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\begin{document}
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{\large\bf The 1991 AMS-MSC: 34C15, 34C20, 58F27, 70H05}
\vspace{7mm}
{\large\bf M.~B.~Sevryuk}
\vspace{5mm}
{\small
Institute of Energy Problems of Chemical Physics
The Russia Academy of Sciences
Lenin prospect~38, Bldg.~2, Moscow~117829, Russia
E-mail: rusin@chph.ras.ru
}
\vspace{5mm}
\baselineskip=18.7pt
{\LARGE\bf Invariant sets \\ of degenerate Hamiltonian systems \\
near equilibria}
\vspace{7mm}
\baselineskip=15.5pt
\hrule
\vspace{3mm}
{\small
For any collection of $n\geq 2$ numbers $\omega_1,\ldots,\omega_n$, we prove
the existence of an infinitely differentiable Hamiltonian system of
differential equations $X$ with $n$ degrees of freedom that possesses the
following properties: 1) $0$ is an elliptic (provided that all the $\omega_i$
are different from zero) equilibrium of system $X$ with eigenfrequencies
$\omega_1,\ldots,\omega_n$; 2) system $X$ is linear up to a remainder flat at
$0$; 3) the measure of the union of the invariant $n$-tori of system $X$ that
lie in the $\varepsilon$-neighborhood of $0$ tends to zero as
$\varepsilon\to 0$ faster than any prescribed function. Analogous statements
hold for symplectic diffeomorphisms, reversible flows, and reversible
diffeomorphisms. The results obtained are discussed in the context of the
standard theorems in the KAM theory, the well-known R\"ussmann and
Anosov--Katok theorems, and a recent theorem by Herman.
}
\vspace{3mm}
\hrule
\baselineskip=18.7pt
\parindent=5mm
\vspace{7mm}{\large\bf
1. Formulation of the results and a discussion
}\vspace{5mm}
According to the KAM (Kolmogorov--Arnol$'$d--Moser) theory, any neighborhood
of an elliptic equilibrium $0$ of a sufficiently smooth Hamiltonian system
of differential equations in $\mR^{2n}$ and any neighborhood of an elliptic
fixed point $0$ of a sufficiently smooth symplectic diffeomorphism
$(\mR^{2n},0)\to(\mR^{2n},0)$ contain invariant Lagrangian $n$-tori carrying
quasiperiodic motions, provided that some nonresonance and nondegeneracy
conditions are met \cite{Arn0}--\cite{SieMos} (see also a detailed discussion
in \cite{MosSymp1}--\cite{BHS}). One usually formulates the nondegeneracy
condition in terms of the coefficients of the Birkhoff normal form of
degree~$4$. For instance, in the case of differential equations, the Birkhoff
normal form of the Hamilton function of degree~$4$ is (up to an additive
constant)
\begin{equation}
H=\sum_{i=1}^n\omega_i\tau_i+\half\sum_{i,j=1}^n\omega_{ij}\tau_i\tau_j
+O\left( |\tau|^{5/2}\right) , \qquad \omega_{ij}=\omega_{ji},
\label{eno}
\end{equation}
where $\tau_i=\half(p_i^2+q_i^2)$ are the local action variables
($1\leq i\leq n$), $|\tau|=\tau_1+\cdots+\tau_n$, and $(p,q)$ are suitable
Darboux coordinates with the origin at $0$. The nondegeneracy condition in
this case is that at least one of the two inequalities
\begin{equation}
\det|\omega_{ij}|\neq 0, \qquad
\det\left|
\begin{array}{cc} \omega_{ij} & \omega_i \\ \omega_j & 0 \end{array}
\right| \neq 0
\label{owt}
\end{equation}
holds. The nonresonance condition consists in the possibility itself of the
reduction of the Hamilton function to normal form~(\ref{eno}), i.e., in the
absence of resonances of order~$4$ and lower among the eigenfrequencies
$\omega_i$. The invariant KAM tori are close to the tori $\tau=\const$,
and the {\em relative\/} Lebesgue measure $\mu_{\varepsilon}$ of their union
in the polydisc
$B_{\varepsilon}=\left\{ \max_{i=1}^n\tau_i\leq\half\varepsilon^2\right\} $
of radius $\varepsilon>0$ centered at $0$ tends to $1$ as $\varepsilon\to 0$.
Analogous results hold for reversible differential equations and mappings
(see \cite{MosStable,BHS,RevSyst} and references therein).
Various estimates of the decay rate for $1-\mu_{\varepsilon}$ as
$\varepsilon\to 0$ are known. If the first of the inequalities~(\ref{owt})
is valid and there are no resonances of order $l\geq 4$ and lower among the
eigenfrequencies $\omega_i$ then \cite{Poeschel}
\[
1-\mu_{\varepsilon}=O\left( \varepsilon^{(l-3)/2}\right) .
\]
For Diophantine collections of eigenfrequencies:
\[
|\langle k,\omega\rangle|\geq\gamma|k|^{-\nu} \quad \mbox{for all} \quad
k\in\mZ^n\prokol
\]
(here $\langle k,\omega\rangle=k_1\omega_1+\cdots+k_n\omega_n$,
$|k|=|k_1|+\cdots+|k_n|$, while $\nu$ and $\gamma$ are positive constants
independent of $k$), an exponential estimate of the difference
$1-\mu_{\varepsilon}$:
\[
1-\mu_{\varepsilon}=O\left(
\exp\left\{ -\left[ \frac{c}{\varepsilon}\right] ^{1/(\nu+1)}\right\} \right) ,
\qquad c=\const>0\mbox{ is independent of }\varepsilon,
\]
is obtained in paper \cite{DelshG} under the conditions that the Hamilton
function is analytic and the second of the inequalities~(\ref{owt}) holds.
In paper \cite{JorbaV}, a similar estimate with exponent $2/(\nu+1)$ in place
of $1/(\nu+1)$ is proven under the conditions that the Hamilton function is
analytic and the first of the inequalities~(\ref{owt}) is valid.
The nondegeneracy requirement for the Birkhoff normal form of degree~$4$ can
be relaxed. For example, consider an area preserving diffeomorphism of the
plane with elliptic fixed point $0$. If the eigenvalues of its linearization
at $0$ are not roots of unity of degree $l\geq 4$ and lower, then this
diffeomorphism is reduced in some Darboux coordinates $(p,q)$ to the Birkhoff
normal form of degree~$l$
\[
z\mapsto z\exp\left[ i\left(
\omega_{(0)}+\omega_{(1)}\tau+\cdots+\omega_{(s)}\tau^s\right) \right]
+O\left( |z|^l\right) ,
\]
where $z=p+iq$, $\tau=\half|z|^2=\half(p^2+q^2)$, and $s=[l/2]-1$. For the
existence of invariant KAM curves in any neighborhood of $0$, it suffices that
at least one of the real coefficients $\omega_{(1)},\ldots,\omega_{(s)}$
does not vanish \cite{Arn0}--\cite{SieMos}.
Nevertheless, one cannot abandon completely the nondegeneracy condition in the
local KAM theory, whatever the eigenvalues of the linearization at $0$ of
the Hamiltonian vector field or symplectic mapping are. This statement follows
from theorems 1 and 2 below. It is discussing and proving those theorems that
the present note is devoted to.
Before formulating theorems 1 and 2, introduce some notation (which in fact
was already used partially). By $|a|=|a_1|+\cdots+|a_N|$, we will always
denote the $l_1$-norm of a vector $a\in\mR^N$, and by
$\langle a,b\rangle=a_1b_1+\cdots+a_Nb_N$, the scalar product of two vectors
$a,b\in\mR^N$. Consider the Euclidean space $\mR^{2n}$ ($n\geq 1$) endowed
with coordinates $(p_1,\ldots,p_n,q_1,\ldots,q_n)$ and symplectic structure
$dp\wedge dq$. In the sequel, we will always have in view this structure
while speaking of Hamiltonian flows and symplectic mappings. Set
\[
p_i=\sqrt{2\tau_i}\cos\varphi_i, \quad q_i=\sqrt{2\tau_i}\sin\varphi_i,
\qquad 1\leq i\leq n,
\]
then $dp\wedge dq=d\tau\wedge d\varphi$. By $P:\mR^{2n}\to\mR_+^n$, denote
the ``projection'' $P:(p,q)\mapsto\tau$. Fix a closed $n$-dimensional cube
\[
\Trho=\left\{ \tau\in\mR^n \;\; \left| \;\;
0\leq\tau_i\leq\half\rho^2, \;\; 1\leq i\leq n\right. \right\}
\]
with edge length $\rho^2/2$ ($\rho>0$). Then
\[
\Brho=\PP(\Trho)
\]
is a $2n$-dimensional polydisc of radius $\rho$ centered at $0$.
Let $\{K_m\}_{m\geq 1}$ be an arbitrary sequence of pairwise disjoint closed
subsets in $\Trho$ possessing the following properties:
\[
K_m\subset\Int\Trho=\left\{ \tau\in\mR^n \;\; \left| \;\;
0<\tau_i<\half\rho^2, \;\; 1\leq i\leq n\right. \right\}
\quad \mbox{for all} \quad m
\]
and
\begin{equation}
\max_{\tau\in K_m}|\tau|\to 0 \quad (m\to+\infty).
\label{ruof}
\end{equation}
A subset of the space $\mR^{2n}$ will be said to be {\em nonexceptional}, if
for any $k\in\mZ^n\prokol$, this subset contains a point $(\tau,\varphi)$
such that $\tau_1>0$, \ldots, $\tau_n>0$, $\sin\langle k,\varphi\rangle\neq 0$.
Nonexceptional sets are exemplified by $n$-tori $\tau=f(\varphi)$ for which
$f_i(\varphi)>0$ for all $i$ and $\varphi$.
\vspace{2mm}
{\sc Theorem~1.}
{\sl Let $n\geq 2$ and let a Hamilton function $H_0:\Brho\to\mR$ depend on
the local action variables $\tau$ only:
\[
H_0(p,q)=F(\tau),
\]
where $F\in\CC(\Trho,\mR)$. If the image of the frequency mapping
\begin{equation}
\Omega:\Trho\to\mR^n, \qquad \Omega(\tau)=\partial F(\tau)/\partial\tau
\label{evif}
\end{equation}
lies in some hyperplane in $\mR^n$ passing through $0$, then there exists a
$\CC$-function $H:\Brho\to\mR$ possessing the following properties:
{\rm i)} the difference $H(p,q)-H_0(p,q)$ is flat at $0$;
{\rm ii)} the Hamiltonian system $X_H$ with Hamilton function $H$ does not
admit nonexceptional invariant sets lying wholly in
$\bigcup_{m=1}^{\infty}\PP(K_m)$.
If function $F$ is linear\/ {\rm (}i.e., $\Omega=\const$ and the system
$X_{H_0}$ with Hamilton function $H_0$ is linear\/{\rm )}, then property\/
{\rm ii)} can be replaced with the following one:
{\rm ii$'$)} system $X_H$ does not admit invariant $n$-tori of the form
$\tau=f(\varphi)$ that intersect
$\Int\left[ \;\bigcup_{m=1}^{\infty}\PP(K_m)\;\right] $.}
\vspace{2mm}
{\sc Theorem~2.}
{\sl Let $A_0:\Brho\to\Brho$ be a symplectic diffeomorphism of the form
\[
A_0:(\tau,\varphi)\mapsto(\tau,\varphi+\Omega(\tau)), \qquad
\Omega(\tau)=\partial F(\tau)/\partial\tau,
\]
where $F\in\CC(\Trho,\mR)$. If the image of the frequency mapping
$\Omega:\Trho\to\mR^n$ lies in some affine hyperplane\footnote{i.e., in a
hyperplane not necessarily passing through $0$} in $\mR^n$, then there exists
a symplectic $\CC$-diffeomorphism $A:\Brho\to\Brho$ possessing the following
properties:
{\rm i)} the difference $A(p,q)-A_0(p,q)$ is flat at $0$;
{\rm ii)} diffeomorphism $A$ does not admit nonexceptional invariant sets
lying wholly in $\bigcup_{m=1}^{\infty}\PP(K_m)$.
If function $F$ is linear\/ {\rm (}i.e., $\Omega=\const$ and diffeomorphism
$A_0$ is linear\/{\rm )}, then property\/ {\rm ii)} can be replaced with the
following one:
{\rm ii$'$)} diffeomorphism $A$ does not admit invariant $n$-tori of the form
$\tau=f(\varphi)$ that intersect
$\Int\left[ \;\bigcup_{m=1}^{\infty}\PP(K_m)\;\right] $.}
\vspace{2mm}
These theorems are proven in section~2.
\vspace{2mm}
{\sc Remark~1.}
Recall that a $\CC$-function defined in a neighborhood of a point $a\in\mR^N$
is said to be {\em flat\/} at $a$, if all the derivatives of this function
of all the orders $l\geq 0$ vanish at $a$.
\vspace{2mm}
{\sc Remark~2.}
The statement of theorem~1 holds for $n=1$ as well, but we have excluded
the case $n=1$ because of its triviality. Indeed, for $n=1$ the hypothesis
of the theorem means that $\Omega\equiv 0$, i.e., $H_0=\const$ (one can set
$H_0\equiv 0$). Let $\chi:\mR\to\mR$ be an arbitrary $\CC$-function flat at
$0$ and such that $d\chi/dx\neq 0$ for $x\neq 0$. Then the only bounded
solutions of the system governed by the Hamilton function $H(p,q)=\chi(pq)$
are equilibria filling up the coordinate axes $\{p=0\}$ and $\{q=0\}$.
\vspace{2mm}
{\sc Remark~3.}
The requirement that the image of the unperturbed frequency mapping should
not lie in any hyperplane (the so called {\em R\"ussmann condition\/}
\cite{Ruess0}--\cite{Ruess90}) is the optimal nondegeneracy condition
in the global KAM theory for analytic systems \cite{BHS,Control}. For the last
several years, this condition and other close requirements have become more
or less standard, see \cite{BHS}, \cite{Ruess0}--\cite{RCD} (an extensive
bibliography is given in \cite{BHS}).
\vspace{2mm}
{\sc Remark~4.}
Theorems 1 and 2 guarantee the absence of {\em any\/} invariant tori of the
form $\tau=f(\varphi)$ with the properties indicated, not only that of tori
carrying quasiperiodic motions (and even not only that of Lagrangian tori).
\vspace{2mm}
{\sc Remark~5.}
If the linear part $\langle\omega,\tau\rangle$ of function $F(\tau)$ in
theorem~1 satisfies some nonresonance conditions, then system $X_H$ possesses
$n$ invariant two-dimensional $\CC$-surfaces passing through $0$ and foliated
into closed trajectories with periods close to $2\pi/|\omega_1|$, \ldots,
$2\pi/|\omega_n|$ (the Lyapunov center theorem, see, e.g.,
\cite{SieMos,Mos68,MMcC}).
\vspace{2mm}
One can choose the sequence of sets $\{K_m\}_{m\geq 1}$ in such a way that
the measure of the difference
\begin{equation}
B_{\varepsilon}\setminus\left[ \;\bigcup_{m=1}^{\infty}\PP(K_m)\;\right]
\label{eerht}
\end{equation}
will decrease as $\varepsilon\to 0$ faster than any prescribed function.
Moreover, within this framework, each of the sets $K_m$ can be chosen to be a
polyhedron. Indeed, let $\chi:[0;\rho]\to\mR_+$ be an arbitrary monotonically
increasing function such that $\chi(0)=0$ and $\chi(\varepsilon)>0$ for
$\varepsilon>0$. Introduce the notation
\[
x_m=\frac{\rho}{m}, \qquad y_m=\chi\left( \frac{\rho}{m+1} \right) .
\]
Fix a polyhedron
\[
K_m\subset\left( \Int T_{x_m^2/2}\setminus T_{x_{m+1}^2/2}\right)
\]
such that the measure of the difference
\[
B_{x_m}\setminus\left[ B_{x_{m+1}}\bigcup\PP(K_m)\right]
\]
is less than
\[
\min_{r=1}^m\frac{y_r}{2^{m-r+1}}.
\]
Then for any $0<\varepsilon\leq\rho$ ($x_{s+1}<\varepsilon\leq x_s$ for some
$s\geq 1$) the measure of difference~(\ref{eerht}) will be less than
\[
\sum_{m=s}^{\infty}\min_{r=1}^m\frac{y_r}{2^{m-r+1}}
\leq y_s\sum_{m=s}^{\infty}\frac{1}{2^{m-s+1}}=y_s\leq\chi(\varepsilon).
\]
Taking this into account, we arrive at the following corollaries of theorems
1 and 2.
\vspace{2mm}
{\sc Corollary~1.}
{\sl Let $n\geq 2$ and let $0$ be an elliptic\/ {\rm (}if all the $\omega_i$
are different from zero\/{\rm )} equilibrium of a linear Hamiltonian system
in $\mR^{2n}$ with quadratic Hamilton function $H_0=\langle\omega,\tau\rangle$.
Then there exists a Hamilton $\CC$-function $H$ such that
{\rm i)} the difference $H(p,q)-H_0(p,q)$ is flat at $0$;
{\rm ii)} the measure of the union of the $n$-tori of the form
$\tau=f(\varphi)$ invariant under the system with Hamilton function $H$
and lying in the polydisc $B_{\varepsilon}$ of radius $\varepsilon>0$ centered
at $0$ tends to zero as $\varepsilon\to 0$ faster than any prescribed
function.}
\vspace{2mm}
{\sc Corollary~2.}
{\sl Let $A_0:\mR^{2n}\to\mR^{2n}$ be a linear symplectic diffeomorphism
$(\tau,\varphi)\mapsto(\tau,\varphi+\omega)$ with elliptic\/ {\rm (}if none of
the ratios $\omega_i/\pi$ is an integer\/{\rm )} fixed point $0$. There exists
a symplectic $\CC$-diffeomorphism $A$ such that
{\rm i)} the difference $A(p,q)-A_0(p,q)$ is flat at $0$;
{\rm ii)} the measure of the union of the $n$-tori of the form
$\tau=f(\varphi)$ invariant under diffeomorphism $A$ and lying in the polydisc
$B_{\varepsilon}$ of radius $\varepsilon>0$ centered at $0$ tends to zero as
$\varepsilon\to 0$ faster than any prescribed function.}
\vspace{2mm}
It is useful to compare these corollaries with the following three statements.
\vspace{2mm}
{\sc The R\"ussmann theorem} \cite{Ruess67}{\sc .}
{\sl Let $A:(\mR^2,0)\to(\mR^2,0)$ be an area preserving analytic mapping
of the plane into itself for which $0$ is an elliptic fixed point. Let the
eigenvalues of the linearization of mapping $A$ at $0$ be equal to
$\exp(\pm i\alpha)$, where angle $\alpha$ is strongly incommensurable with
$2\pi$, i.e., there exist positive constants $\nu$ and $\gamma$ such that
\[
|s\alpha-2\pi r|\geq\gamma s^{-\nu} \quad \mbox{for all} \quad
s\in\mN, \;\; r\in\mZ.
\]
Suppose that by a formal canonical transformation, one can reduce mapping $A$
to its linear part, i.e., the rotation by angle $\alpha$. Then this mapping
is reducible to its linear part by a convergent canonical transformation as
well. An analogous\/ {\rm (}and even a stronger\/{\rm )} statement holds for
analytic Hamiltonian systems of differential equations as well\/ {\rm (}with
any number $n$ of degrees of freedom\/{\rm )}. Namely, if the collection
of the eigenfrequencies $\omega_1,\ldots,\omega_n$ of the system is
Diophantine and the formal Birkhoff normal form of the Hamilton function is
a function of its linear part $\langle\omega,\tau\rangle$ {\rm (}e.g.,
coincides with $\langle\omega,\tau\rangle${\rm )}, then one can reduce the
Hamilton function to this normal form by a convergent canonical
transformation.}
\vspace{2mm}
{\sc Remark~6.}
In paper \cite[\S~12]{Bryuno}, various generalizations (pertaining to the
case of differential equations) of the R\"ussmann theorem are obtained.
\vspace{2mm}
{\sc The Herman theorem} \cite{Herm98}{\sc .}
{\sl Let $A:(\mR^2,0)\to(\mR^2,0)$ be an area preserving $\CC$-mapping
of the plane into itself for which $0$ is an elliptic fixed point. Let the
eigenvalues of the linearization of mapping $A$ at $0$ be equal to
$\exp(\pm i\alpha)$, where angle $\alpha$ is strongly incommensurable with
$2\pi$. Then in any neighborhood of point $0$, mapping $A$ possesses
invariant circles\footnote{i.e., invariant closed curves surrounding $0$}
whose union is of positive measure. An analogous statement holds for
Hamiltonian $\CC$-systems of differential equations with two degrees of
freedom as well.}
\vspace{2mm}
{\sc A particular case of the Anosov--Katok theorem}
\cite{Anosov1,Anosov2}{\sc .}
{\sl In any neighborhood\/ {\rm (}in the $\CC$-topology\/{\rm )} of the
rotation of a two-dimensional disc by any angle $\alpha$, there are area
preserving $\CC$-diffeomorphisms of this disc that are ergodic.}
\vspace{2mm}
We arrive at the following description of invariant curves of symplectic
diffeomorphisms of the plane. Let $A:(\mR^2,0)\to(\mR^2,0)$ be an area
preserving $\CC$-mapping of the plane into itself, and let the linear part at
$0$ of this mapping be the rotation by angle $\alpha$ strongly incommensurable
with $2\pi$. Consider the formal Birkhoff normal form of mapping $A$ at $0$:
\begin{equation}
z\mapsto z\exp\left[ i\left(
\alpha+\sum_{s=1}^{\infty}\omega_{(s)}\tau^s\right) \right] ,
\qquad z=p+iq, \;\; \tau=\half|z|^2.
\label{evlewt}
\end{equation}
We are interested in the relative measure $\mu_{\varepsilon}$ of the union
of invariant circles of mapping $A$ in the $\varepsilon$-neighborhood
of fixed point $0$. If at least one of the coefficients $\omega_{(s)}$ in
(\ref{evlewt}), $s\geq 1$, is different from zero then
$\mu_{\varepsilon}\to 1$ as $\varepsilon\to 0$ according to the standard KAM
theory. On the other hand, if $\omega_{(s)}=0$ for all $s$ then still
$\mu_{\varepsilon}>0$ for each $\varepsilon>0$ (by the Herman theorem), but
$\mu_{\varepsilon}$ can {\em decrease\/} as $\varepsilon\to 0$, and faster
than any prescribed function (according to corollary~2). In both the cases,
fixed point $0$ of mapping $A$ is Lyapunov stable. Finally, if
$\omega_{(s)}=0$ for all $s$ but $A$ is analytic then by the R\"ussmann
theorem, $\mu_{\varepsilon}\equiv 1$ (to be more precise, some punctured
neighborhood of point $0$ is wholly foliated into invariant circles of
mapping $A$).
The linear parts of Anosov--Katok ergodic diffeomorphisms are rotations
by angles $\alpha$ that are not strongly incommensurable with $2\pi$.
What can one say about area preserving $\CC$-mappings
$A:(\mR^2,0)\to(\mR^2,0)$ such that the eigenvalues $\exp(\pm i\alpha)$ of
the linearization of $A$ at $0$ are roots of unity (the case opposite to
the strong incommensurability of $\alpha$ with $2\pi$)? No analogue of the
Herman theorem exists for such mappings because for $\alpha=2\pi r/s$
($s\in\mN$, $r\in\mZ$, $2r/s\notin\mZ$) fixed point $0$ can be unstable even
for analytic mappings $A$ \cite{SieMos}. On the other hand, the R\"ussmann
theorem is carried over to the case $\alpha=2\pi r/s$. If $\alpha$ is
commensurable with $2\pi$ then the Birkhoff normal form\footnote{to be more
precise, the Birkhoff--Gustavson normal form, see \cite{Mos68}} of mapping $A$
is, generally speaking, more complicated than (\ref{evlewt}). However,
exploiting the methods developed in \cite{Bryuno,Pliss} one can easily show
that if an area preserving analytic mapping $A$ with $\alpha=2\pi r/s$ is
formally linearizable then it is linearizable by a convergent canonical
transformation as well \cite{Gong98}. In Gong's paper \cite{GongHelv} devoted
to the normalization of some special reversible diffeomorphisms
characterized by a parameter $\lambda\in\mC$, $|\lambda|=1$, the case where
$\arg\lambda$ is strongly incommensurable with $2\pi$ and the case where
$\lambda^s=1$ are treated in parallel.
\vspace{2mm}
{\sc Remark~7.}
Other examples of situations where flat perturbations of Hamiltonian
$\CC$-flows or symplectic $\CC$-diffeomorphisms change the dynamics
drastically can be found in paper \cite{Douady}.
\vspace{2mm}
{\sc Remark~8.}
Let $A_{\varepsilon}$ be the mapping of a two-dimensional annulus of the form
$A_{\varepsilon}:(\tau,\varphi)\mapsto(\tau,\varphi+\varepsilon\tau)$,
where $0<\varepsilon\ll 1$. An interesting criterion for the absence of
invariant curves of the form $\tau=f(\varphi)$ for perturbations of the
mappings $A_{\varepsilon}$ is obtained in recent paper \cite{Ortega}.
\vspace{5mm}{\large\bf
2. Proofs
}\vspace{5mm}
In the global KAM theory, the analogue of theorems 1 and 2 is the following
proposition: if the image of the unperturbed frequency mapping lies in some
hyperplane (passing through $0$ in the case of flows) then one can destroy all
the invariant tori by an arbitrarily small perturbation of the initial
integrable system. This statement is proven in \cite{BHS,Control} for
Hamiltonian flows and in \cite{Control} for symplectic diffeomorphisms. By
some complication of the constructions of papers \cite{BHS,Control}, one may
prove local theorems 1 and 2 of the present note.
We will prove in detail theorem~1 only.
\vspace{2mm}
{\sc Proof of theorem~1.}
Due to condition~(\ref{ruof}), the sets $K_m$, $m\geq 1$, admit pairwise
disjoint neighborhoods. We can therefore choose two other sequences of closed
subsets $\{S_m\}_{m\geq 1}$ and $\{Q_m\}_{m\geq 1}$ in $\Trho$ possessing the
following properties:
\vspace{2mm}
a) $K_m\subset\Int S_m$, $S_m\subset\Int Q_m$, $Q_m\subset\Int\Trho$;
b) the sets $Q_m$, $m\geq 1$ are pairwise disjoint;
c) $\max_{\tau\in Q_m}|\tau|\to 0$ ($m\to+\infty$).
\vspace{2mm}
The condition that the image of mapping~(\ref{evif}) lies in some hyperplane
in $\mR^n$ passing through $0$ means that
$\langle c,\Omega(\tau)\rangle\equiv 0$ for some $c\in\mR^n\prokol$.
A new Hamilton function $H(p,q)$ will be looked for in the form
\[
H=H_0+\sum_{m=1}^{\infty}h_m,
\]
where each $\CC$-function $h_m:\Brho\to\mR$ vanishes outside $\PP(Q_m)$. Fix
$m\in\mN$ and a $\CC$-function $\psi:\Brho\to\mR$ equal to unit in $\PP(S_m)$
and equal to zero outside $\PP(Q_m)$; it suffices to consider a partition of
unity $\psi$, $\Psi$ of class $\CC$ subordinate to the covering
$\PP(\Int Q_m)$, $\mR^{2n}\setminus\PP(S_m)$ of space $\mR^{2n}$ \cite{Hirsch}.
Choose a matrix $M\in{\rm GL}(n,\mR)$ sufficiently close to the identity
matrix (in particular, subject to the condition $\MM Q_m\subset\Trho$) and
such that $Mc$ is proportional to some integer vector $k\in\mZ^n\prokol$. Set
\begin{equation}
\Onew(\tau)=\frac{\partial}{\partial\tau}F(\MM\tau)=\trans\Omega(\MM\tau)
\label{thgie}
\end{equation}
(the superscript t means transposing), then
$\langle k,\Onew(\tau)\rangle\equiv 0$. We will look for function $h_m$ in
the form
\begin{equation}
h_m=[\; F(\MM\tau)-F(\tau)+\delta\cos\langle k,\varphi\rangle \;]\psi,
\label{enin}
\end{equation}
where $\delta\in\mR\prokol$. Having chosen matrix $M$ to be sufficiently
close to the identity matrix and then number $\delta\neq 0$ to be sufficiently
small, one can satisfy the inequalities
\begin{equation}
\left| \frac{\partial^{|\kappa|}h_m(p,q)}
{\partial p_1^{\kappa_1}\cdots\partial p_n^{\kappa_n}
\partial q_1^{\kappa_{n+1}}\cdots\partial q_n^{\kappa_{2n}}}\right|
\leq\frac{1}{m} \quad \mbox{for all} \quad \kappa\in\mZ_+^{2n}, \;\;
0\leq |\kappa|\leq m
\label{xis}
\end{equation}
in $\PP(Q_m)$. These inequalities imply that the function
$\sum_{m=1}^{\infty}h_m$ is flat at $0$.
In $\PP(S_m)$, the Hamilton function $H_0+h_m$ is equal to
$F(\MM\tau)+\delta\cos\langle k,\varphi\rangle$ and affords the system of
differential equations
\[
\dot{\tau}=\delta k\sin\langle k,\varphi\rangle, \qquad
\dot{\varphi}=\Onew(\tau),
\]
for which
\[
\frac{d}{dt}\langle k,\varphi\rangle=\langle k,\Onew(\tau)\rangle\equiv 0.
\]
Suppose that the system with Hamilton function $H_0+h_m$ admits a
nonexceptional invariant set lying wholly in $\PP(K_m)$. On that set, choose
a point $(\tau^0,\varphi^0)$ for which $\sin\langle k,\varphi^0\rangle\neq 0$.
Along the trajectory passing through this point, one has
\[
\dot{\tau}=\delta k\sin\langle k,\varphi^0\rangle=\const\neq 0.
\]
The contradiction obtained proves the first part of theorem~1. Note that we
nowhere used that $\psi\equiv 1$ in $\PP(S_m)$; it sufficed for our purposes
that $\psi\equiv 1$ in $\PP(K_m)$.
Let now the function $F(\tau)=\langle\omega,\tau\rangle$ be linear, so that
$\Omega=\const=\omega$. Choose a sufficiently small vector $\vartheta\in\mR^n$
such that the vector $\omega+\vartheta$ is proportional to an integer one:
$\omega+\vartheta=\Lambda L$ where $\Lambda>0$ and $L\in\mZ^n$. Let
$\langle k,L\rangle=0$ for some integer vector $k\in\mZ^n\prokol$ (recall that
$n\geq 2$). Then $\langle k,\omega+\vartheta\rangle=0$. We will look for
function $h_m$ in the form
\begin{equation}
h_m=[\; \langle\vartheta,\tau\rangle+
\delta\cos\langle k,\varphi\rangle \;]\psi,
\label{net}
\end{equation}
where $\delta\in\mR\prokol$. Having chosen vector $\vartheta$ and number
$\delta\neq 0$ to be sufficiently small, one can satisfy the
inequalities~(\ref{xis}). Fix a number $\sigma>0$ such that $\tau'\in S_m$
whenever $\tau\in K_m$ and $|\tau'-\tau|\leq\sigma$, and impose the
additional condition that
\begin{equation}
|\delta|\leq\frac{\Lambda\sigma}{2\pi|k|}.
\label{neves}
\end{equation}
In $\PP(S_m)$, the Hamilton function $H_0+h_m$ is equal to
$\langle\omega+\vartheta,\tau\rangle+\delta\cos\langle k,\varphi\rangle$ and
affords the system of differential equations
\[
\dot{\tau}=\delta k\sin\langle k,\varphi\rangle, \qquad
\dot{\varphi}=\omega+\vartheta,
\]
for which
\[
\frac{d}{dt}\langle k,\varphi\rangle=\langle k,\omega+\vartheta\rangle\equiv 0.
\]
Suppose that the system with Hamilton function $H_0+h_m$ admits an invariant
torus of the form $\tau=f(\varphi)$ intersecting $\Int\PP(K_m)$. On this torus,
choose a point $(f(\varphi^0),\varphi^0)$ inside $\PP(K_m)$ for which
$\sin\langle k,\varphi^0\rangle\neq 0$. Due to inequality~(\ref{neves}), the
trajectory $(\tau(t),\varphi(t))$ starting at this point remains in
$\PP(S_m)$ for $|t|\leq 2\pi/\Lambda$. One has:
\[
\renewcommand{\arraystretch}{1.375}
\begin{array}{ll}
\varphi(0)=\varphi^0, \quad &
\varphi(2\pi/\Lambda)=\varphi^0+2\pi L=\varphi^0\bmod 2\pi, \\
\tau(0)=f(\varphi^0), \quad &
\tau(2\pi/\Lambda)=f(\varphi^0)+
2\pi\Lambda^{-1}\delta k\sin\langle k,\varphi^0\rangle
\neq f(\varphi^0).
\end{array}
\renewcommand{\arraystretch}{1.}
\]
The contradiction obtained proves the second part of theorem~1.
\vspace{2mm}
{\sc Proof of theorem~2} runs entirely similarly to the proof of theorem~1,
and we will point out just the necessary changes in our construction.
The condition that the image of mapping~(\ref{evif}) lies in some affine
hyperplane in $\mR^n$ means that
$\langle c,\Omega(\tau)\rangle\equiv c_0$ for some $c\in\mR^n\prokol$,
$c_0\in\mR$. For a fixed $m\geq 1$, choose a matrix $M\in{\rm GL}(n,\mR)$
sufficiently close to the identity matrix and such that the pair $(Mc,c_0)$
is proportional to some pair $(k,2\pi k_0)$ with $k\in\mZ^n\prokol$,
$k_0\in\mZ$. Introduce the mapping $\Onew(\tau)$ by formula~(\ref{thgie}) and
the function $h_m$ by formula~(\ref{enin}). Then
$\langle k,\Onew(\tau)\rangle\equiv 2\pi k_0$. The desired symplectic
diffeomorphism $A$ is determined by the generating function
$F+\sum_{m=1}^{\infty}h_m$, i.e., in $\PP(Q_m)$
\[
A:(\tau,\varphi)\mapsto(\tau',\varphi'),
\]
where
\[
\tau=\tau'+\frac{\partial h_m(\tau',\varphi)}{\partial\varphi}, \qquad
\varphi'=\varphi+\Omega(\tau')+
\frac{\partial h_m(\tau',\varphi)}{\partial\tau'}.
\]
One can prove by the standard methods \cite{Arn3,Arn2} that $A$ is well
defined for sufficiently small $h_m$. In particular, for $\tau'\in S_m$ we
have
\[
\tau'=\tau+\delta k\sin\langle k,\varphi\rangle, \qquad
\varphi'=\varphi+\Onew(\tau'),
\]
and
\[
\langle k,\varphi'\rangle=
\langle k,\varphi\rangle+\langle k,\Onew(\tau')\rangle=
\langle k,\varphi\rangle+2\pi k_0=
\langle k,\varphi\rangle\bmod 2\pi.
\]
One easily sees that $A$ cannot admit nonexceptional invariant sets lying
wholly in $\PP(K_m)$.
Let now the function $F(\tau)=\langle\omega,\tau\rangle$ be linear, so that
$\Omega=\const=\omega$. Choose a sufficiently small vector $\vartheta\in\mR^n$
such that the vector $\omega+\vartheta$ has the form
$\omega+\vartheta=2\pi L/L_0$ with $L\in\mZ^n$ and $L_0\in\mN$. Let
$\langle k,L\rangle=L_0k_0$ for some $k\in\mZ^n\prokol$ and $k_0\in\mZ$. Then
$\langle k,\omega+\vartheta\rangle=2\pi k_0$. We will look for function $h_m$
in the form~(\ref{net}) imposing the additional condition
\begin{equation}
|\delta|\leq\frac{\sigma}{L_0|k|}.
\label{nevele}
\end{equation}
In $A^{-1}[\PP(S_m)]$, the mapping $A:(\tau,\varphi)\mapsto(\tau',\varphi')$
has the form
\[
\tau'=\tau+\delta k\sin\langle k,\varphi\rangle, \qquad
\varphi'=\varphi+\omega+\vartheta.
\]
If a point $(\tau^0,\varphi^0)$ lies in $\PP(K_m)$ then its image
$(\tau^{\ast},\varphi^{\ast})$ under the $L_0$-th iteration $A^{L_0}$ of
diffeomorphism $A$ lies in $\PP(S_m)$ due to inequality~(\ref{nevele}) and
\[
\tau^{\ast}=\tau+L_0\delta k\sin\langle k,\varphi^0\rangle, \qquad
\varphi^{\ast}=\varphi^0+2\pi L=\varphi^0\bmod 2\pi.
\]
Hence, $A$ cannot admit invariant tori of the form $\tau=f(\varphi)$ that
intersect $\Int\PP(K_m)$.
\vspace{2mm}
{\sc Remark~9.}
For Lagrangian tori of the form $\tau=f(\varphi)$ invariant under a
symplectic diffeomorphism, Herman has obtained, in some cases, estimates
of the derivatives $\partial f_i/\partial\varphi_j$ in terms of the entries
of the Jacobi matrix of the diffeomorphism \cite{HermExist,HermApriori}.
The dynamics on such tori was studied by Herman in papers
\cite{HermDyn1,HermDyn2}.
\vspace{2mm}
Theorems 1 and 2 (as well as corollaries 1 and 2) can be carried over to
reversible differential equations and reversible diffeomorphisms. The reversing
involution is $G:(p,q)\mapsto(p,-q)$, or
$G:(\tau,\varphi)\mapsto(\tau,-\varphi)$. We will not formulate the reversible
counterparts of theorems 1 and 2 here and will just point out briefly the
key changes in the proofs. In the sequel, as before, $M$ denotes a matrix in
${\rm GL}(n,\mR)$ close to the identity matrix.
In the case of flows, one has a $G$-reversible system of differential equations
\[
\dot{\tau}=0, \qquad \dot{\varphi}=\Omega(\tau),
\]
where now the mapping $\Omega:\Trho\to\mR^n$ is, generally speaking, not a
gradient. Let $\langle c,\Omega(\tau)\rangle\equiv 0$ for some
$c\in\mR^n\prokol$, and let $Mc$ be proportional to $k\in\mZ^n\prokol$. Set
\[
\Onew(\tau)=\trans\Omega(\tau), \qquad \langle k,\Onew(\tau)\rangle\equiv 0.
\]
The perturbed $G$-reversible system of differential equations in $\PP(Q_m)$ has
the form
\[
\dot{\tau}=\eta[\; \sin\langle k,\varphi\rangle \;]\psi, \qquad
\dot{\varphi}=\Omega(\tau)+[\; \Onew(\tau)-\Omega(\tau) \;]\psi,
\]
where $\eta\in\mR^n\prokol$ is small while a buffer function
$\psi(\tau,\varphi)$ is even in $\varphi$. In $\PP(S_m)$, this system takes
the form
\[
\dot{\tau}=\eta\sin\langle k,\varphi\rangle, \qquad
\dot{\varphi}=\Onew(\tau).
\]
In the case of mappings, one has a $G$-reversible diffeomorphism
\[
A_0:(\tau,\varphi)\mapsto(\tau,\varphi+\Omega(\tau)).
\]
Let $\langle c,\Omega(\tau)\rangle\equiv c_0$ for $c\in\mR^n\prokol$,
$c_0\in\mR$, and let the pair $(Mc,c_0)$ be proportional to a pair
$(k,2\pi k_0)$ with $k\in\mZ^n\prokol$, $k_0\in\mZ$. Set
\[
\Onew(\tau)=\trans\Omega(\tau), \qquad
\langle k,\Onew(\tau)\rangle\equiv 2\pi k_0.
\]
The perturbed $G$-reversible diffeomorphism $A$ in $\PP(Q_m)$ is determined as
\[
A=GWGW^{-1},
\]
where the diffeomorphism
$W:(\tau,\varphi)\mapsto(\widetilde{\tau},\widetilde{\varphi})$ has the form
\[
\renewcommand{\arraystretch}{1.5}
\begin{array}{l}
\widetilde{\tau}=\tau-\half\eta
[\; \sin\langle k,\widetilde{\varphi}\rangle \;]
\psi(\tau,\widetilde{\varphi}), \\
\widetilde{\varphi}=\varphi-\half\Omega(\tau)-\half
[\; \Onew(\tau)-\Omega(\tau) \;]\psi(\tau,\varphi)
\end{array}
\renewcommand{\arraystretch}{1.}
\]
with small $\eta\in\mR^n\prokol$. It is not hard to verify that for
$\psi\equiv 1$, the diffeomorphism $A:(\tau,\varphi)\mapsto(\tau',\varphi')$
takes the form
\[
\tau'=\tau+\eta\sin\langle k,\varphi\rangle, \qquad
\varphi'=\varphi+\Onew\left( \half(\tau+\tau')\right) .
\]
\vspace{2mm}
I am indebted to V.~I.~Arnol$'$d who had taught me the KAM theory. Special
thanks go to M.~R.~Herman who has let me know a number of his results prior
to publication, as well as to Xianghong Gong and A.~I.~Ne\u{\i}shtadt for
interesting discussions on the dynamics of symplectic and reversible
mappings of the plane.
\newpage
\baselineskip=15.5pt
\parindent=0mm
{\small
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}
\end{document}