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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 that violate the twist condition in a very
strong sense: $\partial q'/\partial p$ changes sign).
These maps appear in a variety of applications,
notably transport in atmospheric Rossby waves.
We show in suitable 2-parameter families the persistence of
critical circles
(invariant circles whose rotation number is the maximum of all the rotation
numbers of points in the map) with Diophantine rotation number.
The parameter values with critical circles of
frequency $\omega_0$ lie on a one dimensional analytic curve.
In contrast with recent progress in KAM theorem with degeneration
in the frequency map, the curves we consider have rotation numbers on
the boundary of the range of the frequency.
Furthermore, we show a partial justification of Greene's criterion:
If critical analytic critical curves
with Diophantine rotation number $\omega_0$
exist, the residue of periodic orbits 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 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
The goal of this paper is to provide rigorous proofs of several
phenomena discovered empirically in \cite{CGM1}.
In that paper, the authors consider the two parameter family of
area preserving maps, called there the ``quadratic standard map''.
$$T_{\omega,\ep} (p,q) = \left(p+\ep \sin (2\pi q) ,q -(p+\ep \sin 2\pi q)^2 +
\omega \pmod{1}\right).
\EQ(standardmap)$$
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 map $T_{\omega,0}$ describes a situation
where particles in a fluid are moving in a laminar flow
whose velocity is faster in the middle 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.
The map $T_{\omega,0}$ is an integrable map, since the
curves $ p = {\rm cte}$ are preserved and the motion in them
is a rigid rotation. 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 extra term modified by $\ep$ is representative of the
maps that arise when one considers the
physical effect of a
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, the invariant circles are
complete barriers for the mixing of the material
in the pole --- one of the edges of $p$ --- with the material
near the equator --- the other edge of $p$.
This barriers are important in the creation of
``ozone holes'' since they isolate the ozone
created in the tropics from the regions
near the poles.
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 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 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.
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 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)).
Moreover, they also find numerical evidence for:
\CLAIM Claim(claimtwo)
Greene's criterion applies.
We recall that the Greene's criterion, introduced in
\cite{Gr}, asserts that
there exists a smooth 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 $\omega_0$.
More interestingly, a followup paper \cite{CGM2} goes on to find scaling relations for the
invariant circles with rotation number $\omega_0$ of
$T_{\omega(\ep),\ep}$ as $\ep$ goes to a critical value where they cease to
exist. Unfortunately, we do not have any rigorous result for this
non-perturbative regime and, hence, we just call
attention to its existence.
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.
The converse (that is, if the residue
goes to zero, one can find a periodic orbit),
to our knowledge, 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 a KAM 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.
We also point out that, for the maps $T_{\omep}$ as in
\equ(standardmap), the
criterion can be implemented numerically very efficiently. These maps
are reversible and, for reversible maps, the
search for periodic orbits in some critical lines --- not all of
the periodic orbits of the map --- can be reduced to finding zeros of
one dimensional functions, a tractable numerical task. Moreover, it
is possible to show that if there are critical circles, it should be
possible to find periodic orbits in these critical lines converging to
them. We will return to these questions later, in Proposition~4.4.
%%fixed label
About the method of proof we note that there are
two basic methods in KAM theory. One is
based on applying successive transformations 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 the 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.
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 a 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, (that is,
$|k \cdot\omega_0 -m|^{-1} \le C |k|^{\theta -1},\forall k \in \integer ,
m\in \integer \setminus \{0\}$),
$f_{\omega,\ep}$ 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
$\Gamma$ 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 $\Gamma$ 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 one or two periodic orbits.
For more general non-degenerate circles,
when $M$ is even we will find
one or two 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 one
irrespective of the sign or $\omega - p_n/q_n$.
\REMARK
For a Diophantine number,
$|\omega_0 - p_n/q_n|^{-1} \ge C q_n^{\tau}$ 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
The proof that the residue goes to zero faster than any power is
significantly easier than the proof with an explicit rate.
\REMARK
We note that the paper \cite{CGM2} finds
scaling relations that suggest that there is a
renormalization group description of these invariant circles
in the case of a golden mean rotation number
$\omega_0 = (\sqrt5 -1)/2$,
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
$|\omega_0 - F_n/F_{n+1} |^{-1} \approx C \omega_0^{-2n}$.
\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
KAM theory for maps. (The definition that appears naturally in
the 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
hypothesis, 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 ---
%%fixed label
we 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 restatement of \equ(hyp3a), which
is one of the hypothesis 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.
Again, 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
$n/m $ 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 $p/q$ such that $p/q<\omega, |p/q-\omega|\ll 1$,
we can find at least 2 periodic orbits of type $p/q$ 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 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 a symplectic and 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 $A$ 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 hypothesis 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 $5/4$
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
Poincar\'e last geometric theorem
to $F^n - (0,m)$ and obtain
one periodic orbit.
In the case that $M$ is even and the
signs are right, since
we can find two rings
we can obtain two periodic orbits.
A different line of argument that produces
sharper results under stronger hypothesis 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
subharmonic 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} \$ 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
In higher dimensions, under
the non-degeneracy hypothesis of 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 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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