestimates)
provided that the inductive hypothesis hold.

\CLAIM Lemma(implicit2) 
Let $\Omega$ be the frequency 
function  \equ(Omega) defined in 
$\Sigma_{\Omega, \alpha, \beta, U}$ as in \equ(Sigma).
Let $\Delta$ be defined as in \equ(Deltagiven)
and let $\sigma$ be a positive number.
Assume that 
\equ(hyp1), \equ(hyp3a), \equ(hyp2),
\equ(hyp5), hold.
Consider $\tilde \Omega$, the 
new frequency function  defined by
$$
\tilde \Omega(\omega,\ep,p) = \Omega(\omega,\ep,p) 
+\int_0^\ep\, ds\, {\partial \over \partial p }  \Delta(\omega, s,p)
\EQ(tildeomegadef). 
$$
Then, for any $\tilde \alpha \le \alpha$ satisfying
$$
K \sigma^{-1} \| E\|_{\Omega,\alpha,\beta,U} \le \tilde \alpha.
\EQ(hyp6)
$$
we have:
\item{$i)$}
$
\left\|\Omega - \tilde \Omega\right\|_{U_\sigma} \le
K \sigma^{-1} \|E\|_{\Omega,\alpha, \beta, U} \le  \tilde \alpha
$
\item{$ii)$}  For $\tilde \alpha$ as before, 
$\tilde  \beta = \beta - \alpha - 4 \sigma$,
$\tilde U \equiv  U_{4 \sigma }$,
we have:
$$
\Sigma_{\tilde \Omega,  \tilde \alpha,  \tilde \beta, \tilde U }
\subset
\Sigma_{\Omega,  2  \alpha , \beta - 4 \sigma, U_{4 \sigma} }
$$
\item{$iii)$}
$$
\left\| \left({\partial^M \over \partial p^M}  \tilde \Omega \right)^{-1}
\right\|_{\tilde U}  \le
\left\| \left({\partial^M \over \partial p^M}  \tilde \Omega \right)^{-1}
\right\|_{U_{4\sigma}} 
\le
\left\| \left({\partial^M \over \partial p^M}   \Omega \right)^{-1} \right\|
_{U} 
+ K \sigma^{-M-1} \|E\|_{\Omega,\alpha,\beta,U}
$$
\item{$iv)$} The inequalities  \equ(goodestimates1) hold. 
That is, for $\tau = 2\nu + 3$
$$
\| \tilde E\|_{  \tilde \Omega, \tilde \alpha, \tilde \beta, \tilde U}
\le K \sigma^{- \tau} \|E\|_{\Omega,\alpha,\beta,U} (
\|E\|_{\Omega,\alpha,\beta,U}  + \tilde \alpha)
$$

The  only difference between the proofs of \clm(implicit)
and \clm(implicit2) is that in \clm(implicit2) we do not need to 
worry about the non-degeneracy in $\Omega$ with respect to $\omega$.
Item $iii)$ in \clm(implicit2) is just an slight generalization 
of the standard implicit function theorem.
\QED

We also note that if
$E$ is $O(p^L)$,
then $G$ is also 
$O(p^L)$ and, as a consequence,
all the terms
in the decomposition of 
$\tilde E$ according to
\equ(breakup) are $O(p^{2L-1})$ except
$(G_\omep\circ i^{-1}_\omep - G_\omep \circ T^0)$
which is only $O(p^{L+M})$.
We note that, for high enough 
$L$, $2L -1 > L + M $ so that, 
for large enough  $L$ the order of tangency 
grows by $M$ in each step.

We can therefore assume that if 
we have performed $n$ steps, the 
resulting non-integrable part 
is $O(p^{Mn - A}) $ where $A$ is a number that 
may depend only 
on $M$ and not on $n$. The number $A$  takes into account 
that in the  first steps of the iteration 
it could happen that $2L -1$ is smaller than $L+M$.

\REMARK
One could have obtained slightly more sophisticated 
estimates taking advantage of the fact that the 
functions we are considering 
vanish with powers of $p$ and  
we can use the sharper \clm(Cauchy2) instead
of \clm(cauchy). As it turns out, this does 
not make an appreciable difference in 
the final answer and it would require
that the estimates leading to 
\clm(estimates2) are redone.

 
%\PROOF
\medskip\noindent{\bf Proof of part $ii)$ of \clm(implicit2): 
Iteration of the inductive step.}
\nobreak\smallskip\noindent

Now we discuss the possibility and the effect of 
iterating the inductive step.
Since the goals are quite different than 
in the iteration leading to the KAM theorem,
the choices that we will make in domain losses
etc. will be also quite different. 
In our case,  we are not  interested in 
having some analyticity domain left 
(the existence of an analytic torus 
is part of the assumptions); 
rather, we are interested in 
obtaining control  of the remainders 
in a wide domain.

We will take take
$$
\alpha_{n+1} = c n^{-\eta}, \qquad
\beta_n = c n^{-\gamma}
\EQ(alphabetachoices)
$$
with  $\eta$, $\gamma > 0 $  chosen in 
such a way that 
$$
(\gamma +1 )\tau  - \eta < 0 
\EQ(exponentcondition)
$$
Note that then, $ \eta > \tau \gamma \ge \gamma +1$ so that the domains 
in the $p$ variable are smaller than those in 
the $q$ variable.
Moreover,
$\sigma_n = (\beta_n - \beta_{n+1})/4 =
 c \gamma  n^{-\gamma -1} + O(n^{-\gamma -2})$
and we can bound
$\sigma_n^{-\tau} \le K n^{\tau (\gamma +1)}$.
Note also that  it also follows from 
\equ(exponentcondition) that $\eta > \tau$
and that given any $\eta > \tau$ 
we can chose $\gamma>0$ in such 
a way that \equ(exponentcondition) is 
satisfied.



We claim that if the
iterative step can be iterated  $N$ times,
and $c$ as in \equ(alphabetachoices) is sufficiently small, 
we have:
$$
\left\|E^N \right\|_{\Omega^N, \alpha_N, \beta_N, U^N}  
\le (N!)^{ (\gamma +1 )\tau - \eta }
\EQ(claimedbounds)
$$
We can proceed by induction.
Note that if \equ(claimedbounds) were 
true,  we could, for $N > N_0(c)$,
obtain the bound 
$ \| E^N \|_{\Omega^N, \alpha_N, \beta_N, U^N}
+ \alpha_{N+1} \le  K \alpha_{N+1} $.
Then, 
$$
\left\| E^{N+1}\right\|_{\Omega^{N+1}, \alpha_{N+1}, \beta_{N+1}, U^{N+1}}
\le   (N!)^{ (\gamma +1 )\tau - \eta } K c (N+1)^{ (\gamma +1 )\tau - \eta }
$$
which implies the result  for $N+1$ when $c$ is small 
enough.

We note, as in the proof of
the KAM theorem,  that all  but one of  the hypotheses of 
the iterative step are satisfied provided that 
$\| E^n\|$ is much smaller than  $\sigma_n$ to
a fixed power. 
The only condition that involves the $\alpha$
is \equ(hyp6). Namely,
$$
K \sigma^{-1} \| E\|_{\Omega,\alpha,\beta,U} \le \tilde \alpha
$$

We note that, if we fix $c$, we have the hypotheses satisfied 
for $N >  N_1 > N_0$.
If we assume that the error is sufficiently small to start
with---which can be assumed if we start in a neighborhood sufficiently
small---, then, we can  perform the $N_1$ steps 
and then, the iteration can continue. Therefore, if the initial 
error $\|E^0\|$ is sufficiently small, we can iterate indefinitely. 
Notice that since $E^0$ vanishes up to order $M$ in $p$ it 
suffices to choose $c$ sufficiently small.

Moreover, the estimates $iii)$ of \clm(implicit2) tell 
us that we can bound from below
$\left|\left( {\partial^M  \over \partial p^M} \Omega \right)^{-1}\right| $
independently of the number of iterates.
Then, the domain  $\Sigma_\delta$ is contained 
in all the domains of the form
$\Sigma_{\Omega^N, K \delta^{1/M}, K \delta^{1/M}, U^N}$
provided that $U^N$ contains a neighborhood 
of the map.

With the choices of $\alpha_n$ $\beta_n$ 
that we have made above in \equ(alphabetachoices),
we see that we can repeat the iterative 
step described in \clm(implicit2)
and obtain control in a $ 2 \delta$ neighborhood of 
the circle
while  $ c N^{-\eta} \ge K \delta^{1/M}$. 
That is, $N \le K^{-1} \delta ^{-1/(M\eta) }$. 

As we have seen in \equ(claimedbounds), 
for $N$ large enough---which is 
implied by $\delta$ small enough---we have:
$$
\|E^N \|_{\Omega^N, \alpha_N, \beta_N, U^N}  
\le (N!)^{ (\gamma +1 )\tau - \eta }
\le \exp\left( -  K^{-1} \delta ^{-1/(M\eta)}  
| \log(  \delta ^{-1/(M\eta)} )| + K\right) 
$$
By worsening slightly the power of $\delta$ in the first 
term, we can suppress the logarithm to 
simplify the expression 
$$
\|E^N \|_{\Omega^N, \alpha_N, \beta_N, U^N}  
\le (N!)^{ (\gamma +1 )\tau - \eta }
\le \exp\left( -  K^{-1} \delta ^{-1/(M\eta)- \zeta} \right) 
\EQ(errorestimates)
$$
for some small $\zeta > 0$.
Now, we note that the system $f_\omep$ is obtained by solving up to 
time $1$   the system
$$
{d \over d \ep } x = {\cal F}^N_\ep(x) = {\cal I}^N_\ep (x) + {\cal E}^N_\ep(x)
$$
Applying Cauchy bounds to  \equ(errorestimates), we 
can obtain 
bounds for 
$ {\cal E^N}$ in a $\delta$ neighborhood of 
the origin which are of the same form 
as \equ(errorestimates) with an slightly bigger $\zeta$ 
and some bigger $K$.

Note that, by definition, ${\cal I}^N$ generates an integrable flow.
Hence, applying the usual estimates for the dependence of
the solutions on the vector field, we obtain the result claimed in 
\clm(reduction).






Note that the argument we have given shows that 
we can take 
$\mu_1 = 1/M(\eta) $
and $\mu_2$ any number strictly smaller.
Since we  only needed $(\gamma +1) \tau - \eta < 0$, 
we can  choose $\eta$ any number bigger than $\tau$
and then choose $\gamma$. Of course, the constants
will be worse.
\QED

\SUBSECTION Proof of \clm(Greene) using \clm(reduction)

A possible proof can be made  following the argument in 
\cite{FL}.

We note  that \clm(Greene) makes statements 
about the trace of derivatives  of $F^n$
at fixed points of $F^n$. 
Since  the trace 
of the derivative of a map 
at a fixed point is invariant under changes of coordinates,
we can study the derivatives of this map 
in the coordinates provided by \clm(reduction).


First, we need to obtain some idea of 
where the periodic orbits could be.
We will need to show that if $|\omega_0 - m/n|$ is 
small, then the orbit is very close to 
the invariant circle so that, 
in the coordinates provided
by \clm(reduction), the orbit is 
close to being the orbit of 
an integrable system. 
Note that for the orbit of 
an integrable system, the derivative 
is  upper triangular with a diagonal
which is the identity (hence, for an integrable
system the trace of the derivative is 
$2$ and the residue is $0$). A second part of 
the argument is a perturbation argument that 
shows that if the system is close to integrable, 
the trace of the derivative is close
to $2$ and, hence, the residue is small.

The first part of the argument is  accomplished by the following

\CLAIM Proposition(location)
For $m/n$ sufficiently close to $\omega_0$,
any orbit  of type $m/n$ should be contained in
annuli of radii
$r \pm O(r^{1 + \ep})$  where $r$ satisfies
$\omega_0 + \kappa_M r^M =  m/n$.


We see that, when $M$ is odd, we 
find one such $r$, namely
$r = \left( (\omega_0 - m/n)/\kappa_M\right)^{1/M}$.
When $M$ is even, if 
$(\omega_0 - m/n)/\kappa_M$ is  positive 
we can find two such $r$, namely
$r =  \pm \left(m/n - \omega_0)/\kappa_M\right)^{1/M}$
and when  
$(m/n - \omega_0)/\kappa_M$ is  negative, 
we can find none. (In general, for each of the values of $r$
that guarantee the existence of periodic orbits, they will
appear in pairs: elliptic and hyperbolic.)

The argument will 
also show that,  when  we cannot find any $r$ solving the 
equation, there are no periodic 
orbits of type $m/n$ in a sufficiently small neighborhood
of the non-degenerate circle.

\PROOF
If we apply the first claim of \clm(reduction) to
order $2M + 2$, we obtain that, 
in an appropriate system of coordinates, 
our map can be 
written as 
$$
(p, q) \mapsto \left(p , q + \Omega(p)\right) + O\left(p^{2M+2}\right)
\EQ(systemagain)
$$
with $\Omega(p) = \kappa p^M + O(p^{M+1})$. 

In the set $  I = [ (9/10) r , (11/10) r] \times \torus ^1$,
the mapping \equ(systemagain) can be considered
as a perturbation of 
an integrable system.

We note that the frequencies present in the integrable 
system in the domain considered are 
$$
\omega_0 + \kappa r^{M} [ (9/10)^{M}, (11/10)^{M}] + O( r^{M+1}).
$$
Note also that ${d \Omega \over d p} >  \kappa M r^{M-1} +  O(r^{M})$.
This lower bound on the derivative is called the twist constant.

We recall that, by standard arguments in Diophantine approximation, 
we can find 
$\omega^*$ such that 
$ \forall \ i \in \integer ,\, \forall \ j \in \natural$, 
$|\omega^* - i/j|^{-1} \le C j^{5/4}$ in 
any  interval  of length bigger than 
$K C^{-1}$. 
(It  suffices to  fix $i,j$
and consider the length of the interval of 
$\omega$  for which the desired inequality fails. 
See e.g. \cite{AA} p. 252.) 

Hence,
we can find two frequencies $\omega_\pm$ such that
\item{a)} They are
Diophantine  with exponent $\theta -1 = 5/4$ 
and with constant $C = r^{-M}$ 
\item{b)} $\omega_- < m/n < \omega_+ $
\item{c)} $\omega_+ - \omega_- \le K r^M$

We now recall the quantitative version of 
the twist mapping theorem \cite{He} that 
states that if we perturb an integrable system
with twist constant $\sigma$
defined in a range of $A$ of diameter $D$,
by a perturbation of ${\cal C}^4$ size $\rho$, 
the invariant circles corresponding to 
a Diophantine frequency of 
constant $C$ persist 
provided that $C^2 \rho $ 
$\rho \sigma^{-1}/D$ are sufficiently 
small. Moreover, 
${\cal C}^1$ distance of these invariant tori
to the unperturbed ones can be bound by $\rho\sigma^{-1}$.

If we apply this to the circles 
of frequencies $\omega_\pm$ in the domain indicated,
we see 
that $\rho = O(r^{2M +2}) $, $C = O(R^{-M})$,
$\sigma^{-1} = O(r^{-M+1} )$, and $D \ge 2/10 r$.

Hence, we conclude that these circles with 
frequency $\omega_\pm$ persist. 
Since in a sufficiently small neighborhood of
the invariant circle, the map is a 
twist map, all the orbits with rotation 
number in $[\omega_-,\omega_+]$ have 
to be contained in the annulus bounded by these
two invariant circles. In particular
those of rotation number $m/n$.

This finishes the proof of \clm(location).
\QED


For the cases where  we can find 
an $r$ such that the rotation number of
the integrable part is $m/n$,
we can apply   
\clm(reduction) 
with $\delta =  2 r$ 
with $r$ as above to 
obtain that $\| R_N \|_\delta$
vanishes to order 
$  K^{-1} |\omega - m/n|^{-\mu_1/M} $
and has size smaller than 
$K \exp( - K^{-1}  |\omega - m/n|^{-\mu_2/M} )$.


The improved Cauchy estimates, 
\clm(Cauchy2),  give us 
that  the entries on the matrix 
$DR$ are 
smaller than 
$$\eqalign{
2^{  - K^{-1} |\omega_0 - m/n|^{-\mu_1/M} }
K \exp&\left( - K^{-1}  |\omega_0 - m/n|^{-\mu_2/M} \right)\cr 
&\le K \exp\left( - K^{-1}  |\omega_0 - m/n|^{-\mu_3} \right)}
\EQ(derivbounds)
$$
for some $\mu_3 > 0$.

We also note that the  derivatives of the integrable 
part are of the form 
$DI =\left( {1\atop 0}\ {a\atop 1}\right)$
with $a$ bounded independently of 
the number of iterates 
that we need to take  in \clm(reduction).

If we  have a periodic orbit of type 
$m/n$, 
by the chain rule we have 
$DF^n(x) = DF(x_{n-1}) \cdots DF(x)$,
where $x_i = F^i(x)$. 
Note that $DF(x_i) = DI(x_i) + DR(x_i)$.

Therefore, we can apply 
the following lemma, 
which appears as
Lemma~3.4 of \cite{FL}.

\CLAIM Lemma(perturbation)
Let $\{ A_i\}_{i=1}^N $ be a set of $2\times 2$ matrices of the
form $A_i =\left( {1\atop 0}\ {a_i\atop 1}\right)$ with $\sup_{1\le i\le N}
|a_i|\le A$.
\vskip1pt
Let $\{ B_i\}_{i=1}^N$ satisfy
$$\sup_{\scriptstyle 1\le i\le N\atop\scriptstyle j,k=1,2} |(B_i)_{jk} -
(A_i)_{jk}|\le\varep \quad \hbox{\rm  with }\varep\le A\ .$$
Then $B= B_1 \cdots B_N$ satisfies
$$|\Tr B-2|\le 2\left[\bigl( 1+3\sqrt{A}\,\sqrt{\varep}\,\bigr)^N -1\right]$$

Applying \clm(perturbation) with  $A_i =  DI(x_i)$, 
$B_i = DF(x_i)$, we obtain that  for sufficiently large 
$n$, recalling that 
\clm(Greene) includes in the assumptions that 
$|\omega_0 - m/n | \le 1/n $ and that 
therefore 
$ K  \exp(- K^{-1} |\omega_0 - m/n|^{\mu_3} ) $ tends to zero
$$
\eqalign{
|\Tr DF^n(x) - 2 | &\le 
2\left[\bigl( 1+ K  \exp(- K^{-1} |\omega_0 - m/n|^{\mu_3} )  \bigr)^n -1\right] 
\cr
& \le 
n K  \exp(- K^{-1} |\omega_0 - m/n|^{\mu_3} ) \le 
 K  \exp(- K^{-1} |\omega_0 - m/n|^{\mu_4} ) 
}
$$

This concludes the proof of \clm(Greene).
\QED


We also remark that the argument that we gave to 
locate the periodic orbits also shows 
that if we have 
a non-degenerate critical circle, then
it is approximated by periodic orbits.

In the cases that we can find an approximate $r$
(i.e. in the case of odd $M$ or, when $M$ is 
even, that the sign of  $\omega_0 - m/n$
is chosen correctly) 
we see that we can apply   
the classical 
Poincar\'e last geometric theorem
\cite{Fr} 
to $F^n  - (0,m)$ and obtain 
that there are two fixed points of 
different index, and, hence
two different periodic orbits of $F$.

In the case that $M$ is even and the 
signs are right, since 
we can find two rings 
we can obtain  four periodic orbits. 

\REMARK 
Note that in order to obtain two periodic
orbits using this argument, we need to 
use the modern version of the Poincar\'e theorem
which includes information about the index of 
the fixed points  of  $F^n  - (0,m)$.
The classical Poincar\'e fixed point theorem 
(See e.g \cite{St} p. 195 does not provide
information about the index and hence, 
we cannot exclude that the two fixed points 
of  $F^n  - (0,m)$ produced by it are 
part of the same orbit for $F$.

\REMARK
In our case, noting that our maps
admit a generating function, we could also
produce the two periodic orbits using 
variational methods. 
(See  \cite{KH} theorem 9.3.7.)


A different line of argument that produces 
quantitative results under stronger hypotheses is 
the following:

In a annulus $p \in [ r - r^{1+\ep}, r + r^{1 +\ep}]$
the map is an small perturbation of an integrable map
that is non-degenerate.
If this perturbation satisfies some
non-degeneracy assumptions,
one can find  two periodic orbits of
type $m/n$. One of them is hyperbolic 
an another one is elliptic.
The first order calculations of 
these periodic orbits is sometimes  called 
sub-harmonic Melnikov theory. 
Formal expansions, including 
non-degeneracy assumptions  that imply that
the expansions predict one pair of 
elliptic and hyperbolic periodic orbits
can be found in \cite{Po} \S 74, \S 79.
A justification of these expansions  for 
finitely differentiable functions
that shows that, under the formal 
conditions derived in \cite{Po} 
one can find indeed the periodic orbits 
with the character predicted by the expansions 
can be found in \cite{LW} chapter 2,
or in \cite{Po} \S 39.

\REMARK
Note that the above argument only requires estimates about the trace of the 
derivative.  The fact that the trace of the derivative can be studied 
requires that $g_N \in {\cal C}^1$. 
The argument that we used to show that, in the coordinates given by $g_N$, 
the periodic orbit of period $m/n$ is at a distance 
not more that $|\omega_0 - m/n|^{1/M}$ requires the twist mapping theorem 
with Lipschitz estimates and hence that $g_N \in {\cal C}^4$.
The rest of the argument
applying \clm(perturbation) only requires that the   map $g_N \in {\cal C}^1$.
Hence we see that if $g_N \in {\cal C}^4$ we have that 
$|\Res (O_{m,n})| \le K | \omega_0 - m/n|^{N/M}$.
Therefore, as we remarked before, to show that the residue 
goes to zero faster than a power, one only needs finite differentiability and 
for ${\cal C}^\infty$ mappings one can show that
the residue goes to zero faster than
any power.

\REMARK
For a Diophantine number \equ(Diophantineeq), it holds that
$|\omega_0 - p_n/q_n|^{-1} \ge C q_n^{\tau}$ for 
some $\tau \ge 2$, if
we take $p_n/q_n$ to be the convergents of
the continued fraction expansion of $\omega_0$.
Hence, the conclusion of  \clm(Greene) can be written as
$$\Res (O_n) \le C_1 \exp (-C_2q_n^{\mu'})$$

\REMARK
A followup paper \cite{CGM2} of \cite{CGM1}
goes on to find scaling relations for the
invariant circles with rotation number $\omega_0=(\sqrt5 -1)/2$ of 
$T_{\omega(\ep),\ep}$ as $\ep$ goes to a critical value where they cease to 
exist.
These scaling relations suggest that there is a 
renormalization group description of these invariant circles
with the KAM circles corresponding to a trivial 
fixed point.
If this was the case (to our knowledge nobody has yet 
worked out a precise formulation and  computed the
trivial  fixed points), the residue 
of a periodic orbit of 
type $F_n/F_{n+1}$ would go to zero 
super-exponentially  fast in $n$, since for the Fibonacci numbers
$F_0 = F_1 = 1$, $F_{n+1} = F_n + F_{n-1}$, one has
$|\omega_0 - F_n/F_{n+1} |^{-1} \approx C \omega_0^{-2n}$.


\REMARK 
In higher dimensions, under 
the non-degeneracy hypotheses of the KAM theorem---which are  weaker than 
twist hypothesis---an argument similar to the one 
given above has been developed in \cite{T1}.
The reduction to integrable normal
form up to a very small error can  be carried out. Similarly, there is
an analogue of \clm(perturbation) that shows that 
products of sufficiently small perturbations of 
Jordan Blocks with identity in the diagonal, still have 
characteristic polynomials 
close to  $(t-1)^{2d}$. Therefore, if there is a periodic orbit in 
a neighborhood of the torus, not only the trace but all the other 
coefficients of the characteristic polynomial have to converge to 
those of the Jordan normal form. 
One important element from our present 
argument  that does not generalize to higher dimensions is the application of 
the twist mapping theorem to conclude that the distance of the periodic orbits
to the invariant circle is bounded by the difference of the rotation numbers.
Nevertheless, it is possible to show that if there is an invariant torus, there 
are periodic orbits that approximate it well and that the characteristic 
polynomial of the derivative converges to $(t -1)^{2d}$.
It has been argued---and implemented numerically
in \cite{T2}---that this convergence of
the coefficients of the characteristic 
polynomial of the derivative  can considered as a test of 
the presence of a KAM torus.

We think that it should be possible to extend the methods presented here to 
establish  one  of the implications of Greene's criterion  for some 
invariant torus that satisfy some hypothesis of non-degeneracy 
weaker than the twist hypothesis.




\SECTION  Acknowledgements

We thank D. Del Castillo and P. Morrison for 
many discussions about their work and for encouragement.
We also thank A. Haro and C. Sim\'o for helpful discussions and
for making their unpublished results available to us.
This work has been partially supported by the NATO grant CRG950273.
Research by A.D. is also supported by
EC grant ERBCHRXCT-940460, the Spanish grant
DGICYT PB94-0215 and the Catalan grant CIRIT 1996SGR--00105.
Research by R.L. is also supported by NSF grants.
We also thank TICAM, UPC  and IMA for invitations that made possible
this collaboration.


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%\note{translated from Zh. \`Eksper. Teoret. Fiz., 91:500--516, 1986

\endref
\end
