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%
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%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def \cite#1{{[#1]}}
%*************** TO GET SMALLER FONT FAMILIES *****************
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def \cite#1{{[#1]}}
\def\cite#1{{\rm [#1]}}
\def\bref#1{{\rm [~\enspace~]}} % blank ref cite
\def\degree{\mathop{\rm degree}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\bT{{\bf T}}
\def\frac#1#2{{#1\over#2}}
\def\Norm{{\~\enspace~\}}
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\TITLE On necessary and sufficient conditions for uniform
integrability of families of Hamiltonian systems.
\AUTHOR Rafael de la Llave
\FROM Department of Mathematics
University of Texas at Austin
Austin TX 787121082
\ENDTITLE
\vskip 3 em
\leftline{\bf To the memory of Ricardo Ma\~n\'e}
\ABSTRACT
In \cite{Po} Ch. V, specially \S 81, H. Poincar\'e
discussed an obstruction to uniform
integrability of families of Hamiltonians. (That is, the
existence of changes of variables analytic in
the parameter $\epsilon$ and in the variables that
make the family of Hamiltonians a function of only
action variables).
We examine his proof and discover that,
for nondegenerate systems, this condition is
also sufficient for the integrability
to first order in the parameter
(That is, there exist analytical
changes of variables, analytic
in $\epsilon$ so that the
family in these new variables depends
only on the action variables up
terms which are $o(\epsilon)$.) This leads
to the existence of obstructions in higher
order. We show that the vanishing of the obstructions
to order $n$ is sufficient for the existence of
analytic and symplectic
changes of variables analytic in the
parameter $\epsilon$ that reduce the system to
integrable
up to errors of order $\epsilon^{n+1}$. Moreover, we
show that the vanishing of all the
obstructions means that the
system is uniformly integrable.
This answers the question posed
by Poincar\'e at the end of his chapter V.
We note that these obstructions
have a geometric meaning and
they are cohomology obstructions
computed on periodic orbits of the
system.
\ENDABSTRACT
\SECTION Introduction
Many problems in Mechanics can be reduced to the study of a family of
Hamiltonian systems $H_\epsilon$ where $H_0$ is an integrable system.
(By an integrable system we mean one whose phase space admits
actionangle variables and such that the Hamiltonian depends only on the
actions.)
At the early times in Mechanics  and more incomprehensively, up to
recently  there was widespread hope that almost all
families
$H_\epsilon$ could be systematically reduced to integrable by a
canonical change of variable. (Found, e.g., solving the HamiltonianJacobi
equations.)
That is, there was hope that one could find a family of canonical
transformations $g_\epsilon$ in such a way that
$$H_\epsilon \circ g_\epsilon = I_\epsilon
\EQ(integrability)$$
where $I_\epsilon$ is only a function of the actions.
The fact that these hopes can not be fulfilled
 even in the very weak sense of
formal power series  was demonstrated very
eloquently in \cite{Po}. The whole of
chapter V is devoted
to finding obstructions
for uniform integrability. In particular,
in \S81 one can find obstructions for the
existence of analytic solutions of \equ(integrability) that depend
analytically on $\epsilon$ (uniform integrability) and in \S84, the
conditions are extended slightly and
verified for the threebody problem.
At the end of \cite{Po} Chapter V (p. 259) the question is raised of
whether these obstructions are sufficient.
Of course, the more recent K.~A.~M. theorem shows that provided
some nondegeneracy conditions are met, one can find solutions of
\equ(integrability) but only on a Cantor set.
The goal of this paper is to reexamine in more modern language the
circle of ideas introduced in \cite{Po}.
We will show that the criterion of nonintegrability introduced in
\cite{Po} can be strengthened and that this strengthening is necessary
and sufficient for uniform integrability when the unperturbed
system is nondegenerate.
We point out that a thorough analysis of the degenerate
situation was undertaken in \cite{Ga}. There, the situation
is quite different and there are examples with systems integrable to all
orders but not uniformly integrable. In \cite{Ga}, one can also
find sufficient conditions for convergence of perturbation
expansions of some degenerate situations.
The obstructions to integrability we will find are integrals over periodic
orbits. They are equivalent to those of \cite{Po} even
if they are usually stated in terms of
Fourier series coefficients.
We try to write everything in terms of
periodic orbits because the geometric meaning is
clearer. In particular, the obstructions can
be interpreted as cohomology obstructions for
existence of cocycles.
The main technical tool to prove the converse of the criterion will be a
result that states that if the obstructions vanish on periodic orbits, a
cohomology equation has a solution.
This is somewhat reminiscent of the celebrated Livsic theorem \cite{Li}
and its use in rigidity results for Anosov diffeomorphisms.
Of course, in our case there is no hyperbolic behavior, hence, the
proofs are very different. In that respect we call attention to
\cite{Ve}, where in is shown that cohomology equations
for linear maps on the torus can be solved using
only obstructions on periodic orbits. The methods, however
are very algebraic and do not generalize to our situations.
We will prove two types of results:
some that have to do with asymptotic expansions and their feasibility and some
more complicated ones that discuss the convergence of these expansions.
Even if the obstructions and asymptotic results need only a finite number
of derivations and simple minded calculations,
the converse of the full set of obstructions 
that is, the theorem that shows
that if all the obstructions vanish, the system is uniformly integrable 
does require analyticity and, as far as we know,
an iterative process similar to those K.~A.~M.
theorem.
The same circle of ideas applies to other related problems that have
appeared in the literature.
We will also discuss the problem of relative integrability.
That is, the existence of a
family of canonical transformations
$g_\epsilon$ and a function of a real variable
$F_\epsilon$ such that:
$$H_\epsilon \circ g_\epsilon = F_\epsilon (H_0)
\EQ(relativeintegrability)$$
This means that $H_\epsilon$ describes the same motions as $H_0$ up to
a change of variables and a reparametrization of the orbits. That is,
the orbits of $H_\epsilon$ are the same as that of $H_0$ under
a change of variables, even if they may
be transversed at different speed. The change of
speed depends only on the energy surface.
We emphasize that the results we present here deal
exclusively with the problem of uniform integrability.
In particular, we do not discuss
related problems such as the integrability for open
domains of the parameter. We also call attention to the
fact that we use integrability in a well
defined sense that involves changes of variables that
are analytic and with analytic inverses.
The literature abounds with weaker notions
(multivalued integrals, integrals with singularities etc.) which are
sometimes also referred as integrability.
We have little to say in this paper about these
notions. It is clear that our methods rely on our
precise definitions. When to avoid cluttering the sentences
we just refer to ``integrability'' we mean one of
our precise definitions that we hope, will be clear from the context.
\SECTION Definitions, preliminary calculations
Even if several of the calculations we will perform will require only a
finite number of derivatives, the main results indeed require analyticity.
Hence, we will assume
that all the Hamiltonians are analytic in all the parameters.
We will consider a phase space of the form $M= U\times \torus^d$ where $U$
is an open set in $\real^d$. (We will also use
complexifications of this phase space.)
Given a function $\eta :U\times \torus^d\to\complex$ we will denote the partial
Fourier expansion as
$$\eta (A,\varphi) = \sum_{k\in \integer^d} \hat\eta_k (A)
e^{2\pi ik\varphi}.$$
We denote the symplectic form by $\gamma$ (and reserve $\omega$ for the
frequencies).
In the actionangle variables we have
$\gamma = \sum_{i=1}^d dA_i\wedge d\varphi_i$.
Note that if the phase space admits actionangle variables
we have $\gamma = d\theta$
where $\theta = \sum_i A_i d\varphi_i$. We will refer
to $\theta$ as the symplectic potential.
(Not all symplectic manifolds have a symplectic
potential, but we are only concerned in this paper with those
having action angle variables as above.)
A transformation $f$ is called symplectic
when $f_* \gamma = \gamma$. Equivalently, if
$d ( f_* \theta \theta) = 0$. If, furthermore,
we have
$ f_* \theta \theta = dS $, we say that $f$ is
exact symplectic. Note that the notion of exact symplectic
maps only makes sense for symplectic manifolds in which
the symplectic form admits a potential.
We say that a Hamiltonian is integrable if it
is a function only of the action
variables. Perhaps it would me more appropriate to
say that a system is integrable when it can be
reduced to one of the above form by a canonical
change of variables. However, the
above notation is quite extended and we will follow it.
We will also be concerned with integrable systems that are nondegenerate
in the following sense:
\CLAIM Definition(nondegenerate)
We say that a Hamiltonian $H_0: M\to\real$ is non degenerate if the mapping
$\omega :U\to \real^d$ defined by
$\omega= {\partial H_0\over\partial A}$
is a diffeomorphism in its range and
$\omega^{1} :\omega (U) \to U$
is a uniformly analytic map.
We will find it convenient to use the deformation calculus for families.
If $\G_\epsilon$ is a family of diffeomorphisms we write:
$${d\over d\epsilon} g_\epsilon = \G_\epsilon \circ g_\epsilon$$
and we will refer to $\G_\epsilon$ as the generator of the family.
Note that, for $C^1$ vector fields, $\G_\epsilon$ and $g_0$ determine
uniquely $g_\epsilon$.
The transformations are symplectic if and only if
${g_\epsilon}_* \gamma =\gamma$.
This happens if and only if
${g_0}_*\gamma=\gamma\quad\hbox{and}\quad
{d\over d\epsilon} ({g_\epsilon}_* \gamma) =0\ .$
Using the formulas for the Lie derivative and the fact that $d\gamma=0$,
we see that a family is symplectic if and only if
%$$\eqalign{
%{g_0}_* \gamma &= \gamma\cr
%d(i(\G_\epsilon)\gamma) & = 0\cr}$$
$$
{g_0}_* \gamma = \gamma; \ \
d(i(\G_\epsilon)\gamma) = 0
$$
This motivates the following well known definitions:
\CLAIM Definition(symplecticvector)
We say that a $C^1$ vector field $\F$ is locally Hamiltonian if and only if
$$d(i(\F)\gamma) =0$$
We say that a $C^1$ vector field $\F$ is globally Hamiltonian if there exist
a function $F$ such that
$$i(\F)\gamma = dF$$
Similarly, we say that $g_\epsilon$ is a locally (globally) Hamiltonian
isotopy if its generator $\G_\epsilon$ is locally (globally) Hamiltonian.
It is a well known result that $g_\epsilon$ is a family of
symplectic mappings if and only if $g_0$ is symplectic and is
a locally Hamiltonian isotopy. Similarly, it is a family
of exact symplectic mappings if $g_0$ is exact symplectic and
$g_\epsilon$ is a globally Hamiltonian isotopy.
Note that given a function $F$, we can compute $dF$ and, because $\gamma$
is nondegenerate, we can find a unique $\F$ such that
$i(\F) \gamma = dF$.
Conversely, given $\F$, $i(\F) \gamma = dF$ determines $F$ up to
an additive constant on every connected component.
\CLAIM Definition(uniformintegrable)
Let $H_0$ be an integrable Hamiltonian, $H_\epsilon$ be an
analytic family agreeing
with $H_0$ when $\epsilon=0$.
We say that $H_\epsilon$ is a uniformly integrable family
if we can find an analytic family
of canonical transformations $g_\epsilon$ and a family of integrable
Hamiltonians $I_\epsilon$ such that
$$\eqalign{
&H_0 = I_0\ ,\qquad g_0 = Id\cr
&H_\epsilon \circ g_\epsilon = I_\epsilon\cr}
\EQ(integrability)$$
In the literature, one sometimes finds the
notion of relatively integrable family.
\CLAIM Definition (relativeintegrable)
We say that $H_\epsilon$ is $H_0$integrable if we can find a family of
canonical transformations $g_\epsilon$ and a $C^r$family of real valued
functions of a real variable $F_\epsilon$ in such a way that
$$\eqalign{
&g_0 = Id\ ,\quad F_0 = Id\cr
&H_\epsilon \circ g_\epsilon = F_\epsilon (H_0)\cr}
\EQ(relativeintegrability)$$
We will refer to the $g_\epsilon$ as the integrating transformations.
The reason to introduce the concept of relative integrability is that if
\equ(relativeintegrability) is met, we can understand
the dynamics of all the Hamiltonians $H_\epsilon$ in terms of the dynamics
of $H_0$.
This is advantageous when the dynamics of $H_0$ is ``well understood.''
Of course, when $H_0$ is integrable, the dynamics is well understood
because of the explicit solution.
Nevertheless, a system may have dynamics for which a great deal is
known even if there are no explicit formulas.
Note also that the problem of relative integrability is geometrically
more natural than the problem of integrability.
It does not use the fact that there exist action and angle variables
and, hence, it makes sense to pose the problem in any manifold.
In particular, there are less stringent notions of
actionangle variables (see e.g. \cite{Du}) where much of
the ideas here could carry over.
In this paper, however, we will only consider the problem of relative
integrability with respect to an integrable system.
The papers \cite{CEG} and \cite{LMM}
consider the problem of relative integrability of Anosov systems,
which certainly do not have action angle variables.
In the relatively integrable case, we will use some more
geometric structures. Note that for $E$ real  or complex 
we can define the energy surface $\Sigma_E = H_0^{1}(E)$.
Recall also that the symplectic form $\gamma$ defines
a volume form $\gamma^d$.
If the energy surface does not
contain critical points of
the Hamiltonian, we can introduce a natural
$2d 1$ form
$\mu$ on the energy surface by $\gamma^d = d \mu_E \wedge d H_0$.
Since both $H_0$ and $\gamma$ are
invariant under the flow of
$H_0$, so is $\mu$. We will assume that the Hamiltonians we consider
are defined on domains $H_0^{1} (\Omega)$
where $\Omega$ is an interval in $\real$ or
a complex domain containing a real interval.
We will furthermore assume that there are no critical
points in this set and, moreover that the
energy surfaces are compact for real values of the energy.
Note that the $I_\epsilon, g_\epsilon$ that integrate a family of
Hamiltonians are not unique.
In effect, if $h_\epsilon (A,\varphi) = (A+\Delta_\epsilon,\varphi)$
then,
$$H_\epsilon \circ g_\epsilon \circ h_\epsilon
= I_\epsilon \circ h_\epsilon = \widetilde I_\epsilon$$
where $\widetilde I_\epsilon$
defined by
$\widetilde I_\epsilon(A) = I_\epsilon( A + \Delta_\epsilon)$
depends only on the actions.
Similarly if $K_\epsilon (A)$ is the generator of $k_\epsilon$,
$k_0=Id$
$$k_\epsilon (A,\varphi) = (A,\varphi +\Gamma_\epsilon (A))$$
(where $\Gamma_\epsilon (A) = \int_0^\epsilon {\partial K_\sigma\over
\partial A} (A)\,d\sigma$)
and again, $H_\epsilon \circ g_\epsilon \circ k_\epsilon
= I_\epsilon \circ k_\epsilon = \ttI_\epsilon$
where $\ttI_\epsilon$ depends only on the actions.
Similarly, for relatively integrable systems, we observe that
if $g_\epsilon$ integrates the system and
$k_\epsilon$ has Hamiltonian $K_\epsilon = K_\epsilon(H_0)$
then $g_\epsilon \circ k_\epsilon $ also integrates the system.
We will use this elementary observations to show that, without
loss of generality the integrating transformations may be
assumed to satisfy some
extra normalizations. This will simplify some of the calculations.
Note also that, unless some normalizations
are imposed, it will be hard to perform
any analysis since these trivial factors may ruin
convergence of any limiting process.
Related to the concept of the integrability but better adapted to
perturbation expansions is the concept of asymptotic integrability.
\CLAIM Definition(asymptotic)
We say that an analytic system $H_\epsilon$ is ``asymptotically integrable
to order $n$'', if we can find a $g_\epsilon$ and an $I_\epsilon$ as
before such that:
$$
H_\epsilon \circ g_\epsilon = I_\epsilon (A) + o(\epsilon^n)
\EQ(asymptint)
$$
Similarly we say that it is ``asymptotically relatively integrable up to
order $n$'' if we can find $g_\epsilon$, $F_\epsilon$ in such a way that
$$
H_\epsilon \circ g_\epsilon = F_\epsilon(H_0) + o(\epsilon^n)
\EQ(asymptrelint)
$$
Note that, if a $H_\epsilon$ is asymptotically integrable to order $n$,
and ${\tilde H}_\epsilon$ is another system such that
$H_i = {\tilde H}_i$ for $i=0,\ldots,n$
then ${\tilde H}_\epsilon$ is also integrable and we can use the same
$g_\epsilon$.
Similarly, if $g_\epsilon$ integrates a system asymptotically to order $n$
and $\tilde {g_\epsilon}$ agrees with $g_\epsilon$ up to order $n+1$
then $\tilde {g_\epsilon}$ also integrates the system.
That is, asymptotic integrability up to order $n$ is a property of the
Taylor expansion in $\epsilon$ up to order $n$ of $H$ and
can be verified using only the Taylor expansion to order
$n$ of $g$.
Since we will have to deal often with truncated Taylor expansions,
we introduce the notation for Hamiltonians
$$H_\epsilon^{[\le n]} = \sum_{i=0}^n H_i \epsilon^n$$
and, similarly, $g^{[\le n]}$ will be the globally Hamiltonian isotopy
starting at $g_0$ and of Hamiltonian $G_\epsilon^{[\le n]}$. We will
also use the notation with other subscripts with inequalities. They
are meant to describe the range of the sum in a power series and,
for deformations the deformation where the Hamiltonian is a sum
of powers of $\epsilon$
similarly restricted.
Note that if a system is integrable, then it is
asymptotically integrable to order $n$ for all $n$.
The converse is certainly not automatic  it is clearly false for
families $C^\infty$ in $\epsilon$.
Take $\widetilde H_\epsilon = H_{\exp 1/\epsilon^2}$ where $H_\epsilon$
is not integrable.
Note also that one system could be integrable to all orders
in $n$ but that the transformations that achieve this
integrability could be quite different  they are not
unique, as we have shown .
Nevertheless one of the main results of this paper is that
analytic Hamiltonians satisfying
\clm(nondegenerate) that are integrable to all orders are uniformly
integrable.
The following proposition will be important to address the
nonuniqueness question.
\CLAIM Proposition (normalization)
Assume that $H_\epsilon$ is integrable (resp.
relatively integrable) to order $n$.
then, it admits an integrating family such that
$$
\int_{\torus^d} G^i = 0 ; \ \ \ 0\le i \le n
$$
$$
({\rm resp.} \int_{\Sigma_E} G^i = 0; \ \ \ 0 \le i \le n )
$$
\PROOF
For the uniformly integrable case, note that if $\tilde g_\epsilon$
integrates the problem to order $n$ then
for any $k_\epsilon$ such that $K_\epsilon = K_\epsilon(A)$,
$g_\epsilon = \tilde g_\epsilon \circ k_\epsilon$
also integrates the problem. We claim that it is possible
to choose $K_\epsilon$ in such a way that the normalization
is satisfied.
Note that the Hamiltonian of $g_\epsilon$ is
$G_\epsilon = {\tilde G}_\epsilon + K_\epsilon \circ {\tilde g}_\epsilon$.
If we expand in powers of $epsilon$, we obtain the the
coefficient of $\epsilon^i$ has the form
$G^i = {\tilde G}^i + K^i + R^i$
where $R^i$ is an expression involving only
terms of lower order.
Therefore, we can recursively pick
$K^i =  \int_{\torus^d} {\tilde G}^i + K^i$.
In the relatively integrable case, the same argument tells us
that it suffices to pick recursively
$ K^i =  \int_{\Sigma_E} {\tilde G}^i + K^i$.
\QED
\SECTION Obstructions to uniform integrability and
statement of results.
Since the solutions of \equ(integrability),
if they exist, are highly nonunique (as illustrated by the remarks
after the definition) to derive obstructions, it will be useful to show
that if there exist solutions,
they can be found in a restricted class.
\CLAIM Proposition(exact)
Assume that $H_\epsilon \circ g_\epsilon = I_\epsilon$ on
$U\times \torus^d$, $g_0=Id$ then, we can find a family $t_\epsilon$ of the
form
$$t_\epsilon (A,\varphi) = (A+\Delta_\epsilon,\varphi)\ ,\qquad
\Delta_0=0$$
in such a way that $H_\epsilon \circ g_\epsilon \circ t_\epsilon
= I_\epsilon \circ t_\epsilon$
and $g_\epsilon \circ t_\epsilon$ is a globally Hamiltonian isotopy.
In other words, if there is an integrating family defined in a certain
domain, we can find a globally symplectic transformation defined in
a slightly different domain.
Hence, to exclude the existence of integrating transformations, it
suffices to exclude the existence of globally Hamiltonian ones.
\PROOF
We denote by $[\ ]$ the cohomology class of a form and by \# the
operator induced on cohomology by a transformation.
Since
$${d\over d\epsilon} g_\epsilon \circ t_\epsilon
= (\G_\epsilon + g_\epsilon * \T_\epsilon) \circ g_\epsilon \circ
t_\epsilon$$
the family $g_\epsilon \circ t_\epsilon $ will be globally Hamiltonian
if and only if
$$[i(\G_\epsilon)\gamma] + g_{\epsilon \#} [i(\T_\epsilon)\gamma] =0
\EQ(nocohomology)$$
Since $g_0=Id$, $g_\epsilon$ is isotopic to the identity
and, therefore, $g_{\epsilon\#}$
is the identity.
Moreover, $\T_\epsilon = (\dot\Delta_\epsilon,0)$ and
$$i(\Tau_\epsilon)\gamma = \sum_i \dot\Delta_{\epsilon,i} \,d\varphi_i$$
since $[d\varphi_i]$ are a basis for the first cohomology in our phase space,
we can compute $\dot\Delta_{\epsilon,i}$ as the component of
$[d\varphi_i]$ of $[i(\G_\epsilon)\gamma]$.
Hence, we can compute $\Delta_\epsilon$ by integration.
Note that, if $\G_\epsilon$ is analytic in $\epsilon$, so is
$\Delta_\epsilon$.
\QED
To obtain obstructions for asymptotic integrability to order 1
 a fortiori
obstructions to uniform integrability , following
\cite{Po} \S 81, we take derivatives with
respect to $\epsilon$ on
\equ(integrability) and obtain
$$(\dot H_\epsilon +\{G_\epsilon,H_\epsilon\}) \circ g_\epsilon
= \dot I_\epsilon + o(\epsilon)
\EQ(derivative)$$
where $\{\quad \}$ denotes Poisson brackets and $\dot{\vphantom HH}$ denotes
derivatives with respect to $\epsilon$.
Evaluating at $\epsilon=0$ we obtain
$H_1 + \{G_0,H_0\} = I_1$.
Furthermore, we note that $\{G_0,H_0\} = L_{\G_0} H_0 =  L_{\H_0} G_0$.
Hence, if $\beta$ is a periodic orbit of the flow generated by $\H_0$,
$$\int_\beta H_1 + \{G_0,H_0\} = \int_\beta H_1  L_{\H_0} G_0 =
\int_\beta H_1$$
because the integral along a periodic orbit of the derivative along the
flow of a function vanishes.
Hence, we have proved, following \cite{Po} \S 81
\CLAIM Theorem(obstruction1)
A necessary condition for the family $H_\epsilon$ to be asymptotically
integrable to order~$2$ is that
$${1\over \beta} \int_\beta H_1$$
is the same for all periodic orbits of $H_0$ with the same action variables.
Moreover, since $I_1$ is a function of the actions only and
the actions are conserved along the flow of $H_0$, if we denote
by $A(\beta)$ the values of the actions and by $\beta$ the length
of the periodic orbit
$\int_\beta I_1 = I_1 (A(\beta)) \beta$.
Hence, we should have $I_1 = {1\over \beta} \int_\beta H_1$
Note that $\{A\} \times \torus^d$ contains a periodic orbit if and only if
$\omega (A) = {1\over T} p$ with $p\in \integer^d$.
In that case $\{A\}\times \torus^d$ is foliated by periodic orbits that
can be parameterized by $\torus^{d1}$.
If ${1\over \beta} \int_\beta H_1$ is independent of all the orbits
by averaging over the extra variables
$$I_1 (A) = \int_{\torus^d} H_1 (A,\varphi)\,d\varphi$$
on a torus where there are periodic orbits.
We will generalize this observation
later.
Similarly, we have
\CLAIM Lemma (obstructionrel1)
A necessary condition for relative integrability up to
first order is that, for any periodic
orbit
$$
{1 \over \beta} \int_\beta H_1 = F_1(H_0(\beta))
$$
\clm(obstruction1) is the main result of \S 81 of \cite{Po}.
Even if the ideas presented here are the same, it is
quite interesting to compare the notation there. In
particular, we call attention to
the fact that Poincar\'e phrases the problem as
finding uniform integrals, that is analytic
first integrals that depend analytically on
the parameter. Of course, due to
LiouvilleArnol'd theorem, the existence
of integrals in involution is the same as
integrability. But the formulation in terms
of the number of integrals is more general.
In Chapter V of \cite{Po} we find many
variants of this idea and applications to
several problems.
At the end of \S 86, we find
\vskip 2 em
{\it
\narrower
Les conditions \'enonc\'ees dans ce Chapitre
\'etant neccesaires main non suffisantes, rien ne prouve
que cette troisi\`eme int\'egral existe;
il convient avant de se prononcer, d'attendre la
publication compl\`ete des r\'esultats de
${\rm M}^{\rm me}$ de Kowaleski.
}
The footnote accompanying this paragraph is a
one of the few paragraphs in a Mathematics
book that are drama. Witness to that the industry
 this paper belongs to it of
working out the sketches of these two giants.
\vskip 2 em
{\it
\narrower
Depuis que ces lignes ont \'et\'e \'ecrites le monde
savant a eu \`a d\'eplorer la
morte pr\'ematur\'ee de
${\rm M}^{\rm me}$ de Kowaleski.
Les notes q'on a retrouv\'ees chez elle
sont malhereusement insuffisantes pour permettre de
reconstituer ses d\'emonstrations et ses calculs.
}
In this paper we will be concerned with stating and proving converses to
this obstruction to uniform integrability.
The first of the main results of this paper is:
\CLAIM Theorem(converse1)
Assume that $H_0,H_1$ are defined on $U\times \torus^d$ and that,
$H_0$ is nondegenerate,
$H_1$ satisfies the necessary conditions if \clm(obstruction1).
Then, we can find $G_0$ defined also in $U\times \torus^d$ in such a way that
$$H_\epsilon^{[\le 1]} \circ g_\epsilon^{[\le 0]}
= I_\epsilon^{[\le 1]} + o(\epsilon^2)$$
In other words, when $H_0$ is nondegenerate, the conditions in
\clm(obstruction1) are not only necessary but also sufficient for
asymptotic
integrability to order~2.
The proof of this result will be postponed till section~4, when we have
developed some technical results needed in the proof.
In that section, we will also show that it is possible to obtain
estimates of the analyticity properties of $G_0$ in terms of the
analyticity properties of $H_1$.
An interesting example that shows that the nondegeneracy conditions
in \clm(converse1) appears in the thesis of
E. Meletlidou \cite{Me}.
\CLAIM Example (tompaidis)
Consider $H_\epsilon = H_0 + \epsilon H_1$,
with $H_0 = {1 \over 2}(A_1^2 + A_2^2)$
$H_1 = \cos(2\varphi_1 \varphi_2)$.
One can verify that this system has the integrals $H_\epsilon$
and $J_1 + 2 J_2$, hence, one would be tempted to call it
integrable.
On the tori invariant by $H_0$ for which
$2 A_1 = 2 A_2$, ${1 \over \beta} \int H_1 = \cos( 2 \varphi_1(0)  \varphi_2(0))$
which is not independent of the starting point.
\QED
The reason why this example is not a contradiction
with \clm(converse1) is that precisely at the tori
with $2 A_1 = A_2$ the gradient of the
second integral fails to
be independent of the gradient of the first and,
indeed, one can check that the
integrating transformation cannot
be analytic there.
This example points to the fact that some of the
assumptions in the theorem are really necessary and,
hence, that talking about "integrability" without
attaching it a precise meaning can lead to confusion.
Once that we have necessary and sufficient conditions for asymptotically
integrability up to order two, we can investigate integrability up to
higher orders.
Clearly, the higher the order, the more necessary conditions will have to be
met. It can happen that a system is asymptotically integrable up to
a certain order but not to a higher one.
(For example, if we take $H_{\epsilon^7} \circ g_\epsilon$ with $H_\epsilon$
not integrable to order~2, we obtain a family integrable to order~6
but not to order~7.)
As in the criterion to exclude
integrability to first order, we can
derive necessary conditions
for integrability up to high orders which are related to periodic
orbits.
\CLAIM Lemma(ordern)
Let $n$ be a natural number bigger than two, $H_\epsilon$ a family of
Hamiltonians. Assume that there exists a $g_\epsilon^{[\le n]}$,
$I_\epsilon^{[0$.
We point out that the condition
in \clm(allconverse)
that $H_0$ satisfies \clm(nondegenerate)
really does belong. In \cite{Ga} p. 369
one can find examples of systems that
fail to satisfy \clm(nondegenerate)
and are not integrable.
A simple example is
$$
H_\epsilon(A_1,A_2, \varphi_1, \varphi_2)
= \alpha A_1 + A_2 + \epsilon (A_2 + f(\varphi_1) f(varphi_2) )
$$
where $\alpha$ is the golden mean.
The same idea can be used to
produce examples
where ${\rm rank} {\partial^2 H_0 \over \partial A \partial A}$
is $1, 2, \cdots, d 1$.
The paper \cite{Ga} studies the problem of
finding analogues to \clm(allconverse) when
${\rm rank} {\partial^2 H_0 \over \partial A \partial A}$
is 1,2. There, one can find conditions  that go
beyond those found in \cite{Po} and discussed here 
that are also necessary and sufficient.
Two questions that we have not been able to explore
are the converses for existence or a reduced number
of first integrals. We also do not know whether
existence of analytic integrating transformations for all
values of $\epsilon$ in a complex neighborhood of
zero imply that the system is uniformly integrable.
\SECTION Existence and regularity for solutions of cohomology equations
We have shown already that the existence of asymptotic integrating
transformations can be reduced to solving a hierarchy of equations of the
form
$$\{G_n,H_0\} + I_n =  (n+1) H_{n+1} + {1\over (n+1)!}
R_n = \Gamma_n
\EQ(equationn)$$
where $H_0$, $H_{n+1}$, $R_n$ are known and we are supposed to find
$G_n$ and $I_n$.
We have already shown that a necessary condition for the existence of
\equ(equationn) is that $\int_\beta \Gamma_n$ depends only on the action.
Similarly, the problem of relative integrability reduces to the study of
a hierarchy of equations of the form
$$\{G_n,H_0\} + F_n (H_0) = R_n$$
To prove \clm(conversen), and the similar result for
relative integrability,
it suffices to show that when the obstructions found in \clm(ordern),
(resp. \clm(orderreln)
are met, we can find analytic $G_n$ and $I_n$.
(resp. analytic $G_n$, $F_n$.)
We first argue that, we can determine $F_n$, $I_n$ in such a way that we
reduce the problem to considering the equation
$$\{G_n,H_0\} = \Gamma_n$$
with the compatibility conditions $\int_\beta \Gamma_n =0$ for every
periodic orbit of $H_0$.
We first observe that finding the $I_n$ (for \equ(integrability)) and the
$F_n$'s is easy.
In effect, if we consider
$$\Gamma_n^t(A,\varphi) = {1 \over t} \int_0^t \Gamma_n (A,\varphi +\omega(A)t)$$
they are equicontinuous functions.
Moreover, if $\omega(A)$ is of the form
${2\pi\over T} (p_1,\ldots,p_d)$ with $p_1,\ldots,p_d\in\integer$ they
converge to the average over the periodic orbit.
If the system is nondegenerate, the set of periodic orbits is dense.
By AscoliArzel\'a theorem, this sequence of functions is
converging uniformly to a limit. By our hypothesis on periodic orbits, the
limit depends only on the actions.
Hence, if the average on the periodic orbits depends only on the actions,
(resp. (on the energy) of a periodic orbit, we obtain that
$\Gamma_n^t (A,\varphi)$ converges uniformly as $t\to\infty$ to a function
of $A$ (resp. the energy) which, of course, agrees with the averages over
periodic orbits.
(Hence, we conclude, in particular, that the averages over periodic orbits
of $\Gamma_n$ have to be continuous.)
Now, we proceed to compute a more convenient form of these averages.
Then,
$$\eqalign{
\int_{\torus^d} d\varphi\ \Gamma_n^t (A,\varphi)
& = {1\over t} \int_0^t \int_{\torus^d} d\varphi\ \Gamma_n
(A,\varphi + \omega (A)t)\cr
& = {1\over t} \int_0^t dt \int d\varphi\ \Gamma_n (A,\varphi)\cr
& = \int_{\torus^d} d\varphi\ \Gamma_n (A,\varphi)\cr
\int_{\Sigma_E} d\mu_E \ \Gamma_n^t (A,\varphi)
& = {1\over t} \int_0^t \int_{\Sigma_E} d\mu_E\
\Gamma_n(A,\varphi +\omega(A)t)\cr
& = {1\over t} \int_0^t \int_{\Sigma_E} d\mu_E\ \Gamma_n\cr
&= \int_{\Sigma_E} d\mu_E\ \Gamma_n\cr}$$
Hence, if $\Gamma_n^t$ is converging uniformly to a function $I_n$ only
of $A$ (resp. $F_n$ only of $H_0$) we see that this function has to be
$$\eqalign{I_n(A) &= \int_{\torus^d} d\varphi\ \Gamma_n (A,\varphi)\cr
\hbox{(resp.}\qquad F_n (E) & = \int_{\Sigma_E} d\mu_E\ \Gamma_n\quad ).\cr}
\EQ(Isolution)$$
These functions have to agree with the average on periodic orbits.
Note that both $I_n(A)$, $F_n(E)$ given by \equ(Isolution)
are analytic functions of the position if $\Gamma_n$ is.
Therefore, if we set $I_n$, (resp. $F_n$) to be given by
\equ(Isolution) we reduce our problem to
$$\leqalignno{\{G_n ,H_0\} & = \Gamma_n  I_n\cr
\noalign{\hbox{(resp.}}
\{G_n,H_0\} & = \Gamma_n  F_n\quad )\cr}$$
where, in both cases the R.H.S. have average zero over periodic orbits.
\REMARK
Since the problem of relative integrability is geometrically natural, the
above reduction can be worked out in much larger generality.
This is done in \cite{LMM}, pp.~549553.
\REMARK
The $I_n$'s and the $F_n$'s play a role very similar to the averages
that need to be taken off in the proofs of K.~A.~M. theory.
Therefore \clm(converse1), \clm(conversen)
are reduced to the proof of:
\CLAIM Theorem(cohomology)
Let $H_0$ be an integrable system satisfying \clm(nondegenerate).
Let $\Gamma$ be an analytic function on $U\times \torus^d$ satisfying:
$$\int_\alpha \Gamma=0$$
for all $\alpha$ periodic orbit of $H_0$.
Then, there exists an analytic function $G$ on $U\times\torus^d$ such that
$$\{G,H_0\} = \Gamma$$. This solution is unique
up to addition of a function of the actions.
\PROOF
Since we will study the cohomology equation
$$\{G,H_0\} \equiv \omega (A) {\partial\over\partial\varphi} G=\Gamma$$
using Fourier analysis, we first work out the implication of the
compatibility conditions for the Fourier coefficients.
We observe that if $\omega (A_0) = {1\over T} p$ with $T\in\real$,
$p\in\integer^d$, $\{A_0\} \times\torus^d$ consists of periodic
orbits of period $T$ and, for all $\varphi \in \torus^d$,
$$0 = {1\over T} \int_0^T \Gamma (A_0,\omega (A_0)t+\varphi)\, dt\ .$$
Using the partial Fourier expansion, we have
$$\eqalign{
0 & = {1\over T} \int_0^T \sum_k \hat\Gamma_k (A_0)
e^{2\pi i {1\over T} p\circ k+k\varphi}\,dt\cr
& = \sum_{k\cdot p=0} e^{2\pi ik\varphi} \hat\Gamma_k (A_0)\cr}$$
We denote by $\P$ the set of $\omega$'s that can be written as
${1\over T}p$ for some $T\in \real^+$, $p\in\integer^d$.
That is, the $\omega$'s that give rise to a periodic orbit.
Hence if the compatibility conditions are met,
$$\hat\Gamma_k (A_0) = 0 \hbox{ if } k\cdot\omega (A_0)=0
\hbox{ and } \omega(A_0) \in \P$$
Now, we argue that the last condition $\omega(A_0)\in \P$ is superfluous.
In effect, note that if $k\in\integer^d$ and $k\ne0$  say $k_d\ne 0$ 
$\omega\cdot k=0$ if and only if $\omega$ can be written as
$(\omega_1,\ldots,\omega_{d1}  {1\over k_d} (k_1\omega_1 +\cdots +
k_{d1} \omega_{d1}))$.
Since given any real numbers $\omega_1,\ldots,\omega_{d1}$ we can
approximate then by rationals, we conclude that any $\omega\in\real^d$
with $\omega\cdot k=0$ can be approximated by one with rational components,
which, taking $T$ as the lowest common denominator, is of the form
${1\over T} p$ with $p\in\integer^d$.
We conclude that if $k\in \integer^d$, $k\ne0$,
$\P \cap \{\omega\in\real^d\mid \omega\cdot k=0\}$
is dense on
$\{\omega\in \real^d\mid \omega\cdot k=0\}$.
(Of course the result is also true when $k=0$.)
Since $\hat\Gamma_k$ is continuous it vanishes when
$\{\omega (A)\cdot k=0\}$ even if $\omega(A) \notin \P$.
We observe that, in terms of Fourier coefficients, the equation reads
$$\left(\omega(A) {\partial\over\partial\varphi} G\right)_k^\wedge
= 2 \pi i ( \omega(A) \cdot k)\hat G_k(A)
= \hat\Gamma_k (A)
\EQ(Fourierhomology)$$
The regularity of the $\hat G_k$ and the $\hat\Gamma_k$'s is easier to
study when we consider them as functions of $\omega$.
Since $\omega^{1}$ is uniformly analytic, we can consider
$\tilde\Gamma_k = \hat\Gamma_k\circ \omega^{1}$,
$\tilde G_k = \hat G_k\circ \omega^{1}$.
Then \equ(Fourierhomology) reads
$$\tilde G_k(\omega)(\omega\cdot k) 2\pi i = \tilde\Gamma_k(\omega)
\EQ(Fourierhomology2)$$
but we have that $\tilde\Gamma_k(\omega) =0$ when $\omega\cdot k=0$.
Hence, the solution
$$\tilde G_k(\omega) = {1\over 2\pi i}
\cases{ (\omega\cdot k)^{1} \tilde \Gamma_k (\omega)\ ,&$\omega\cdot k\ne0$\cr
{1\over k^2} (k\cdot\partial_\omega) \tilde\Gamma_k(\omega)\ ,
&$\omega\cdot k=0$\cr}
\EQ(gform)$$
is the only continuous solution.
Certainly $\tilde G_k(\omega)$ is analytic if $\tilde\Gamma_k$ is
(and hence $\hat G_k(A)$ is analytic if $\hat\Gamma_k$ is).
Recall that, by Cauchy inequalities we can bound
$$\Big (\partial_\omega)^n\tilde\Gamma_k (\omega)\Big
\le n!K_\omega a_\omega^n e^{\deltak}\ \hbox{ where }\ a<1,\ \delta >0$$
It follows from \equ(gform) that $\tilde G_k$ satisfies similar bounds
and hence, is analytic.
This finishes the proof of \clm(cohomology) and, hence the proof of
\clm(converse1), \clm(conversen).
\QED
The proof of convergence will be obtained, not by estimating the recursion
outlined before, but by using a quadratically convergent scheme of the
K.~A.~M. type.
This convergent scheme will, of course, require quantitative versions
of \clm(cohomology) that not only claim the existence of the solutions.
As it turns out, this convergent scheme will need to be implemented on
spaces of families of functions.
Hence, our first task is to introduce appropriate spaces of families of
functions and formulate quantitative versions of \clm(cohomology) for them.
\CLAIM Definition(analyticnorms)
Given an open set $U\subset \real^d$ we can consider it also as a subset
of $\complex^d$ and define, for $\delta>0$
$$U^\delta = \{z\in\integer^d \mid d(z,U)\le \delta\}$$
Given a nondegenerate frequency function we will denote $V=\omega (U)$
$$\\eta\^{\omega,U,\delta}
= \sup_{\scriptstyle z\in V^\delta\atop \scriptstyle \Im \varphi\le\delta}
\eta (\omega^{1}z,\varphi)$$
The set of functions for which the above norm is finite forms a Banach
space which we will denote by $\chi^{\omega,U,\delta}$.
Similarly, given an analytic family of nondegenerate frequency functions
$\omega_\epsilon$ we define for
$$\\Gamma\^{\omega,U,\delta}
= \sup_{{\scriptstyle z\in V^\delta\atop\scriptstyle \Im \varphi\le\delta}
\atop \scriptstyle \epsilon\le e^\delta}
\Gamma_\epsilon (\omega_\epsilon^{1} x,\varphi)$$
and denote by $\Phi^{\omega,U,\delta}$ the Banach space of families for
which the above norm is finite.
The reason to introduce the norms in \clm(analyticnorms)
is that, for the solution of the cohomology equation,\equ(gform)
the variables $\omega$ are more natural than the angle variables.
This has the inconvenient that the norms depend on the
frequency $\omega$, but the dependence is slight as we
will prove later.
Note that we denote the radius of the balls of the
parameter $\epsilon$ by $e^\delta$, where
$\delta$ is the same parameter that measures the
complex component in the complex extension of the torus.
The exponential notation is justified so that the
Cauchy inequalities look the same. The
use of the same parameter for the two variables
is justified just to keep down the number of
parameters even it it leads to
wasteful estimates.
We also introduce the norms
$$\eqalign{
\\eta\_\infty^{U,\delta}
&= \sup_{\scriptstyle z\in U^\delta\atop \scriptstyle \Im \varphi\le\delta}
\eta (z,\varphi)\cr
\noalign{\vskip6pt}
\\Gamma\_\infty^{U,\delta}
& = \sup_{\scriptstyle z\in U^\delta\atop\scriptstyle \Im\varphi\le\delta}
\Gamma_\epsilon)\cr}$$
and denote the corresponding Banach spaces by $\chi_\infty^{U,\delta}$,
$\Phi_\infty^{U,\delta}$.
When the functions take values on matrices or on linear spaces we
substitute for the absolute value in the R.H.S.\ of the above definitions
the appropriate norms.
Note that we have that if $V = \omega(U)$
we have
$$
R^{\omega,U,\delta} =  R\circ \omega^{V,\delta}_\infty
\EQ(trick)
$$
We clearly have that
$$\\eta\^{\omega, U, \delta} \le \\eta\^{\omega,U,\delta+\sigma}
\ \hbox{ for }\ \sigma>0$$
since we are taking the supremum over a larger set.
Note that $d(z,U)\le\alpha$ implies
$$d(\omega(z),V) \le \alpha \D\omega\_\infty^{U,\alpha}$$
Hence,
$$\\eta\_\infty^{U,\delta} \le \\eta\^{\omega,U,\delta
\D\omega\_\infty^{U,\alpha}}$$
and since $d(w,V)\le\alpha$ implies $d(\omega^{1}(w),U)\le
\alpha\D\omega\_\infty^{V,\alpha}$ we have
$$\\eta\^{\omega,U,\delta}
\le \\eta\_\infty^{U,\delta \D\omega^{1}\_\infty^{\omega, U,\delta}}$$
The following are well known Cauchy bounds applied to the
angular variables and the parameters. (We do not state the
much better known inequalities for derivatives
with respect to the $A$.)
\CLAIM Lemma(Cauchy)
$$\eqalign{
\\hat\eta\^{\omega,U,\delta}
& \le e^{k\delta} \\eta\^{\omega,U,\delta}\cr
\\hat\eta_k\_\infty^{U,\delta}
& \le e^{k\delta} \\eta\_\infty^{U,\delta}\cr
\left\ {\partial^{j}\over\partial\varphi^j}\eta\right\^{\omega,U,\delta
\sigma}
&\le K\sigma^{j} \\eta\^{\omega,U,\delta}\cr
\\eta^{n}\^{\omega,U,\delta}
& \le K e ^{n \delta} \\eta\^{\omega,U,\delta} \cr
\\eta^{[\le N]}\^{\omega,U,\delta\sigma}
& \le K\sigma^{1} \\eta\^{\omega,U,\delta} \cr
\\eta^{[\ge N]}\^{\omega,U,\delta\sigma}
& \le K\sigma^{1} e^{N\delta} \\eta\^{\omega,U,\delta}\cr}$$
Before we embark in the proof of \clm(allconverse), let us state another
technical lemma that will control the dependence of the
norm $\ ^{\omega, U, \delta}$ on $\omega$.
The main point of the lemma is that the
change of the norms, when we change $\omega$ is controlled
by the change in $\omega$. We need, however to make sure that the
domains match and that the change is not so drastic as to
make the new frequency function noninvertible. In view
of the observation \equ(trick) we see that the
estimates that we need are just the customary
estimates for the composition of functions that appear
in almost all the proofs of K.~A.~M. theorems
based in transformations.
\CLAIM Lemma (omegadependence)
Assume that $\omega$, $\tilde \omega$
are nondegenerate frequency functions.
\vskip 1pt
Assume moreover
\item{i)} $\tilde U$ is such that
$d(\tilde U , \complex^d  U) >
K  \omega  \tilde \omega^{U,\delta}_\infty + K \sigma$
\item{ii}$  \omega  \tilde \omega^{U, \delta}_\infty < 1/K$
\vskip 1pt
where $K$ depends only on
$ \omega^{U,\delta}_\infty$,
$ \tilde \omega^{U,\delta}_\infty$
$ \omega^{1}^{U,\delta}_\infty$,
$ \tilde \omega^{1}^{U,\delta}_\infty$
and can be chosen uniformly when these norms
can be bounded uniformly.
\vskip 1pt
Then,
$$
\big R^{\omega,\tilde U, \delta}
 R^{\omega,\tilde U, \delta} \big
\le C \sigma^{1}  R^{\omega,U,\delta}\omega  \tilde \omega^{U,\delta}_\infty
$$
Analogous results hold for families.
\PROOF
We have:
$$
\big R^{\omega,\tilde U, \delta}
 R^{\omega,\tilde U, \delta} \big
\le
 R \circ \omega^{1}  R\circ {\tilde \omega}^{1}^{V,\delta}_\infty
$$
where $V$ is any domain such that $\omega(U) \subset V$,
$\tilde \omega(U) \subset V$.
If $\tilde V$ is such that
$\lambda\omega^{1}(x) + (1\lambda) \tilde \omega^{1}(x) \in \tilde V$
for $x \in V$ and $\lambda \in [0,1]$
we can bound this by
$$
 \nabla R^{\tilde V,\delta}_\infty
 \omega^{1}  \tilde \omega^{1}^{V,\delta}_\infty
$$
Finally, if $\tilde {\tilde U}$ is such that
$d( \tilde U, \complex^d  \tilde {\tilde U}) \ge \sigma$
we can use Cauchy
bounds to bound the gradient. Moreover, given uniform
bounds on the functions we can bound the difference of the
inverses by the difference of the functions
using the implicit function theorem.
We therefore obtain the desired result.
The result for families is obtain by just applying this
result for each value of $\epsilon$ and then taking
suprema.
\QED
Since in the proof of the theorem we will have to give up domains
in a controlled way repeatedly, we adopt the convention
that a domain with a $\tilde{}$ means a domain related to the
original one by a relation such as $i)$ above.
The following is the main result of this section. It provides
quantitative estimates for the solutions that were shown to exist in
\clm(cohomology)
\CLAIM Theorem(cohomologyestimates)
Let $I_\epsilon$ be an analytic family of integrable systems with
nondegenerate frequency function on $U$, with $U$
a complex extension of a domain in the reals.
Let $R_\epsilon$ be an analytic family in $\Phi^{\omega, U,\delta}$
Assume that $\delta$ is sufficiently
small and that $U$ is sufficiently close to the real
and that $R$
satisfies the compatibility conditions
$\int_{\beta_\epsilon} R_\epsilon =0$
for every $\beta_\epsilon$ periodic orbit of $I_\epsilon$.
\vskip 0 pt
Let $G_\epsilon$ be the solution of
$$\{I_\epsilon,G_\epsilon\} = R_\epsilon$$
which also satisfies
$\int_{\torus^d} G_\epsilon (A,\varphi)=0$.
Then,
$$\G\^{\omega,U,\delta\sigma}
\le \R\^{\omega,U,\delta} \sigma^{d} K$$
where $K$ depends only on the dimension.
\REMARK
Note that a nondegenerate
system will not have periodic
orbits outside of the real part of $U$ if
the extension is small enough.
Clearly a frequency with an imaginary frequency
will not lead to a periodic orbit.
Once we fix a neighborhood, there is an
open set of Hamiltonians with the same property.
We will assume that all the perturbations are
small enough that all our constructions
do not leave this neighborhood.
\PROOF
By Cauchy estimates we have
$$\\hat R_k\^{\omega,U,\delta}
\le \R\^{\omega,U,\delta} e^{k\delta}$$
We also recall that the solution $G$ was given by \equ(gform)
$$\hat G_{k\epsilon} \circ \omega_\epsilon^{1}(\alpha)
= {1\over2\pi i}
\cases{(\alpha\cdot k)^{1} \hat R_{k,\epsilon} \circ \omega_\epsilon^{1}
(\alpha)\ ,&$\alpha\cdot k\ne0$\cr
{1\over k^2} (k\cdot \partial_\alpha) \hat R_{k,\epsilon}(\alpha)\ ,
&$\alpha\cdot k=0$\cr}$$
Hence, the supremum of $\hat G_k\circ\omega^{1}$ is bounded by a constant
times the supremum of ${1\over k}\nabla (\hat R_k\circ \omega^{1})$.
This can be bounded as follows.
If $\tilde U$ is a domain in $\complex^d$
such that $\tilde U \subset U$ and
$d( \tilde U, \complex^d U) > \sigma$
we have:
$$\eqalign{\\hat G_k\^{\omega,\tilde U,\delta\sigma}
& \le K {1\over k} \\nabla \hat R_k\^{\omega,\tilde U,\delta\sigma}
\le K\sigma^{1} {1\over k} \\hat R_k\^{\omega,U,\delta}\cr
& \le K\sigma^{1} {e^{\sigma k}\over k} \R\^{\omega,U,\delta}\cr}$$
Then
$$\eqalign{
\G_k\^{\omega,U,\delta\sigma}
&\le \sum_{k\in\integer^d\{0\}} e^{(\delta\sigma)k}
\\hat G_k\^{\omega,U,\delta}
\le \sum_{k\in\integer^d\{0\}} e^{\sigmak} {1\over k}
\\hat R\^{\omega,U,\delta}\cr
&\le K\sigma^{1} \sum_{k\in\natural \{0\}} e^{\sigma k} k^{d2}
\R\^{\omega,U,\delta}
\le K\sigma^{d} \R\^{\omega,U,\delta}\cr}$$
\QED
\REMARK
It is quite possible that the previous estimates are rather wasteful
and that the negative power of $\sigma$ can be greatly reduced.
It seems that since \equ(gform) only loses one derivative
we should only have a factor $\sigma^{1}$ in the
estimates above. Part of the problem is due to the use of
Fourier series and supremum norms in the way that we have done it.
Note that if we wanted to estimate the identity operator
using the sloppy bounds that we have employed, we would have obtained
a $\sigma^{(d 1)} $ factor, which, obviously does not
belong. These issues are important to improve the smoothness conditions
in the usual K.~A.~M. theorem. Since the main results in
this paper have no
hope of working in any regularity
substantially smaller than analytic we will refrain from such
improvements.
\SECTION Proof of \clm(allconverse)
As we have indicated before the proof will be done using a Newton algorithm
to obtain quadratic convergence.
We proceed to describe the inductive steps.
At step $n$ we will have a family of
Hamiltonians $H_\epsilon^{(n)}$
$$\eqalign{
H_\epsilon^{(n)} & = I_{\epsilon}(A)^{(n)} + R_\epsilon^{(n)}\cr
H_\epsilon^{(n)} & = F_\epsilon^{(n)} (H_0) + S_\epsilon^{(n)}\cr}
\EQ(beginning)$$
where $R_\epsilon^{(n)}$ and $S_\epsilon^{(n)}$ are $o(\epsilon^{2^n})$.
Since the splitting of the Hamiltonian in these parts
would not be unique otherwise, we
assume that $I^{(n)}_\epsilon$ contains
only terms up to $O(\epsilon^{2^n1})$
and that
$ \int_{\torus^d} R^{[ \le 2^n 1]}_\epsilon d \varphi = 0$.
The inductive step will consist in determining
$$G_\epsilon^{(n)} = \sum_{i=2^n}^{2^{n+1}1} G^{(n),i} \epsilon^i$$
(and $I_\epsilon^{(n+1)}$, $F_\epsilon^{(n+1)}$) in such a way that
setting $H^{(n+1)}_\epsilon = H^{(n)}_\epsilon\circ g^{(n)}_\epsilon$
we can also find $R^{(n)}_\epsilon$, $I^{(n)}_\epsilon$
(resp. $S^{(n)}_\epsilon$, $F^{(n)}_\epsilon$ ) in
such a way that
they satisfy \equ(beginning) with $n+1$ in place of $n$.
We will describe the procedure to determine the $G^{(n)}_\epsilon$
once all the other elements are known and, then, we will
develop enough bounds that imply convergence
As it is customary in K.~A.~M. theory, the
functions $H^{(n)}_\epsilon$ will
be controlled in domains $U^{(n)}\times \torus^d$
which will be decreasing as the iteration proceeds,
(but which will not become empty). We will denote by
$\omega^{(n)}$ the frequency function corresponding to
$I^{(n)}$.
It is quite important to remark that because of the
way that $H^{(n+1)}_\epsilon$ is produced, it will be
integrable to all orders (resp.
relatively integrable to all orders) if $H^{(n)}_\epsilon$ is.
The first step of the iteration
will be obtained by taking
$H^{(0)}_\epsilon = H_{\Lambda \epsilon}$
where $\Lambda$ is a sufficiently large real
number. Note that such $H^{(0)}_\epsilon$
is integrable (resp relatively integrable)
to all orders in $H_\epsilon$ is. Moreover,
by choosing $\Lambda$ large enough, we can obtain
that $ R^{(0)}_\epsilon^{\omega,U,\delta}$
(resp. $ S^{(0)}_\epsilon^{\omega,U,\delta}$) is
small enough. Of course $H^{(0)}_0 = H_0$ so that it
is integrable and nondegenerate.
The quadratically convergent procedure
will be based in finding a $g^{n}_\epsilon$
in such a way that $ H^{(n+1)}_\epsilon \equiv
H^{(n)}_\epsilon \circ g^{(n)}_\epsilon$
is much closer to an integrable family.
First, we will derive heuristically what are the equations
that the $G^{(n)}_\epsilon$ have to satisfy,
show that these equations admit solutions and
estimate the new error. Then, it will become
quite an standard procedure in K.~A.~M. theory to
show that the $g^{(0)}_\epsilon \circ \cdots \circ g^{(n)}_\epsilon$
converge and the limit integrates the system.
The fact that we can find our $g^{(n)}_\epsilon$
will use the fact that that $H^{(n)}_\epsilon$
is integrable to all orders. Note that, by the
definition of $H^{(n+1)}_\epsilon$,
it will be integrable to all orders if
$H^{(n)}_\epsilon $ is.
For the experts in K.~A.~M. theory we note that,
even if it follows from our result that the set
of families integrable to all orders is a
manifold, we cannot use it in the proof.
In particular, we cannot use the formulation
of K.~A.~M. theory as implicit function theorems
and just use the fact that all the steps of
the iterative procedure respect the set of
families that are integrable to all orders.
This is very similar to the problem of
rotations of the circle as originally solved in
in \cite{Ar}, where we do not know a priori that the
set of maps with the same rotation number is a manifold  it is
consequence of the theorem. Similar arguments happen
in the papers  much closer to our problem  \cite{CEG}
and \cite{Ga}.
We will find it convenient to introduce a second
auxiliary parameter $\mu$. This may be justified
by noting that we are talking about small in two
different senses. One is of course, that $\epsilon$ is
small, but we also have to consider distances
between families whose parameter is $\epsilon$.
If we consider the family $H^{(n)}_\epsilon \circ g^{(n)}_{\mu \epsilon}$
for $\mu \in [0,1]$ and apply the
mean value theorem
$F(1) = F(0) + F'(1) + E$, where $E \le {1\over2} \sup F''$
we obtain  in the region where the flow
does not leave the domain of definition of $H^{n}_\epsilon$ 
$$
\eqalign{
H^{(n)}_\epsilon \circ g^{(n)}_\epsilon
& = H^{(n)}_\epsilon +
\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon
+ E
\cr
& =
H^{(n)}_\epsilon +
\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}
+[
\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon

\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon] + E
}
\EQ(meanvalue)
$$
where $E$ is an error
term that, by the mean value theorem can be bounded by the
supremum of the second derivative along the flow.
We just recall that the second
derivative along the flow can be expressed in terms
of Poisson brackets
$$
{d^2 \over d \mu^2 } H^{(n)}_\epsilon \circ g^{(n)}_{\mu \epsilon}
=
\{
G^{(n)}_\epsilon,
\{
G^{(n)}_\epsilon,
H^{(n)}_\epsilon\} \} \circ g^{(n)}_{\mu \epsilon}\ .
\EQ(secondderivative)
$$
Substituting \equ(beginning) in \equ(meanvalue)
we obtain for the problem of uniform integrability
\def\nep#1{ {#1}^{(n)}_\epsilon}
$$
\eqalign{
\nep{H} \circ \nep{g}
& = \nep{I} + {\nep{R}}^{[\le 2^{n +1}]}
+ {\nep{R}}^{[> 2^{n +1}]}
+ \epsilon\{ \nep{G}, \nep{I} \} +
\epsilon \{ \nep{G}, \nep{R}\} + \cr
& +[
\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon

\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}] + E
}
\EQ(identityuniform)
$$
and, for the problem of relative integrability:
$$
\eqalign{
\nep{H} \circ \nep{g}
& = \nep{I} + {\nep{R}}^{[\le 2^{n +1}]}
+ {\nep{S}}^{[> 2^{n +1}]}
+ \epsilon\{ \nep{G}, \nep{F}(H_0) \} +
\epsilon \{ \nep{G}, \nep{S}\} + \cr
& +[
\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon

\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}] + E
}
\EQ(identityrelative)
$$
Now,
we work out the consequences of the
system being integrable to all orders.
We note that if there was a $g_\epsilon$ that integrated the
system,
we argued n \clm(normalization) that we could
find one integrating $g_\epsilon$
such that $\int_{\torus^d} G_\epsilon = 0$.
If we apply \equ(identityuniform)
to this $g_\epsilon$, substituting $G_\epsilon$ for $\nep{G}$ in
\equ(identityuniform)
we obtain:
$$
\{ G^0, I^{(n)}_0 \} = 0
$$
and, that and the normalization condition,
allows us to conclude that $G^0 \equiv 0$ by the
uniqueness statement in \clm(cohomology)
In other words, if $\nep{H}$ can be integrated
 by a system with normalized Hamiltonian, as
we showed without loss of generality  then
$G^0 = 0$. Similarly, for
terms whose order in $\epsilon$ is not more
that $2^{n} 1$ we obtain that
$$
\{G^i, I^{(n)}_0 \} = \Gamma_i
$$
where $\Gamma_i$ is an expression involving
$G^0,\cdots, G^{i1}$ that vanishes if
$G^0,\cdots,G^{i1}$ vanishes.
(This is because the terms involving the second derivatives,
looking at \equ(secondderivative) have order
in $\epsilon$ at least $2 + 2r$ if the
$G_\epsilon$ is order $r$ in $\epsilon$.
That is, we have shown that if the system is
integrable to all orders then, any
integrating normalized transformation
has to satisfy:
$ G^0 = G^1 = \cdots = G^{2^n1} = 0$.
Once we know that the coefficients in the
expansion of order up to
$2^{n} 1$ have to be zero,
equal to zero, we realize that the
only terms in \equ(identityuniform)
that contain terms of order less or equal
than $2^{n+1} $ are the first, second and
third.
In particular, we obtain that
${\nep{R}}^{[\le 2^{n+1}]} + \epsilon \{ G, \nep{I} \}
= I(A) + O(\epsilon^{2^{n+1} +1})$
Proceeding as in \clm(obstruction1), integrating
along periodic orbits of $\nep{I}$, we obtain that
if the system is integrable to all orders, then,
$ {1 \over \beta_\epsilon} \int_{\beta_\epsilon} {\nep{R}}^{[\le 2^{n+1}]}$
is only a function of $A$.
We note that these are the compatibility conditions
for the existence of $\nep{ \Delta I }$
$\nep{G}$ solving
$$
\nep{ \Delta I } + {\nep{R}}^{[\le 2^{n+1}]} +
\epsilon\{ \nep{G}, \nep{I} \} = 0
\EQ(tosolveuniform)
$$
Analogously, we find that if $\nep{H}$ was relatively integrable
to all orders, the compatibility conditions
for the equation for the equation
$$
\nep{\Delta F}(H_0) +{\nep{S}}^{[\le 2^{n+1}]} +
\epsilon\{ \nep{G}, \nep{F}(H_0) \} = 0
\EQ(tosolverelative)
$$
This is the step that makes all the method work
because those are the main terms in
\equ(identityuniform) and \equ(identityrelative).
We want to obtain estimates on the resulting $\nep{G}$'s, then
estimate the remainder  in an
slightly smaller region  and finally prove that the
whole procedure converges.
We will write down with the notation
of uniform integrability since the
remaining of the procedure is exactly the same
for the relative integrability case.
As we have shown before, both cases
rely on solving the same equation and
the two identities \equ(identityuniform),
\equ(identityrelative) are
completely similar.
To obtain estimates for $\nep{G}$ we observe
that by Cauchy estimates \clm(Cauchy)
we have
$
 {\nep{R}}^{[ \le 2^{n+1}]}^{\omega, U^{(n)}, \delta  \sigma} \le
K \sigma^{1} \nep{R}^{\omega, U^{(n)}, \delta  \sigma}
$.
Then, applying \clm(cohomologyestimates)
we obtain:
$$
\nep{G}^{\omega^{(n)}, \widetilde {U^{(n)} }, \delta  2 \sigma}
\le K \sigma^{ (d+1)}
 \nep{R}^{\omega, U^{(n)}, \delta}
\EQ(Gbounds)
$$
Also, in an slightly smaller domain we will have:
$$
\eqalign{
 \nabla \nep{G}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)} }}, \delta  3 \sigma}
& \le K \sigma^{ (d+2)}
 \nep{R}^{\omega, U^{(n)}, \delta} \cr
 \nabla^2 \nep{G}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)} }}, \delta  3 \sigma}
& \le K \sigma^{ (d+3)}
 \nep{R}^{\omega, U^{(n)}, \delta}
}
\EQ(gradientbounds)
$$
Since $\nep{G}$ is chosen in such a way that the
main terms in \equ(identityuniform)
cancel, the new remainder will be estimated
by estimating systematically all the terms
in \equ(identityuniform) that are not involved
in \equ(tosolveuniform).
Using \equ(gradientbounds), provided that
$\sigma \ge K e^\delta \sigma^{ (d+2)}
 \nep{R}^{\omega, U^{(n)}, \delta}
$ the solutions of the Hamilton equations
corresponding to $\nep{G}$ that start in
$\widetilde {\widetilde {\widetilde {U^{(n)}}}} \times
 {\rm Im} \varphi 
\le \delta  4 \sigma$
will not leave
$\widetilde {\widetilde {U^{(n)}}] \times
 {\rm Im} \varphi 
\le \delta  3 \sigma}$
for a "time" $\epsilon$ $\epsilon \le e^\delta$.
Then, we can estimate as follows:
$$
\eqalign{
 \epsilon\{\nep{G}, \nep{R}\}^
{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma}
& \le K  \nabla \nep{G}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma}
\nabla \nep{R}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma} \cr
& \le K \sigma^{(d+3)} \left(  \nep{R}^
{\omega^{(n)}, {U^{(n)}}, \delta }
\right)^2
}
\EQ(error1)
$$
Similarly, we have
$$
\eqalign{
 \nabla \epsilon\{\nep{G}, \nep{H}\}^
{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma}
& \le K  \nabla^2 \nep{G}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma}
\nabla \nep{H}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}},
\delta  2 \sigma} \cr
&\le K \sigma^{(d+5)}
 \nep{R}^{\omega^{(n)}, {U^{(n)}}, \delta }
 \nep{H}^{\omega^{(n)}, {U^{(n)}}, \delta }; \cr
}
$$
$$
\eqalign{

\epsilon \{ G^{(n)},& H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon

\epsilon \{ G^{(n)}, H^{(n)}_\epsilon\}\circ g^{(n)}_\epsilon
^{\omega^{(n)}, \widetilde {\widetilde {\widetilde {U^{(n)}}}}, \delta  3 \sigma} \le \cr
& \le
 \nabla \epsilon\{\nep{G}, \nep{H}\}^
{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma}
 \nep{g}  {\rm Id}^{\omega^{(n)}, \widetilde {\widetilde {\widetilde {U^{(n)}}}}, \delta  3 \sigma}\cr
&\le K \sigma^{(d+5)}
 \nep{R}^{\omega^{(n)}, {U^{(n)}}, \delta }
 \nep{H}^{\omega^{(n)}, {U^{(n)}}, \delta }
 \nabla \nep{G}^{\omega^{(n)}, \widetilde {\widetilde {U^{(n)}}}, \delta  2 \sigma} \le \cr
& K \sigma^{ (2d+7) }
\nep{H}^{\omega^{(n)}, {U^{(n)}}, \delta }
\left( \nep{R}^{\omega^{(n)}, {U^{(n)}}, \delta }\right)^2
}
\EQ(error2)
$$
$$
\eqalign{
 \epsilon^2
\{ \nep{G}, \{ \nep{G}, \nep{H}\} \} & \circ \nep{g}^
{\omega^{(n)}, \widetilde {\widetilde {\widetilde {U^{(n)}}}}, \delta  3 \sigma} \cr
& \le
 \epsilon^2
\{ \nep{G}, \{ \nep{G}, \nep{H}\} \} ^
{\omega^{(n)}, \widetilde {\widetilde{U^{(n)}}}, \delta  2 \sigma}\cr
&\le K  \nabla^2 \nep G^{\omega^{(n)}, \widetilde {\widetilde{U^{(n)}}}, \delta  2 \sigma}
 \nabla^2 \nep{H}^{\omega^{(n)}, \widetilde {\widetilde{U^{(n)}}}, \delta  2 \sigma} \cr
& \le K \sigma^{(2d+12)}
\nep{H}^{\omega^{(n)}, {U^{(n)}}, \delta }
\left( \nep{R}^{\omega^{(n)}, {U^{(n)}}, \delta }\right)^2
}
\EQ(error3)
$$
$$
 {\nep{R}}^{[ \ge 2^{n+1}]}
^ {\omega^{(n)}, \widetilde {\widetilde {\widetilde {U^{(n)}}}},
\delta  3 \sigma}
\le K e^{\delta 2^{n+1}}
 \nep{R}^{\omega^{(n)}, {U^{(n)}}, \delta }
\EQ(error4)
$$
Substituting these estimates in \equ(identityuniform)
we obtain
$$
 R^{(n+1)}_\epsilon ^{\omega^{ (n)}, U^{(n+1)}, \delta  3 \sigma}
\le K \sigma^{a}  \nep R^{\omega^{ (n)}, U^{(n)}, \delta}
\left(  \nep R^{\omega^{ (n)}, U^{(n)}, \delta}
+ K e^{\delta 2^{n+1}} \right)
\EQ(quadraticestimates)
$$
Finally, we note that since
$$
\omega^{(n+1)} = \omega^{(n)} + \nabla \int_{\torus^d} R^{(n)}
$$
we can obtain applying \clm(omegadependence)
as well as Cauchy bounds  again provided that smallness conditions are
satisfied .
$$
R_\epsilon^{(n+1)}^{\omega^{(n+1)}, U^{(n+1)}, \delta  3 \sigma}
\le
K \sigma^{a}
( R^{(n)}^{\omega^{(n)},U^{(n)},\delta\sigma})^2
(1 + \sigma^{2}
 R^{(n)}^{\omega^{(n)},U^{(n)},\delta\sigma})
\EQ(goodestimates)
$$
Once we have estimates such as those above, it is quite standard
in K.~A.~M. theory that choosing at each iterative step
$\sigma^{(n)} = \sigma_0 2^{n}$, with $\sigma_0$
small enough so that the domain
controlled does not reduce to
the empty set, if the initial remainder is small enough
we obtain a procedure that is converging
quadratically. It is also possible to show that the
smallness hypothesis that one needs are also satisfied.
Once we have this quadratic convergence, it is also
quite straightforward to show that the composition
$g^{(0)}_\epsilon \circ g^{(1)}_\epsilon \circ
\cdots \circ g^{(n)}_\epsilon $
converges in a non empty domain.
\SECTION Acknowledgements
I thank S. Tompaidis for several comments on a preliminary
version of the manuscript and E. Meletlidou
for providing \clm(tompaidis). This research has been supported
by NSF and TARP grants as well as an AMS Centennial
Fellowship and and a URI from U.T. Austin.
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\end