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\TITLE A RIGOROUS PARTIAL JUSTIFICATION OF GREENE'S CRITERION
\footnote{${}^{\rm 1}$ } {\rm This preprint is available from the
math-physics electronic preprints archive.
Send e-mail to {\tt mp\_arc@math.utexas.edu} for instructions}
\ENDTITLE
\AUTHOR Corrado Falcolini
\footnote{${}^2$}{ Permanent address: Dipt. di Matematica,
II Universit\'a degli Studi di Roma ``Tor Vergata'',
Via del Fontanile di Carcaricola, 00133 Roma, Italia}
\footnote{${}^3$}{ e-mail address: {\tt FALCOLINI\%40085.decnet.cern.ch}},
Rafael de la Llave
\footnote{${}^4$}{Supported in part by National Science
Foundation Grants}
\footnote{${}^5$}{ e-mail address: {\tt llave@math.utexas.edu}}
\FROM Dept. of Mathematics
University of Texas at Austin
Austin TX 78712
\ENDTITLE
\ABSTRACT
We prove several theorems that lend support for Greene's
criterion for existence or not of invariant circles in twist maps.
In particular, we show that some of the implications
of the criterion are correct when the
Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic.
We also suggest a simple modification that can work
in the case that the Aubry-Mather sets have non-zero Lyapunov exponents.
\ENDABSTRACT
\SECTION Introduction
In a remarkable paper, [Gr] proposed a criterion for existence
of non-trivial invariant circles in twist mappings. Using it, he
was able to compute the critical value at which golden circles
ceased to exist with an accuracy that even today is unsurpassed
and that, at the time of its appearance was almost impossible to believe.
The purpose of this paper is to present some mathematically rigorous
results that serve as a partial justification of Greene's criterion.
We recall that, given any number $\omega$, Aubry-Mather
theory establishes the existence of
at least one set on which the motion is semiconjugate to a rotation of
angle $\omega$ in the circle. Such sets enjoy several
remarkable properties; among them, they are either Cantor sets
or Lipschitz circles ( we refer to [Ma2] for a review).
From the practical point of view, it is quite important to distinguish
between these two possibilities, since
an invariant circle is a complete barrier to
long scale transport and Cantor sets are not.
Greene's criterion asserts that an invariant circle exists if and
only if a certain limit is positive. We show that if the circle exists and is
analytic or sufficiently differentiable, the number is $0$
and the limit is reached exponentially fast.
If, on the other hand, there is an
Aubry-Mather set
with the conjectured
rotation number and positive Lyapunov exponent, the liminf is positive.
In case that the Aubry-Mather set is uniformly hyperbolic,
there is a positive limit.
The practical importance of Greene's criterion is that the
limit is computed on periodic orbits
which are quite easy to compute.
There is considerable evidence that Greene's criterion
is correct, at least in some cases.
First there is the agreement between the
quantitative values obtained rigorously.
In recent times, the methods of {\sl
``computer assisted proofs ''} have been applied to the problem of
computing the range of applicability of K.~A.M. theorem.
For particular examples, there are positive results on values for which the theorem
does apply ([CC] [CC2], [Ra], [LR]) and well as
results on values for which the
conclusions of the theorem are false ([Ju],[Ma],[MP]).
Notice that the values in [Ju], [LR] differ by
about 7\% and that the value obtained by Greene's method
is in the allowed interval. The value of [Ju] agrees
to several decimal places with the value of [Gr].
The paper [OS] introduces another method that not only establishes
non-existence of invariant circles, but also that the invariant set of
golden mean rotation is hyperbolic. Even if the implementation of the
criterion in [OS] is not completely rigorous because
it ignores the effects of round--off error, the authors
have performed a very careful analysis that makes the
results of the paper quite close to a proof.
It seems that the algorithm proposed is within reach of ``computer-assisted
proofs'' technology. The agreement of these numerical results with Greene's
value is quite remarkable and lends support to the conjecture that,
for the standard family, as
soon as the invariant circle disappears it becomes a hyperbolic
invariant set.
Besides the rigorous numerical results indicated above, there
are arguments based on the renormalization group that
lend credence to the Greene's method.
There is considerable evidence that the
phenomena of break-up of invariant tori
can be described for a large class of families
by a renormalization group picture [McK1], [McK2].
(Indeed the arguments for existence
of a fixed point and linearization of the
spectrum
of these two papers are
quite close to being a proof.)
This renormalization group picture implies
that all dynamical quantities have {\sl ``bulk properties''}
and that to compute the parameter value at
which a transition occurs
we can use as indicator whatever property is more
convenient to measure. (This is quite similar to the
fact that we can measure
the boiling point of water by examining electric, magnetic properties, density, etc.)
The scaling properties predicted by the
renormalization group for periodic orbits
can be displayed quite dramatically in the
{\sl "fractal diagrams"} [SBi] and can be used to
improve the numerical effectiveness of Greene's method.
([McK1]\S 4.6.2,\S 4.6.3).
We should, nevertheless, point out that the
renormalization group picture gets considerably more
complicated when the familes are slightly different
from the standard one.(~[W1], [W2], [KMcK]~)
which can be explained by saying that the
dynamics of the renormalization operator
has basins in which the dynamics is
controlled by a more complicated landmark than a simple fixed
point as exhibited in [McK1], [McK2]. This matter
merits further investigation.
In this paper, we present some rigorous
results which are independent of
the renormalization group picture, but rather
use standard techniques from K.A.M. and from
hyperbolic perturbation theories.
We consider Greene's methods as part
of a long tradition in mathematics
of using periodic orbits,
the simplest landmark that organizes the long term behaviour
as the skeleton on which to study dynamical properties.
Perhaps the forerunner of this approach could
be Poincar\'e ( see e.g. [Po], vol I, p. 82).
\vskip 2 em
{\it
\narrower
Il y a m\^eme plus: voici un fait que je n'ai pu
demontrer rigorousment, mais qui me parait
pourtant tr\`es vrisemblable.
\narrower
\`Etant donn\'ees des
\`equations de la forme
definie dans le ${\rm n}^{\rm o}$ 13 [ Hamilton eq. ]
et unne solution particuli\`ere quelconque de ces equations,
on peut toujours trouver une solution p\'eriodique
(dont la p\'eriode peut, il est vrai,\^etre tr\`es longue),
tel que la diff\'erence entre les deux solutions
soit aussi petite qu'on le veut, pendant un temps, assi long
qu'on le veut. D'allieurs, ce qui nous rende ces solutions
p\'eriodiques si precieuses, c'est qu'elles sont, pour
ansi dire, la seule br\`eche par o\`u nous puissions
essayer de p\'en\'etrer dans une place jusqu'ici r\'eput\'ee
inabordable.
}
\SECTION Notation and statement of results.
Let $f:\torus^1\times\real\to\torus^1\times\real$ be an
analytic, area preserving map.
Let $x\in\torus^1\times\real$ satisfy $f^N(x) =x$. We say that it
is a periodic orbit of type $M/N$, $N,M\in\integer$ if, denoting by $\tilde f$
and $\tilde x$ the lifts of $f$ and $x$ to the universal cover of
$\torus^1\times\real$, we have $\tilde f^N (\tilde x) =\tilde x + (M,0)$.
We denote the orbit of a periodic point by $o(x)$.
For such an orbit Greene defined the ``residue'' by
$$R(x) = {1\over4}\Bigl(\Tr\bigl( Df^N (x)\bigr) - 2\Bigr)
\EQ(residue)$$
Greene defined the ``mean residue'' to be $[R(x)]^{1/N}$ and
observed numerically that,
if $M_i/N_i$ were the continued convergents of an irrational number $\omega$
and $x_i$ are points of type $M_i/N_i$, $[R(x_i)]^{1/N_i}\to\rho(\omega)$
and that $\rho (\omega) >1$ when there is no invariant circle and that
$\rho (\omega) <1$ when there is an invariant circle.
The practical importance of this criterion lies on the fact that
there are quite efficient methods for the computation of periodic orbits.
Moreover, by computing the residues of a significative number of
periodic orbits,
we can get an idea of the set of rotation numbers for which there is an
invariant circle.
We notice that
$$R(x) = {1\over4}\Bigl(\Tr\bigl( Df(f^{N-1} (x))\ldots Df(x)\bigr) -2
\Bigr) $$
so that, using the invariance of the trace under cyclic permutations the
residue is the same for all points in an orbit.
Notice that the residue of a periodic
orbit can be easily related to the
eigenvalues of the derivative of the return map.
If one eigenvalue is $\lambda$, by the preservation of
area the other one should be $1/\lambda$ and the trace
is $\lambda + 1/\lambda$. If $\lambda\approx\exp(\gamma M)$, the
mean residue should be $\approx\gamma$.
Hence, it is natural that the Lyapunov exponents come into play
when the residue grows exponentially fast.
We recall that a number $\omega$ is called Diophantine
if, for every $p,q\in\natural$ we have:
$$
|\omega - p/q | \ge K |q|^{-\nu}
\EQ(Diophantine)
$$
These numbers play an important role in
K.~A.~M. theory. We also recall that
the convergents of the continued fraction expansion
of a number $\omega$ satisfy: $\left|\omega - p/q\right|\le K/q^2$,
so the best exponent $\nu$ we can hope to have in
\equ(Diophantine) is $2$. The numbers for
which it is possible to satisfy
\equ(Diophantine) with $\nu =2$ are called
{\sl ``constant type numbers'' } and
even if they have measure zero, they
include all quadratic irrationals and, in particular,
are dense. If we take any $\nu > 2$, the
set of Diophantine numbers with this
exponent has full measure.
\CLAIM Theorem(main1)
Assume that $f$ as above admits a topologically non-trivial analytic
invariant circle and that the motion on it is analytically conjugate to
a rotation $\omega$ such that:
$$\lim {1\over N}\sup_{q\le N}\log |\omega - p/q| =0
\EQ(weakdiophantine)
$$
Then, for every $k\in\natural$, we can can find $C_k >0$, depending on $f$ and
on the
circle, such that for
every $N,M$ such that $|\omega - M/N|\le 1/N$ and any periodic point $x$
of type $M/N$, we have
$$|R(x)|\le C_k\left|\omega - {M\over N}\right|^k N$$
In particular, if
$\left|\omega - {M_i\over N_i}\right|\le {K\over (N_i)^2}$
({\it e.g.}, if $M_i/N_i$ are the continued fraction convergents to $\omega$),
then $\limsup |R(x_i)|^{1/N_i}\le 1$.
\REMARK
The same method of proof establishes that if $\omega$ is Diophantine
and
the circle and the map are $C^r$ then,
if $x$ is a periodic orbit of type $M/N$,
we have $R(x)\le C_k |\omega - M/N|^k $
for all $k\le k^*(r)$, where $k^*(r)$ depends on the
exponent $\nu$ in \equ(Diophantine), but
$k^* (r)\to\infty$ as $r\to\infty$.
For Diophantine numbers, the previous result can be improved from the
residue being smaller than any power to be exponentially small.
\CLAIM Theorem(main1improved)
Let $f$ be as before, $\omega$ as
in \equ(Diophantine). Assume that
$\sup_{|\Im \varphi|\le\delta} |f(A,\varphi)|\le\Gamma\le\infty$,
$\sup_{|\Im\varphi|\le\delta} |f^{-1} (A,\varphi)|\le\Gamma\le\infty$,
and that there is a mapping $K:\torus^1\to\torus^1\times\real$
with $f(K(\varphi)) = K(\varphi+\omega)$ and that $\sup_{|\Im\varphi| }
\le\delta|K(\varphi) |\le\Gamma $.
\vskip 0 em
Then, there exists a constant $D>0$ -- depending on the
Diophantine properties of the
number $\omega$ -- such that
for every periodic orbit $x$ of
type $M/N$ with $|\omega -M/N|\le 1/N $
$$|R (x)|^{1/N}\le D e^{-D\Gamma\delta^{1+\nu} |\omega - M/N| }$$
\REMARK
The fact that the residues converge exponentially fast to zero
when there is an analytic invariant circle is
one of the predictions of
the renormalization group analysis.
Notice that, when one knows that the convergence of
a sequence to its limit is exponentially fast, it is
possible to use Aitken extrapolation [SBu] \S 5.10
to compute the limit more effectively.
This leads to more effective implementations of
Greene's method. This idea is suggested in
[McK1]\S 4.6.3.
\REMARK
The conclusion of \clm(main1improved)
suggests that there is a relation between the
exponent of decrease of the
residue and the analyticity domain
of the circle.
Unfortunately, the statement we have proved
is not enough to conclude that. Notice that the
coefficient also depends on $\Gamma$ wich depends on the
analyticity properties of the circle.
The main reason to conjecture that such a relation should exist
is that both of them scale with the renormalization group in the same way.
We now proceed to state our results
for the case in which the
Aubry-Mather sets are hyperbolic.
\CLAIM Theorem(main2)
Assume that $\Gamma$ is a hyperbolic Aubry-Mather set of rotation number
$\omega$ and that $\{ x_n\}$ is a sequence of periodic points of type
$M_n/N_n$ such that $o(x_n)$ converges to $\Gamma$.
Then, for sufficiently large $n$, $|R(x_n)|^{1/N_n} >\lambda >1 $.
Actually, if the hyperbolic set has Lyapunov
exponent $\gamma$,
$\lim_n R(x_n)^{1/N_n} = e^\gamma$
\CLAIM Theorem(Lyapunov)
Let $f$ be a $C^2$ twist mapping as above and let
$\Gamma$ be an Aubry Mather Cantor set
with rotation number $\omega\notin\rational$.
If $f|_{\Gamma}$ has a positive Lyapunov exponent $\gamma$,
then
\item{a)} For any sequence ${x_n} $ of periodic periodic orbits
of type $M_n/ N_n$ converging to $\Gamma$
$\liminf_n R(x_n)^{1/N_n} \ge e^\gamma$.
\item{b)} There exists a sequence of
periodic points ${x_n}$ of type $M_n / N_n$
converging to $\Gamma$
such that $\liminf_n R( x_n)^{1/N_n} = e^\gamma.$
\REMARK
In principle, when we use Lyapunov exponents, we should specifiy with respect
to which ergodic measure we take them.
Nevertheless, as we will
discuss in the proof
of \clm(main2) and \clm(Lyapunov),
for Aubry-Mather sets with irrational rotation number there is only
one invariant measure
with support in the set, so that the notation is unambiguous.
\REMARK
Notice that from the point of view of practical applications,
claim $a)$ is stronger since it makes an assertion about all
possible sequences of orbits.
It implies that, if there is a Cantor set with positive Lyapunov
exponents, any sequence of orbits we take will succeed
in excluding the existence of an invariant circle. Claim $b)$
establishes that by looking at the mininum of mean residues
we can guess the Lyapunov exponent of the Cantor set.
We do not know whether it is possible to find
examples in which a sequence $x_n$ of periodic orbits
satisfies $\liminf_n R(x_n)^{1/N_n} > e^\gamma$.
Previous experience with non-uniformly hyperbolic
systems suggest that this will be the case.
\REMARK
Claim $a)$ is much easier to prove than claim $b)$.
In fact, claim $a)$ is an abstract statement about
uniquely ergodic systems. (Notice that it does not claim that
periodic orbits exist.) Claim $b)$, on the other hand,
uses methods of Pesin theory and establishes
existence of periodic orbits.
We remark that an sketch of a method of proof
of \clm(main1) and \clm(main1improved) has been available for a
long time.
In particular it was suggested by John Mather
as early as 1982 (see e.g. [MacK] p. 1.3.2.4).
Nevertheless, we thought it would be worth publishing a detailed account of
these arguments since fairly quantitative results
are needed in subsequent numerical work by the authors [FL].
The method we present here is optimized for
computability and it does not require
to perform succesive changes of variables. It is also
written in such a way that it readily generalizes to higher number
of variables or to the case when the values of
some of the parameters are complex. The later is used
essentially in [FL].
The proof of \clm(main2) is a standard result of perturbation
of hyperbolic structures.
\clm(Lyapunov) is a basic result about approximation of
non-uniformly hyperbolic dynamical systems by periodic orbits.
Except for the quantitative results of the
Lyapunov exponent,
part b) is the main lemma in [Ka].
Related results appear in [Ma\~ne].
The proof we presented here
is based on a shadowing lemma for
partially hyperbolic systems,
which has other applications.
The method of proof is inspired by the
treatment of hyperbolic sets in [La].
Results related to ours have been proved in [McK3].
\SECTION Proof of the results
\SUBSECTION Proof of \clm(main1) and \clm(main1improved)
The basic idea in the proof of \clm(main1) is to show that given $k\in
\natural$, we can find a complex neighborhood $U_k$ of the invariant circle
$\Gamma$, an integrable mapping $I_k$ and a constant $C_k$ in such a way
that
$$\| f- I_k\|\le C_k\dist (x,\Gamma)^k$$
Then, an elementary perturbation
argument would allow to estimate the
trace of the derivatives of
orbits that stay close to the invariant
circle. It will be a corollary of
Moser's twist mapping theorem that
the maximum distance of a periodic
orbit to the invariant circle can be estimated
-- in the appropriate coordinates-- by the
difference between the rotation numbers
of the orbit and the circle.
The construction of an integrable system will be done by finding
an approximate integral.
It will simplify the notation to choose an appropriate system of coordinates
\CLAIM Proposition(coordinates)
Let $f:\torus^1\times\real$ be as in \clm(main1) and $\Gamma$ be an
invariant circle $f|_\Gamma$ analytically conjugate to a rotation $\omega$.
Then, we can find a globally canonical analytic mapping $h$ defined in a
neighborhood of $\Gamma$, with an analytic inverse in a neighborhood of
$\Gamma$ and such that
$$h\circ f\circ h^{-1} (A,\varphi) =\bigl( A+A^2u (A,\varphi) ,
\varphi +\omega +Av (A,\varphi)\bigr)$$
with $u,v$ analytic,
$${\partial Au\over \partial A}\ge\alpha >0\hbox{ for }
|A|\le\varep\ ,\varphi\in\torus^1$$
\PROOF
By Birkhoff's theorem [Ma], [Fa] we know that $\Gamma$ is the graph of
an analytic function $\gamma :\torus^1\to\real$.
The transformation $h_1 :\torus^1\times\real\hookleftarrow$
defined by
$$h_1 (A,\varphi) =\bigl( A+\gamma (\varphi),\varphi\bigr)$$
is globally symplectic and sends the circle $\torus^1\times\{0\}$
into the graph of $\gamma$. Hence, $h\circ f\circ h_1^{-1}$ leaves invariant
the circle $\torus^1\times\{0\}$. Hence
$$h_1\circ f\circ h_1^{-1} (A,\varphi) =
\bigl(Au_1 (A,\varphi), v_1 (A,\varphi)\bigr)$$
Since the motion on this circle is conjugate to a rotation, there exists
an analytic $\delta :\torus^1\to\torus^1$ with an analytic inverse
(hence $\delta' (\varphi) \ne 0$) such that $v_1 (0,\delta(\varphi)) =
\delta (\varphi+\omega)$.
The transformation $h_2 (A,\varphi) = (A/\delta'(\varphi),\delta(\varphi))$
is globally canonical and
\vskip 0 em
$h_2^{-1}\circ h_1\circ f\circ h_1^{-1}\circ h_2$
is of the form
$$(A,\varphi)\to (A',\varphi')\equiv\bigl( Au_2 (A,\varphi),
\varphi +\omega + Av_2 (A,\varphi)\bigr)$$
Since the map preserves volume,
$$\det\pmatrix{
{\partial A'\over\partial A} & {\partial A'\over\partial\varphi}\cr
{\partial\varphi'\over\partial A}&{\partial\varphi'\over\partial\varphi}\cr}
=1 $$
and since
$$
{\partial\varphi'\over\partial\varphi}\Big|_{A=0} = 1\quad ,\quad
{\partial A'\over\partial\varphi}\Big|_{A=0} = 0
$$
we should have:
$${\partial A'\over\partial A}\Big|_{A=0} = 1\ .$$
That is the form of the map claimed in \clm(coordinates).
It is a simple calculation to show that the transformations $h_1,h_2$
preserve the positive twist condition. Hence, the last inequality
in the claim is established.
\QED
\CLAIM Lemma(integral)
Let $f$ be as in \clm(coordinates), $\omega$ Diophantine.
\vskip1pt
Given any $k\in\natural$, we can find analytic functions $H_0 (\varphi),
\ldots,H_k(\varphi)$ so that
$H=\sum_{i=0}^k A^i H_i (\varphi)$
satisfies
$|H\circ f-H|\le C_{k+1} A^{k+1}$
\PROOF
We will derive a hierarchy of equations and show that we can solve
them recursively.
We observe that
$$
H\circ f ( A,\varphi) =\sum\left(A + A^2u(A,\varphi)\right)^i
H_i(\varphi +\omega + Av(A,\varphi))
\EQ(Hoff)
$$
Moreover, if we expand $H_i(\varphi +\omega + Av(A,\varphi)) $
using Taylor's formula
in $A$, we obtain:
$$
H_i\bigl(\varphi +\omega +Av(A,\varphi)\bigr)
=\sum_{i = 0 }^N H_i^j(\varphi)A^{i+j} +\OO( A^{i+N+1})
\EQ(Hexpansion)
$$
where $H_i^0(\varphi) = H_i(\varphi +\omega)$,
$H_i^1(\varphi) = H'_i(\varphi +\omega) v(0,\varphi)$.
For higher $j$, $H_i^j$ is an expression involving derivatives of
$H_i$ and of $v$. We observe that the derivatives entering in
$H_i^j$ are of order up to $j$ and that the derivatives of $H_i$ enter linearly.
If we substitute \equ(Hexpansion) into \equ(Hoff)
we obtain
$$
H\circ f ( A,\varphi) = \sum_{i=0}^N A^i\left( H_i(\varphi +\omega) +
H_{i-1}(\varphi +\omega)u(0,\varphi) + L_i(\varphi)\right) +\OO(A^{N+1})
\EQ(Hinpowers)
$$
where $L_i$ is an expression involving
$H_0,\cdots,H_{i-2}$
and their derivatives
of order
up to $i$ as well as derivatives of $H_{i-1}$.
We emphasize that $H_{i-1}$ only enters in $L_i$
in the form of derivatives, so that if $H_{i-1}$
changes by a constant, $L_i$ remains unaltered.
If we equate the term of $A^i$ in \equ(Hinpowers)
with that in the expansion for $H$,
we are lead for $i > 0$ to a hierarchy of equations of the form:
$$
H_i (\varphi +\omega) +u (0,\varphi) H_{i-1} (\varphi +\omega)
+L_i (\varphi) = H_i (\varphi)
\EQ(hierarchy2)
$$
We recall the following:
\CLAIM Proposition(cohomology)
Let $\eta :\torus^1\to\real$ be an analytic function, $\int_{\torus^1}
\eta =0$.
Let $\omega$ be a Diophantine number.
Then, there exist $H:\torus^1\to\real$ analytic satisfying
$$H(\varphi +\omega) - H(\varphi) =\eta (\varphi)$$.
Moreover, $H$ is unique up to an additive constant. In particular,
all the derivatives of $H$ are uniquely determined.
\PROOF
The proof of \clm(cohomology) is quite well-known and is obtained
just matching the Fourier coefficients. Details can be found, among
other places, in [Ar]\S 12, [SM]\S32
\QED
Using \clm(cohomology) it is possible to solve all the equations in
\equ(hierarchy2).
We assume inductively that $H_0,\ldots,H_{i-2}$ are determined and
that $H_{i-1}$ is determined up to an additive constant.
Since $L_i$ depends only on $H_0,\cdots, H_{i-2}$
and the derivatives of $H_{i-1}$, we see that
$L_i$ is determined. Using the twist condition, we have
$\int u(0,\varphi)\ne0$ so that it is possible to determine
uniquely the
additive constant in $H_{i-1}$ by imposing
$$
\int u(0,\varphi) H_{i-1}(\varphi) +\int L_i (\varphi) =0\ .
$$
Using \clm(cohomology), $H_i$ is determined up to an additive constant,
so that we recover the induction hypothesis
with $i-1$ replaced by $i$.
The first step of the induction reduces to an obvious identity.
\QED
Notice that, if $H$ is a conserved quantity so is any function of $H$.
Observe also that the curves $H=h$ for small $|h|$, are
homotopically non-trivial since $H$ is a small perturbation of $A$.
We can define
$$\widetilde H (h) =\int_{H=h} A\, d\varphi\ .$$
The function $H^* =\widetilde H (H)$ will be conserved up to
$O(A^{k+1})$ and it has the property that
$$\int A\, d\varphi = h$$
$H^* = h$.
We can now define a canonical transformation in such a way that $H^*$
becomes the action variable.
In effect, if we can find an $S$ in such a way that
$$\eqalign{ H^* & = A- {\partial S (H^*,\varphi)\over \partial\varphi}\cr
\varphi' & =\varphi + {\partial S\over\partial H^*} (H^*,\varphi)\cr}
\EQ(generating)$$
then, the transformation $(A,\varphi)\to (H^*,\varphi')$ will be
canonical.
Using the first equation of \equ(generating) we can determine $S$ up to
addition of a function of $H^*$.
We can determine this additive function in such a way that
$\varphi' (A,O) =0$.
Expressed in the coordinates $(H,\varphi')$ the mapping $f$ has the form
$$(H,\varphi')\buildrel {\tilde f }\over\longrightarrow
\bigl( H,\varphi' +\omega + H\Delta (H)\bigr) + R(H,\varphi')$$
where $|R|\le C_N H^N$.
We emphasize that, since all the changes of variables are analytic,
the estimates on the remainder remain true in a complex
neighborhood of $(\torus^1\times\{ 0\})$ of the form $\{|\Im\varphi'|
\le\xi$, $|A|\le\xi\}$. As a consequence $\| DR\|\le CH^{N-1}$. Hence,
$$D\tilde f (H,\varphi') =\pmatrix{1&\Gamma (H)\cr 0&1\cr} + O(H^{N-1})$$
We notice that the trace of the derivative of a periodic point ---
hence the residue --- can be computed in any system of coordinates.
Since
$$D\tilde f^N (H,\varphi') = D\tilde f\bigl(\tilde f^{N-1} (H,\varphi')\bigr)
D\tilde f\bigl(\tilde f^{N-2} (H,\varphi')\bigr)\cdots
D\tilde f(H,\varphi')$$
we will find it useful to estimate eigenvalues of products
of matrices close to upper triangular.
\CLAIM Lemma(perturbation)
Let $\{ A_i\}_{i=1}^N $ be a set of $2\times 2$ matrices of the
form $A_i =\left( {1\atop 0}\ {a_i\atop 1}\right)$ with $\sup_{1\le i\le N}
|a_i|\le A$.
\vskip1pt
Let $\{ B_i\}_{i=1}^N$ satisfy
$$\sup_{\scriptstyle 1\le i\le N\atop\scriptstyle j,k=1,2} |(B_i)_{jk} -
|(A_i)_{jk}|\le\varep \quad \hbox{\rm with }\varep\le A\ .$$
Then $B= B_1 ,\ldots,B_N$ satisfies
$$|\Tr B-2|\le 2\left[\bigl( 1+3\sqrt{A}\,\sqrt{\varep}\,\bigr)^N -1\right]$$
\PROOF
Given any norm on 2-vectors, if we define $\| C\| =\sup_{v\in\real^2}
\| Cv\|/\|v\|$, clearly all eigenvalues of $C$ have modulus not bigger than $\| C\|$.
Hence, for a $2\times 2$ matrix $C$, $\Tr C\le 2\|C\|$.
If we define $\| v\| = |v_1|\delta + |v_2|$, then
$$\left\|\pmatrix{C_{11}&C_{12}\cr C_{21}&C_{22}\cr}\right\|\le
\max (|C_{11}| +\delta^{-1} |C_{21}|\ ,\ |C_{21}|\delta + |C_{22}|)\ .$$
In particular, for matrices such as those in the hypothesis of
\clm(perturbation) and for $\delta\le 1$
$$\eqalign{\| A_i\| &\le 1+ |a_i|\delta\le 1+A\delta\cr
\| A_i - B_i\| &\le\varep\max\left ( (1+\delta^{-1}) , ( 1+\delta)\right) =
\varep(1 +\delta^{-1} )
\cr}
\EQ(norms)$$
We can write
$$\leqalignno{B \equiv &B_1\cdots B_N =\cr
&=\bigl( A_1 + B_1 -A_1)\bigr)\bigl( A_2+(B_2-A_2)\bigr)\cdots
\bigl( A_N +(B_N-A_N)\bigr)\cr
\noalign{\hbox{Expanding and grouping by the same factors $(B_i-A_i)$}}
B & = A_1\cdots A_N +\cr
&\qquad +\sum_i A_1\cdots A_{i-1} (B_i-A_i) A_{i+1}\cdots A_N +\cr
&\qquad +\sum_{i,j} A_1\dots A_{i-1} (B_i-A_i) A_{i+1}\cdots A_{j-1}
(B_j -A_j) A_{j+1}\cdots A_N\cr
&\qquad +\cdots\cdots\cr
&\qquad + (B_1-A_1)\cdots (B_N-A_N)\cr}$$
The trace of the first term is 2 and the trace of the other terms can be
bounded by twice the norm. Using the estimates of the norms in \equ(norms)
and bounding the norms of the products by the product of the norms of
the factors,
we can bound the residue by:
$$\eqalign{&2 {N\choose 1} (1+A\delta)^{N-1} (1+\delta^{-1}\varep
+ 2{N\choose 2} (1+A\delta)^{N-2}\bigl( (1+\delta^{-1})\varep\bigr)^2
+\cr
&+\quad\quad\cdots + 2{N\choose N} (1+\delta^{-1}\varep)^N =\cr
&\qquad = 2\Bigl[\bigl( 1+A\delta + (1+\delta^{-1})\varep\bigr)^N -1\Bigr]\cr}$$
If we choose $\delta =\sqrt{{\varep}/A}$,-- which is smaller than $1$--
the upper bound for the residue
we just computed becomes
$$2\Bigl[\bigl( 1+\sqrt{A}\,\sqrt{\varep} +\varep +\sqrt{A}\,\sqrt{\varep}
\,\bigr)^N -1\Bigr]$$
Since $\varep\le A$, $\varep\le\sqrt{A}\,\sqrt{\varep}$ and we
obtain the bound in the claim of the lemma.
\QED
The next ingredient in the proof is an argument that tells that
periodic orbits of rotation number close to that of $\Gamma$ are
contained in a small strip near $H=0$.
Notice that, even if it is not difficult to show that most of the points
should be close enough (otherwise the twist would force the rotation
to be much bigger), we want the much stronger property that all the points
of the orbit are close to the invariant circle.
\CLAIM Lemma(closeto)
If $|\omega - M/N|$ is small enough, all orbits of type $M/N$ are
contained in the strip
$$|H|\le\left|\omega - {M\over N}\right| K$$
where $K$ depends only on the system and on the circle.
\PROOF
By Moser's twist theorem we can find invariant circles whose rotation
numbers $\omega\in\Omega$.
Moreover, ${\mu (\Omega\cap [-\Delta_0+\omega,\Delta+\omega])\over 2\Delta}
\to 1$ where $\mu$ denotes the Lebesgue measure.
It follows that if $M/N$ is close enough to $\omega$, there are going to be
points $\omega'$ of $\Omega$ in $[{M\over N},{M\over N}+|\omega -{M\over N}|]$.
Furthermore, since the mapping that to a rotation number associates the
invariant circle of this rotation number is Lipschitz, the circle of
rotation number $\omega'$ is contained in
$$|H|\le K|\omega-\omega'|\le 2K\left|\omega - {M\over N}\right|$$
By the twist property the orbit of rotation number $M/N$ has to be
contained between the circles of rotation number $\omega'$ and $\omega$.
\QED
\REMARK
Notice that the dependence of $K$ on the system and on the circle
is rather weak. It is, roughly, the Lipschitz constant in the mapping
that to a rotation numbner asociates a K.~A.~M. circle when we topologize the
circles with the $C^0$ norm. In particular, it can be chosen uniformly in
a sufficient $C^5$ neighborhood of the integrable case.
If we know that a map has a sufficiently differentiable circle,
it can be chosen uniformly in a $C^5$ neighborhood.
Using \clm(perturbation), \clm(closeto), it follows that, for every $k$,
$$
R_{M/N}\le 2\left[\left( 1+C_k K\left|\omega - {M\over N}\right|^k
\right)^N -1\right]
\EQ(mainbound)
$$
Were $K$ and $C_k$ are the constants respectively in \clm(closeto),
\clm(integral).
If $\quad |\omega -M/N|^k C_k K N \quad $ is sufficiently small
and $N$ is sufficiently large,
we can bound the R.H.S. of \equ(mainbound) by:
$$
8 C_k K\left|\omega- {M\over N}\right|^k N
\EQ(finalbound)
$$
Since $C_k$ is an arbitrary constant,
multiplying it by $8$ does not change anything,
so that we can denote it by the same letter.
This finishes the proof of \clm(main1).
\QED
\REMARK
The method carried out above can be generalized to higher dimensions.
First, the normal form given by \clm(coordinates) can be carried out with
the only modification that, rather than using the determinant of the
transformation
being $1$, we have to use the preservation of the symplectic form.
Moreover, it is possible to use an analogue of \equ(hierarchy2)
to compute as many independent approximate conserved quantities as the
dimension of the tori.
We point out that an alternative apporach to compute similar normal forms
can be found in [SZ] based on the use of generating functions and succesive
transformations. Even if from the point
of view of theoretical calculations both methods could be used,
the method explained here lends itself to quite efficient computer implementations
so that it should be possible to obtain
good estimates of the residues in concrete cases as well as
estimates of the times of escape from neighborhoods of the tori in higher
dimensions.
Notice that, for any $k$, \equ(finalbound) produces a valid estimate
of the residue.
The proof of \clm(main1improved) will consist only in
estimating explicitely the $C_k$ so that
for given $N,M$ we can choose the $k$ that gives the best bound.
We recall that we had to solve for $H_i$ in
$$
H\circ f =\sum_{i=0}^k\bigl( A+A^2 u(A,\varphi)\bigr)^i
H_i\bigl(\varphi +\omega +Av(A,\varphi)\bigr) =
\sum_{i=0}^k A^i H_i(\varphi) +\OO(A^{k+1})
\EQ(Hmatch)
$$
If we write $\theta =\varphi +\omega + Av(A,\varphi) $,
\equ(Hmatch) can be written as
$$
H\circ f =\sum_{i=0}^k\bigl( A+A^2\tildeu(A,\theta)\bigr)^i
H_i\bigl(\theta)\bigr) =
\sum_{i=0}^k A^i H_i(\theta -\omega - A\tildev(A,\theta) ) +\OO(A^{k+1})
\EQ(Hmatch)
$$
where $\tildeu$, $\tildev$ are analytic functions whose domain
of analyticity depends only on the properties of $u$, $v$. Also
$\tildeu(0,\phi) > 0$.
We will assume that they are defined in a domain of
the form $\{\theta\big| |\Im(\theta )|\le\delta\}$
and that their absolute values there are bounded by a constant $K$.
If $H:\torus^1\mapsto\complex $ is an analytic
function we will denote by:
$$ ||H||_\eta =\sup_{|\Im\theta|\le\eta} | H(\theta)|$$
As before, we can solve \equ(Hmatch) recursively.
Expanding both sides in powers of $A$
and equating the factors of $A^i$, we obtain:
$$
H_i(\theta) + H_{i-1}(\theta) u(0,\phi) + L^1_i(\theta) = H_i(\theta -\omega) + L^2_i(\theta)
\EQ(hierarchyreduced)
$$
where, as before, $L^1_i$, $L^2_i$ are expressions involving
$H_0,\cdots , H_{i-2}$ and their derivatives and
the derivatives of $H_{i-1}$.
The procedure to solve \equ(hierarchyreduced) is very similar to the one
that we used in the proof of \clm(main1).
We assume inductively that $H_0,\cdots,H_{i-2}$ are determined completely and
that $H_{i-1}$ is determined up to an additive constant.
Then, we determine the additive constant
in $H_{i-1} $in such a way that
$$
\int\left( H_{i-1} u(0,\varphi) + L^1_i(\varphi) - L^2_i(\varphi)\right) d\varphi =0
$$
Then, using \clm(cohomology), we can determine $H_i$ up to a constant.
We denote by $\barH_i =\int H_i(\theta) d\theta$
and by $\tildeH_i(\theta) = H_i(\theta) -\barH_i$
\CLAIM Lemma(inductivesolutions)
The equations \equ(hierarchyreduced)
can be solved recursively.
For $\omega$ Diophantine,
if $\delta - k\eta > 0$,
we have:
$$
\eqalign{
&||\tildeH_i||_{\delta - i\eta}\le E\left( D \right)^i\cr
&|\barH_i|\le E\left( D \right)^i
}
\EQ(inductivehypothesis)
$$
Where $D $ is a number of the form $D =\tildeK {\eta^{-1 -\nu}}$
and $\tildeK$ depends on the system but can be taken uniformly in a
$\|\quad \|_\delta$ neighborhood. Similarly for $E$.
\PROOF
The quantitative statements in \equ(inductivehypothesis)
will be obtained by estimating all the steps in the above construction.
We recall that, by definition,
$$
L^2_i(\varphi) =
\sum_{j=1}^i {1\over j!}
\left( {\partial\over\partial A}\right)^j
H_{i-j}(\theta -\omega -A\tildev(A,\theta) )\bigg|_{A=0}
$$
If we denote by $K =\sup_{ |A|\le\delta\atop |\Im(\theta)|\le\delta}
|\tildev( A,\theta) |$,
we can bound:
$$
\sup_{ |A|\le\eta/2K\atop |\Im(\theta)|\le \delta - (i -{1/2} )\eta}
| H_{i-j}(\theta -\omega -Au(a,\theta) )|\le
|| H_{i-j}||_{\delta - (i -1/2)\eta + 1/2\eta} \le
|| H_{i-j}||_{\delta - j\eta}
$$
Using Cauchy estimates in the variable $A$,
we obtain that :
$$
\sup_{ |\Im(\theta)|\le \delta - (i -{1/2} )\eta}
\left|{1\over j!}
\left( {\partial\over\partial A}\right)^j
H_{i-j}(\theta -\omega -A\tildev(A,\theta) )\bigg|_{A=0}\right|
\le
|| H_{i-j}||_{\delta - j\eta}\eta^{-j}(2 K)^{j}
$$
Hence, if we substitute
the induction hypothesis we obtain
that
$$
||L^2_i||_{\delta -i\eta +\eta/2 }\le
\sum_{j=1}^i\left( {2 K\over\eta}\right)^j E D^{i-j} =
D^{i-1} E {2 K\over\eta}\sum_{j=1}^i \left({2 K\over D \eta}\right)^{j-1}\le
D^{i-1} E {4 K\over\eta}
$$
Similarly, we can obtain bounds for $L^1_i$.
We observe that
$$
L^1_i(\theta) =\sum_{j=2}^i H_{i-j} {1\over j!}
\left( {\partial\over\partial A}\right)^j
(A + Au(A,\theta) )^j\big|_{A = 0}
$$
We can estimate the derivatives using Cauchy estimates to obtain:
$$\left|\quad {1\over j!}
\left( {\partial\over\partial A}\right)^j
(A + Au(A,\theta) )^j\big|_{A = 0}\quad \right|\le\left({{K+1}\over\eta}\right)^j
$$
when $|\Im(\theta )|\le\delta$.
Hence,
$$
|| L^1_i||_{\delta - i\eta + {1/2}\eta}\le
\sum{j=2}^i D^{i-j}\left( {{K+1}\over\eta}\right)^j
\le D^{i-1} { 2{K +1}\over\eta}
$$
Since $ ||\tildeH_{i-1}i||_{\delta - i\eta} \le D^{i-1} $
we see that we can determine
$\barH_{i-1}$ and that it satisfies:
$$ |\barH_{i-1}|\le\tildeK /\eta $$
where $\tildeK$ depends only on the suprema of
$\tildeu$,$\tildev$ and can be chosen uniformly
as $\eta$ is arbitrarily small.
We can now apply a
quantitative version of
\clm(cohomology) that
is proved in the same references quoted before.
\CLAIM Lemma(cohomologyquantitative)
Let $\omega$ satisfy \equ(Diophantine).
Then, for every $L:\torus^1\mapsto\complex $ analytic,
satisfying $\int L(\theta) d\theta = 0$,
we can find a unique $H:\torus\mapsto\complex$
satisfying:
$$
\eqalign{ & H(\theta) - H(\theta +\omega) = L(\theta)\cr
&\int H(\theta) d\theta = 0.}
$$
Moreover, for any $\eta > 0$ we have:
$$
|| H ||_{\delta -\eta}\le C\eta^{-\nu} || L||_{\delta}
$$
Applying \clm(cohomologyquantitative)
we obtain:
$||\tildeH_{i}||_{\delta -i\eta}\le D^{i-1}\tildeK\eta^{-1 -\nu}$
So, we see that the induction hypothesis are recovered.
To conclude the proof of \clm(main1improved), we just observe that
if we perform $k$ operations, we can take $\eta$ as big as
$\delta/2k $ and still satisfy the condition that
$\delta -k\eta > 0$.
We also observe that the same argument that we used to bound
$L^1_i(\theta) + H_{i-1}u(0,\theta) - L^2_i(\theta)$
serves to bound the $i^th $ derivative with respect to $A$
of $H\circ f - H $ in a complex neighborhood for $A$.
In the notation of \clm(main1), we have established that
$C_k \le\tildeK\left( {k\over\delta }\right)^{k ( 1 +\nu)}$
An elementary computation of maxima
shows that for a positive
number $B$,
$$
\max_k\left(k\over\delta \right)^{k ( 1 +\nu)} B^k
$$
is reached when $\log k ={1\over 1 +\nu} \log(\delta^{1+\nu} /B e^{1+\nu} )$
and takes the
value
$\exp ( - B^{-1}\delta^{- 1 -\nu} e^{1+\nu})$.
This establishes the desired result.
\SECTION Proof of \clm(main2)
The proof of \clm(main2) is a perturbation theory for hyperbolic structures.
We recall that
\CLAIM Definition(hyperbolic)
We say that a closed set $\Omega\subset M$ is a hyperbolic set for $f:M\to M$
if $f\Omega =\Omega$. We can find $C>0$, $\lambda <1$ and a splitting
$T_xM= E_x^s\oplus E_x^u$ such that
$$\eqalign{
&\| Df^n (x)v\|\le C\lambda^n\|v\|\hbox{ if }\
n\ge 0\ ,\ v\in E_x^s\cr
&\| Df^n (x)v\|\le C\lambda^n\| v\|\hbox{ if }\
n\le 0\ ,\ v\in E_x^u\ .\cr}$$
\REMARK
It follows from the definition that the subspaces $E_x^s$, $E_x^u$ are
uniquely determined and that $Df(x) (E_x^s) = E_{f(x)}^s$, $Df(x)(E_x^u) =
E_{f(x)}^u $.
Moreover, the mapping $x\to E_x^s$, $x\to E_x^u$ are continuous.
The following result is stated and proved in [LW].
\CLAIM Lemma(hyperb-pert)
Let $\Omega$ be a closed hyperbolic set. $\Omega'$ be an invariant set
contained in a sufficiently small neighborhood of $\Omega$.
Then, $\Omega\cup\Omega'$ is a hyperbolic set and it is possible to
extend the bundles $E_x^s$, $E_x^u$ to $\Omega'$ in such a way that
\clm(hyperbolic) is satisfied for some other $\lambda$. If the
neighborhood of $\Omega$ containing $\Omega'$ is small enough, the
$\lambda$ can be chosen as close as desired to that on $\Omega$.
For a periodic point $x$ of period $N$, the remark after \clm(hyperbolic)
implies that
$$\eqalign{
&\| Df^{Ni}(x)|_{E_x^s}\|\le C\lambda^{Ni}\cr
&\| Df^{-Ni} (x)|_{E_x^u}\|\le C\lambda^{Ni}\cr}$$
This implies by the spectral radius formula that all eigenvalues of
$Df^N(x) |_{E_x^s}$ have modulus less than $\lambda^N$ and that all
eigenvalues of $Df^N (x)|_{E_x^u}$ have modulus bigger than $\lambda^{-N}$.
Therefore, $|R(x)|\ge -\lambda^N -2-\lambda^{-N}$.
To prove that the Lyapunov exponents
of the periodic orbit converge to those of the set
we refer to the proof of a similar statement in the
proof of \clm(Lyapunov).
\QED
\SUBSECTION Proof of \clm(Lyapunov)
The two claims of \clm(Lyapunov) are
general results about systems of
positive Lyapunov exponents.
Claim a), which is much easier,
is a statement about
Lyapunov exponents of
uniquely ergodic measures.
We recall that a mapping defined on a set
is called uniquely ergodic if it leaves invariant only one measure.
It is well known , (see e.g. [AA] p. 138) that
an irrational rotation on the circle leaves invariant
the standard Lebesgue measure and no other, so
it is a uniquely ergodic system.
Since the motion on an Aubry Mather Cantor set is
semi--conjugate to a rotation, that is, we can find a
continuous
$h: \Gamma \mapsto\torus $ such that $h\circ f | _\Gamma = h\circ R_\omega$
we see that the only measure defined on
$\Gamma$ invariant under $f$ is the pull back under $h$
of the Lebesgue measure on the circle. We will denote such a measure
by $\delta_\Gamma$.
The fact that Aubry-Mather sets are
uniquely ergodic justifies that
we speak of the Lyapunov exponent of the set
without specifying explicitly the
ergodic invariant measure with respect to which
it is considered.
If $ o(x_n)\equiv\{ x_n, f(x_n),\cdots, f^{N_n-1 }(x_n)\} $ is an orbit of period
$N_n$, the measure that assigns weight $1/N_n$
to each of the points in the orbit is invariant
under $f$. We will denote such measure by $\delta_{o({x_n})}$.
Given a sequence of orbits
$\{ o(x_n)\}_{n=0}^\infty$
converging to $\Gamma$,
by the Banach- Alaoglu theorem,
we can extract a subsequence $\{ o(x_{n_i})\}$
such that the measures $\delta_{o(x_{n_i})}$ converge to a measure
$\delta_\infty$.
Since each of the measures is invariant $f^*_{\delta_{o(x_{n_i}} } =
\delta_{o(x_{n_i})}$ and the pull-back is
continuous in the weak-* topology ,
we conclude that
the $f^*\delta_\infty =\delta_\infty $. On the other hand,
it is easy to see that $\delta_\infty$ has support in
$\Gamma$. By the unique ergodicity
discussed before, we conclude that
$\delta_\infty =\delta_\Gamma$.
We also recall that the largest Lyapunov exponents of an ergodic measure are
lower semicontinuous with respect to the ergodic invariant measures.
This can be easily seen by noticing that
the largest Lyapunov exponent is
computed by appealing to the
{\sl subaditive ergodic theorem }( see e.g [Ru] p. 30 ).
If we denote by $\gamma(f,\mu)$ the
Lyapunov exponent of a measure $\mu $ ergodic for $f$,
we have :
$$
\gamma( f,\mu) =\lim {1\over n }\ln\int || Df^n (x) ||d\mu(x) =
\inf_n {1\over n}\int || Df^n (x) ||d\mu(x)
\EQ(infimum)
$$
From\equ(infimum), it follows
immediately that\ $\inf_i \gamma( f ,\delta_{o({x_{n_i}}) }) \le
\gamma(f,\delta_\Gamma)$
This finishes the proof of claim a) of \clm(Lyapunov).
We emphasize that the proof works word for word for
any set on which the motion is uniquely ergodic.
The proof of claim b) is much more complicated.
It will be a trivial consequence of the following theorem
which we state in full generality since
it can be applied in other contexts.
\CLAIM Theorem(approximation)
Let $f:M\mapsto M$ be a $C^2$ diffeomorphism
leaving invariant the ergodic measure $\mu$.
Assume that, with respect to this measure,
$f$ has no zero Lyapunov exponents.
Then, for almost every point $x_0$ in the support of
$\mu$, it is possible to find
a sequence $\{ x_n\}_{n=0}^\infty $
of periodic points
which converge to $x_0$.
Moreover, the sequence of orbits can be chosen in
such a way that the Lyapunov exponents of
$x_n$ converge to the Lyapunov exponents
of $x_0$.
\REMARK
Results similar to \clm(approximation)
appear in [Ka] (see Th. (4.1) ) [Ma\~n\'e].
They are usually called {\sl ergodic closing lemmas}.
\PROOF
The proof we present here, as the proofs above, will rely on a
shadowing lemma for partially hyperbolic orbits.
The argument will start by proving a constructive version of
a shadowing lemma and then we will show that partially
hyperbolic systems satisfy the hypothesis.
We emphasize that the version of the shadowing lemma
we prove
does not assume any global hyperbolic properties of the
dynamical system but only hyperbolicity properties of the
pseudo orbit considered. Such statements are useful in other contexts.
For example, they are useful when one wants to
verify rigorously that near a computer periodic orbit there
is a true orbit. In that case, even if one has the approximate orbit
quite explicitely,
one does not have much control about the global properties of the dynamical
system.
We will prove the shadowing lemma by systematically
analyzing sequences of orbits.
We will adopt the convention of
denoting sequences in bold-face and their
components by the same letter with a subindex.
\CLAIM Definition (pseudoorbit)
Let $M$ be a manifold and $f:M\mapsto M$
be a diffeomorphism.
We say that a sequence $\{ x_n\}_{n = -\infty}^\infty$
is an $\epsilon$--pseudoorbit if
$d( x_i, f(x_{i-1})\le\epsilon$.
This is equivalent to saying that
we can find mappings $g_i $
defined in a neighborhood $U_i$ of $x_i$
in such a way that $g_i(x_i) = x_{i-1}$,
$|| f - g_i ||_{C^0}\le\epsilon$
\CLAIM Definition(pseudohyperbolic)
We say that an $\epsilon$-pseudoorbit
is $\epsilon$ --pseudo hyperbolic
if we can find a decomposition
$T_{x_i} = E^s_{x_i}\oplus E^u_{x_i}$
and mappings $g_i$ defined
in neighborhoods $U_i$ of
$x_i$ and such that :
\item{$i)$} $g_i(x_i) = x_{i-1} $
\item{$ii)$} $|| f - g_i ||_{C^1}\le\epsilon$
\item{$iii)$} $$
\eqalign{
&|| Dg_{i+n}(x_{i+n}) Dg_{i+n-1}(x_{i+n-1})\cdots Dg_i(x_i) ||\le C\,\lambda^n
||v||\quad {\rm if } n > 0 , v\in E^s_i\cr
&|| Dg^{-1}_{i-n}(x_{i-n}) Dg^{-1}_{i-n+1}(x_{i-n+1})\cdots Dg^{-1}_i(x_i) ||\le
C\lambda^n ||v||\quad {\rm if } n > 0 , v\in E^u_{i-n}\cr
}
$$
We will refer to $\epsilon, C ,\lambda$ above as the parameters of
hyperbolicity.
\def\bfv{{\bf v}}
\def\bfx{{\bf x}}
\def\bfy{{\bf y}}
\def\bfa{{\bf a}}
\def\bfw{{\bf w}}
\def\tildef{{\tilde f}}
\def\tildeg{{\tilde g}}
\def\gtilde{\tildeg}
\def\ftilde{\tildef}
\def\Tau{{\cal T}}
If $\bfx \equiv \{ x_n\}_{n = -\infty}^{\infty}$ is a sequence,
we can pick neighborhoods $U_i$ around
$x_i$ and choose coordinate systems $\Phi_i: U_i\mapsto\real^d$
in such a way that the coordinate mappings are uniformly
$C^\infty$ and that $\Phi_i( x_i) = 0$.
( A geometrically natural way of doing this is
using the exponential mapping of Riemannian geometry
$\Phi_i( y ) =\exp_{x_i}^{-1}( y)$ .)
If we define $\tildeg_i\equiv\Phi_{i+1}\circ g_i\circ\Phi_i^{-1}$
$\tildef_i\equiv\Phi_{i+1}\circ f \circ\Phi_i^{-1}$ they are mappings mapping a
neighborhood
of $0\in\real^d$ to another neighborhood of $0 \in \real^d$. Moreover,
$\gtilde_i ( 0 ) = 0$.
Following [La], we consider the
space $\Xi =\{ {\bfy\in ( \real^d )^\natural | \sup_i | y_i | <\infty}\} $.
Clearly, $\Xi $ is a Banach
space under the norm $\| \bfy\| \equiv\sup_i | y_i|$.
Notice that, for some $\delta > 0$, $||\bfy||\le\delta$
implies that $y_i\in\Phi_i( U_i) $.
On a sufficiently small neighborhood of $0$,
we can define
the operators $\Tau_f$ by:
$\Tau_f(\bfy)_i = f_{i-1}( y_{i-1})$,
Notice that $\Tau_f(\bfy) =\bfy$
if and only if ${\Phi_i^{-1} (y_i )\}_{i=-\infty}^\infty}$
is an orbit for $f$ and that
$\bfy$ is an $\epsilon$--pseudorbit
if and only if $ K^{-1} \epsilon ||\Tau(\bfy) -\bfy||\le K \epsilon$
where $K$ is a bound on the derivatives of $\Phi_i$ and
$\Phi_i^{-1}$.
\CLAIM Proposition(differentiable)
If $f$ is uniformly differentiable, then $\Tau$ is differentiable
in a neighborhood of the origin
and we have :
$$[ D\Tau(\bfx)\bfa]_n = Df_{n-1} ( x_{n-1} ) a_{n-1} $$
If $f$ is uniformly $C^2$, then $\Tau$ is $C^2$ and
we can bound $|| D^2\Tau(\bfx)|| $ uniformly
in a neighborhood of the origin.
\PROOF
To establish the first claim we just have to bound
$$
||\Tau(\bfx + \bfa) -\Tau(\bfx) - D\Tau(\bfx)\bfa||
\EQ(toestimate)
$$
and show that it converges to zero with $||\bfa||$ faster than $||\bfa||$.
We recall that $f$ is uniformly differentiable
if one can find an increasing function
$\eta:\real^+\mapsto\real^+$ with $\eta(0) = 0$ and
$\lim_{t\to 0}\eta(t) /t = 0$ such that
$| f(x + a) - f(x) - Df(x)a |\le\eta( |a|)$.
If the function $f$ that we used to construct $\Tau_f$
is uniformly differentiable -- this is automatic if the manifold is compact
or if $f$ has uniformly bounded first derivatives,
using the fact that
the mappings $\Phi_i$ and their
inverses have uniformly bounded derivatives,
we conclude that for some $\eta: \real^+ \to \real^+$ increasing and
$\eta(0) = 0$, we have
$| f_i(x + a) - f_i(x) - Df_i (x)a_{i-1} |\le\eta( |a_{i-1}|)$.
Using the definition of the norm,
the quantity
\equ(toestimate) that we have to estimate
is just
$\sup_n || f_{n-1}( x_{n-1} + a_{n-1} ) - f_{n-1} ( x_{n-1} ) -
Df_{n-1}( x_{n-1} ) a_{n-1}||$.
Using the uniform differentiability,
we obtain
that this can be bounded by
$\sup_n\eta( |a_{n-1}|)\le\eta( \sup_n | a_{n-1}|) =\eta(||\bfa||)$
which is what we wanted to establish.
The argument for the second derivative is very similar and we leave the
details to the reader.
\QED
The following lemma provides us with a characterization of the
hyperbolicity of orbits by properties of the derivative of the operator
$\Tau_f$ at ${\bfx}$
Their usefulness comes from the fact that they allow us to prove properties
that are true for whole orbits --- uniformly on the time --- by doing
soft analysis on the operator $\Tau_f$.
They are non-autonomous versions of the characterizations in [Ma3] and
the proofs are, actually, quite similar. We point out that i) will not
be used in this paper but we included it because it fits nicely in the circle
of ideas discussed here. Since the spectral theory on Banach spaces
is much more natural on complex spaces
we will consider$\chi$ the natural complexification of $\Xi$.
We leave to the reader that elementary task of checking that, when the problem
considered has real data, the results are real.
\CLAIM {Lemma}(spectrum)
Let $\bf x$ be a fixed point of $\Tau_f$ as before. Then
\vskip1pt
\item{i)} The spectrum of $D\Tau_f(\bfx)$ is invariant under rotations, i.e.,
$$z\in\spec\bigl( D\Tau_f(\bfx)\bigr)
\Rightarrow\forall\theta\in\real\ ,\
e^{i\theta}\in\spec\bigl(D\Tau_f (\bfx)\bigr)$$
\item{ii)} Assume that for $0<\mu_- <\mu_+$
$$\spec\bigl( D\Tau_f(\bfx)\bigr)\cap
\{ z\in\complex\mid\mu_-\le |z|\le\mu_+\}$$
Then, we can find a sequence of subspaces $E_i^{[>]},E_i^{[<]}$ in such a
way that
\vskip1pt
\itemitem{a)} $\real^d = E_i^{[>]}\oplus E_i^{[<]}$
\itemitem{} $\hbox{\rm angle} (E_i^{[>]}, E_i^{[<]})\ge\alpha >0$
\itemitem{b)} $\| Df_{i+m} (x_{i+m}) ,\ldots, Df_i(x_i) \big|_{E_i^{[<]}}\|
\le C\mu_+^m$
\itemitem{} $\| Df_{i-m}^{-1} (x_{i-m}),\ldots, D f_i^{-1}(x_i)
\big|_{E_i^{[>]}}\|\le C\mu_-^{-m}$
\itemitem{c)} $Df_i (x_i) E_i^{[<]} = E_{i+1}^{[<]}$
\itemitem{} $Df_i (x_i) E_i^{[>]} = E_{i+1}^{[>]}$
\vskip1pt
\item{iii)} Conversely, if we can find $E_i^{[>]}, E_i^{[<]}, C,\mu_+,
\mu_-$ satisfying ii.a), ii.b), ii.c) as before, then
$$\spec\bigl( D\Tau_f (\bfx)\bigr)\cap
\{ z\in\complex\mid\mu_-\le |z|\le\mu_+\} =\emptyset$$
\PROOF
To prove i) we recall that a number
$z \in \complex $ is in the spectrum of $(D\Tau_f (\bfx))$ if and
only if there exist a sequence $\{ \bfv_n\}_{n=0}^\infty$,
$\| \bfv_n\|=1$,
$\lim_n\| D\Tau_f (\bfx) \bfv_n - z \bfv_n\|=0$.
So, to prove the theorem it suffices to show that if we have $\bfv\in \Xi$ such
that $\|D\Tau_f (\bfx) \bfv-z\bfv\|\le\varep$ we can find $\bfw$ with
$$\| D\Tau_f (\bfx) \bfw-ze^{i\theta} \bfw\|\le\varep$$
Expressed in components, the hypothesis means that
$$|Df_i (x_i) v_i - zv_{i+1}|\le\varep, \quad
\sup |v_i| =1$$
If we set $V$,
$w_n = e^{-n i\theta} v_n$ we have:
$$\eqalign{
&\sup_n |w_n| = \sup_n |V_n| =1\cr
&\sup |Df_n(x_n) w_n - ze^{i\theta} w_{n+1} | =\cr
&=\sup_n |e^{in\theta} (Df_n(x_n) V_n - ZV_{n+1}|\cr
&=\varep\ .\cr}$$
This finishes the proof of i).
To prove ii) we observe that, by the spectral theorem on Banach spaces,
we can find a $\chi =\chi^{[<]}\oplus\chi^{[>]}$ invariant under
$D\Tau_f (\bfx)$ in such a way that:
$$\eqalign{
&\spec\bigl( D\Tau_f (\bfx) |_{\chi^{[<]}}\bigr)
\subset\{ z\in\complex\mid |z|\le\mu_-\}\cr
&\spec\bigl( D\Tau_f (\bfx)^{-1}\mid_{\chi^{[>]}}\bigr)
\subset\{ z\in\complex\mid |z|\le\mu_+^{-1}\}\cr}
\EQ(spectrumcontained)$$
We also recall that another corollary of the spectral theorem is that
$$\eqalign{
&\bfv\in\chi^{[<]}\Leftrightarrow\cr
&\lim_{n\to\infty}\| [D\Tau_f (\bfx) ]^n\bfv\|^{1/n}\le\mu_-\cr
&\Leftrightarrow\forall\ n\| [D\Tau_f (\bfx) ]^n\bfv\|
\le C\mu_-^n\| \bfv\|\cr
&\bfv\in\chi^{[>]}\Leftrightarrow\cr
&\lim_{n\to\infty}\| [D\Tau_f(\bfx)]^{-n}\bfv\|^{1/n}\le\mu_+^{-1}\cr
&\Leftrightarrow\forall\ n\in\natural
\| [D\Tau_f (\bfx)]^{-n}\bfv\|\le C\mu_-\| \bfv\|\cr}
\EQ(characterizespace)$$
where, in the second characterization, we understand implicitly that
$\bfv\in\Dom (D\Tau_f (\bfx))^{-n}$.
Both characterizations say roughly that $\bfv\in\chi^{[<]}$ if iterates
of $D\Tau_f$ decrease faster than exponential of rate $\mu_-$.
Notice that the two precise definitions of ``exponential of rate
$\mu_-$'' are not equivalent
in general, the first one being weaker even for one vector (notice that
$C$ in the second characterization is independent of the vector).
The proof of \equ(characterizespace) uses essentially the spectral
properties of $D\Tau_f (\bfx)$ we assumed.
We want to prove that $\chi^{[<]}$ is of the form
$$\chi^{[<]} =\{\bfv\in\chi\mid v_i\in E_i^{[<]}\}\ ,$$
that is, whether $\bfv$ belongs or not to the space $\chi^{[<]}$
can be ascertained by testting succesively the
components.
This will be true, basically because to compute one coordinate of
$D\Tau_f (\bfx)\bfv$, we only need to know one component of $\bfv$.
\CLAIM Proposition(coordinates)
The vector $\bfv\in\chi$ belongs to $\chi^{[<]}$ if and only if,
for every $i$, the vector $\bfv^i\equiv (\ldots 0\ldots,v_i,0\ldots
0\ldots)\in\chi^{[<]}$.
\PROOF
We will use \equ(characterizespace) to prove both implications.
$$\eqalign{
&\| (D\Tau_f (\bfx)^n\bfv^i\| =
|Df_{i+n} (x_{i+n})\ldots Df_i (x_i) v_i| =\cr
&|([D\Tau_f (\bfx)]^n\bfv)_{i+n}|\le
\| D\Tau_f (\bfx)]^n\bfv\|
\le C\mu_-^n\|\bfv\|\cr}$$
To prove the if part
$$\eqalign{
&\| [D\Tau_f (\bfx) ]^n\bfx\| =\cr
& =\sup_i |([D\Tau_f (\bfx)]^n\bfv)_{i+n}| =\cr
&=\sup_i\| [D\Tau_f (\bfx)]^n\bfv^i\|\le\cr
&\le\sup_i C\mu_-^n\|v_i\| = C\mu_-^n\| v\|\cr}$$
Hence, we can define
$$E_i^{[<]} =\{ v\in\real^d |\quad{(\ldots,0,\ldots,0,v,0,\ldots,0,\ldots)
\in\chi^{[<]}}\}$$
Since
$\Tau_f\bfx) (\ldots,0,\ldots,0,v,0,\ldots,0,\ldots) =
(\ldots,0,\ldots,0,0,Df_i(x_i)v,\ldots,0,\ldots)$, we see that
$v\in E_i^{[<]}\Leftrightarrow Df_i (x_i)\in E_{i+1}^{[<]}$.
So that ii.c) of \clm(spectrum) is established.
To prove ii.b) we observe that
$$\eqalign{
&\| [\Tau_f (\bfx)]^n (\ldots,0,\ldots,0,v,0,\ldots,0,\ldots,0)\| =\cr
& =\| (\ldots,0,\ldots,0,0,Df_{i+n} (x_{i+n}) Df_{i+n-1} (x_{i+n-1})
\cdots Df_i(x_i)v,0,\ldots)\|\cr
&= |Df_{i+n} (x_{i+n}\ldots Df_i (x_iv|\cr
&\le C\mu_+^n\| (\ldots,0,\ldots,0,v,0,\ldots,0)\| =\cr
&= C\mu_+^n |v|\cr}$$
Analogous argument and definitions work for $\chi^{[>]}$.
To prove a) we observe that clearly $E_i^{[<]}, E_i^{[>]}$ are linear
spaces. Moreover, their intersection is $\{0\}$ since the intersection
of $\chi^{[>]},\chi^{[<]}$ is the null vector. If there was a vector
$v\in\real^d$ $v\notin E_i^{[<]}\oplus E_i^{[>]}$, we see that
the vector $(\ldots,0,\ldots,0,v,0,\ldots,0)\in\chi^{[<]}\oplus
\chi^{[>]}$.
To prove a) we observe taht if we could find a vector $w\in\real^d$,
$w\notin E_i^{[<]}\oplus E_i^{[<]}$.
To show that the angle between spaces is bounded from below, we recall
that as a consequence of the spectral theorem, $\Pi^{[<]},\Pi^{[>]}$, the
spectral projections onto $\chi^{[<]},\chi^{[>]}$ are bounded.
By \clm(coordinates)
$$\Pi^{[<]} (\ldots,v_i,v_{i+1},\ldots) =
(\ldots\pi_i^{[<]} v_i,\pi_{i+1}^{[<]}v_{i+1}\ldots)$$
where $\pi_i^{[<]},\pi_i^{[>]}$ are the projections associated to the
decomposition $\real^d = E_i^{[>]}\oplus E_i^{[>]}$.
Since the spectral projections $\pi^{[<]},\pi^{[>]}$ are bounded, we have:
$$\eqalign{
&|\pi_i^{[<]} v| =\|\Pi^{[<]} (\ldots,0,\ldots,0,v,0,\ldots,0)\|\le\cr
&\le\|\Pi^{[<]}\|\
\| (\ldots,0,\ldots,0,v,0,\ldots,0,\ldots)\| =\cr
&=\|\Pi^{[<]} 0\|\ |v|\cr}$$
A similar argument shows $|\pi_i^{[>]} v|\le\|\Pi^{[>]}\|\, |v|$.
Hence, $\|\pi_i^{[<]}\| ,\|\pi_i^{[>]}\|$ are bounded
independently of $i$. This is equivalent to
saying that the angle between $E_i^{[<]}$
and $E_i^{[>]}$ is uniformly bounded from below.
This finishes the proof of ii) of \clm(coordinates).
To prove iii) it suffices to show that the equation
$$D\Tau_f (\bfx)\bfv - z\bfv =\bfw
\EQ(inverse)$$
can be solved in $\bfv$ for any $\bfw$, $\mu_- < |z| <\mu_+$
and that $\|\bfv\|\le C\|\bfw\|$.
Taking components, \equ(inverse) is equivalent to:
$$Df_i (x_i) v_i - z v_{i+1} = w_{i+1}
\EQ(inversehierarchy)$$
If $\pi_i^{[<]}$ and $\pi_i^{[>]}$ are the projections associated to the
splitting $\real^d = E_i^{[<]} + E_i^{[>]}$, ii.a) implies
$$\eqalign{
& Df_i (x_i)\pi_i^{[<]} =\pi_{i+1}^{[<]} Df_i(x_i)\cr
& Df_i (x_i)\pi_i^{[>]} =\pi_{i+1}^{[>]} Df_i(x_i)\cr}$$
Hence, decomposing into the components along $E_{i+1}^{[<]} ,E_{i+1}^{[>]}$,
\equ(inversehierarchy) is equivalent to
$$\eqalign{
& Df_i(x_i)\pi_i^{[<]} v_i - z\pi_{i+1}^{[<]} v_{i+1}
=\pi_{i+1}^{[<]} w_{i+1}\cr
& Df_i(x_i)\pi_i^{[>]} v_i - z\pi_{i+1}^{[>]} v_{i+1}
=\pi_{i+1}^{[>]} w_{i+1}\cr}
\EQ(hierarchycomponents)$$
We claim that these two equations can be solved by setting:
$$\eqalign{
&\pi_{i+1}^{[<]} v_{i+1} = - {1\over z}\pi_{i+1}^{[<]} w_{i+1}
\sum_{j=0}^\infty {-1\over z^{j+1}} [Df_i(x_i)\ldots Df_{i-j} (x_{i-j}) ]
\pi_{i=j}^{[<]} w_{i-j}\cr
&\pi_{i+1}^{[>]} v_{i+1} =\sum_{j=0}^\infty z^j [(Df_i(x_i))^{-1}
\ldots Df_{i+j} (x_{i+j})^{-1}]\pi_{i+1+j}^{[>]} w_{i+1+j}\cr}
\EQ(solutions)$$
In effect, we see that, using ii.b),
$$
| {1\over z^{j+1}} [Df_i (x_i)\ldots Df_{i-j}]
\pi_{i-j}^{[<]} w_{i-j}|\le
{1\over |z|^{j+1}} C\mu_-^j |\pi_{i-j}^{[<]} w_{i-j}|\le
{C\over z}\left| {\mu_-\over z}\right|^j\|\pi^{[<]}\|\
\| \bfw\|$$
and analogously for the other equation
$$| z^j [(Df_i(x_i))^{-1}\ldots (Df_{i+j} (x_{i+j}))^{-1}]
\pi_{i+j+1}^{[>]} w_{i+j+1}|\le
\left| {z\over\mu_+}\right|^j C\|\pi^{[>]}\|\
\| \bfw\|$$
Hence, the two series in \equ(solutions) converge uniformly and, by
rearranging terms it is easy to verify that they indeed are solutions.
Moreover, since the right hand sides of \equ(solutions) have norms that
can be bounded by $C\|\bfw\|$ independently of $i$, we see that the sum
will also be bounded in the same form, hence $z$ is in the resolvent.
This finishes the proof of \clm(spectrum).
\QED
\CLAIM Lemma(fixedpoint)
Let $\Tau$ be a $C^2$ function on a Banach space, $\bfx$ be a point. Let
$M$ be a linear operator in the Banach space.
\vskip1pt
Assume
\vskip1pt
\item{i)} $\|\Tau (\bfx) -\bfx\|\le\varep$
\item{} $\| D\Tau (\bfx) - M\|\le A$
\item{} $\| (M-I)^{-1}\|\le B$
\item{} $\| D^2\Tau (\bfy)\|\le C$ if $\|\bfx -\bfy\|\le\rho$
\vskip1pt
Then, if
$$\eqalign{ & K\equiv AB + B \le \rho <1\cr
&\varep + K\rho <\rho\cr}$$
There is one fixed point $x^*$ of $\Tau$ in the set $\{ y\mid\| y-x\|\le
\rho\|$. Moreover $\| x-x^*\|\le\varep /1-K$.
\PROOF
Consider
$$\Phi (\bfy) = - (M-I)^{-1} (\Tau (\bfy) -\bfy) +\bfy$$
A simple calculation shows that a fixed point of $\Phi$ is also a fixed
point of $\Tau$.
Moreover $\Phi$ is twice differentiable and we have
$$\eqalign{
D\Phi (\bfy) &= - (M-I)^{-1} (D\Tau (\bfy) - I) + I =\cr
&= - (M-I)^{-1} (M-I + (D\Tau (\bfx) -M) +
(D\Tau (\bfy) - D\Tau (\bfx))) +I\cr
&= -(M-I)^{-1} (D\Tau (\bfx)-M ) +
-(M+1)^{-1} D\Tau (\bfy) - D\Tau (\bfx)\cr}$$
If we bound the first term using the second part of $i)$ and the second
term by
the mean value theorem we obtain
$$\| D\Phi (\bfy) \|\le AB + B\le\rho$$
Hence, $\Phi$ is a contraction in the ball around $\bfx$ of radius $\rho$.
The second inequality implies that this ball is mapped into itself.
From that, we can apply the elementary argument for the
contraction mapping principle.
\QED
We also recall some facts from the theory of non-uniformly hyperbolic
systems. Our ultimate goal is to show that, for systems with positive
Lyapunov exponents, we can construct pseudo-orbits with approximate
inverses such that \clm(fixedpoint) applies.
\CLAIM Theorem(osledec)
Let $\mu$ be Borel measure invariant under $f$. Then, for $\mu$-almost
all $x$, we have
\vskip1pt
\item{i)} $T_xM =\bigoplus_i E_i(x)$
\vskip1pt
\noindent with
\vskip1pt
\item{ii)} if $v\in E_i$
$$\lim_{n\to\infty} {1\over n}\log |Df^n (x)v| =
\lim_{n\to\infty} {1\over n}\log |Df^{-n} xv| =\lambda_i (x)$$
\item{iii)} $\Delta_{ij}(x)$ the angle between $E_i(x),E_i(j)$, $i\ne j$
satisfies
$$\Delta_{ij} (f^n(x))\ge e^{-|n|\varep} (x) C_\varep (x)$$
for some measurable function $C_\varep (x)\ne 0$.
\PROOF
There are several proofs of this theorem in the literature. The original
one is in [Os]. A more modern one can be found in [Ru].
There, iii) is proved explicitly as corollary~3.3.
Notice that $\lambda_i (f(x)) =\lambda_i(x)$. Hence, if $\mu$ is ergodic
under $f$, $\lambda_i$ is a constant. Also, $Df(x) E_i(x) = E_i(f(x))$.
Once we have these results, it is easy to prove
\CLAIM Corollary(uniformset)
Let $\mu$ be a Borel probability measure,
invariant under $f$ and ergodic.
Given $\varep >0$ we can find $\ell >0$ and a set $\Lambda_{\varep,\ell}$
such that:
\vskip1pt
\item{i)} $\mu (\Lambda_{\varep,\ell})\ge 1-\varep$
\item{ii)} $\Lambda_{\varep,\ell}$ is closed.
For all $x\in\Lambda_{\varep,\ell}$ we have
\vskip1pt
\item{i)} If $n,m\in\integer$, $v\in E_i(f^m x)$
$$\ell^{-1} e^{\lambda_in} e^{\varep |n|+|m|}\le |Df^n (f^mx)v|
\le\ell e^{\lambda_i n} e^{E(|n|+|m|)}$$
\item{ii)} The spaces $E_i(x)$ depend continuously on $x$ when
$x\in\Lambda_{\varep,\ell}$.
\item{iii)} $\Delta_{ij} (f^n(x))\ge\ell^{-1} e^{\varep |n|}$
\item{iv)} The sets $\Lambda_{\varep,\ell}$ can be chosen in such a
way that $\Lambda_{\varep,\ell}\subset\Lambda_{\varep,\ell'}$ if
$\ell' >\ell$ and $\Lambda_\varep =\bigcup_{\ell}\Lambda_{\varep,\ell}$
has full measure.
Now we can go back to the proof of \clm(Lyapunov).
If $x\in\sup (\mu_\Gamma)$, we observe that we can find an $\varep,\ell$
in such a way that $x\in\sup (\mu|_{\Lambda_{\varep,\ell}})$.
Hence, for every $\delta >0$
$$\mu (\Lambda_{\varep,\ell}\cap B_\delta (x)) >0$$.
By Poincar\'e recurrence theorem, we can find $x_0,\ldots, x_N$ in such a
way that $f(x_i) = x_{i+1}$, \ $x_0\in\Lambda_\varep\cap B_\delta (x)$,
$x_N\in\Lambda_\varep\cap B_\delta (x)$.
If $\delta$ is small enough, we can take coordinate patches $U_i$ around
$x_0,\ldots,x_{N-1}$ as before
in such a way that $x_N$ will be on the coordinate patch of $x_0$.
Denote by $\tilde x_N$ the coordinate representation of $x_N$ on the
patch $U_0$.
Denote also $\tilde E_j^i (x) = D\Phi_i (\Phi_i^{-1}x) E_j(\Phi_i^{-1}x)$ .
(That is $\tilde E_j^i (x)$ is the coordinate representation of the spaces
corresponding to the $j^{th}$ Lyapunov exponent.)
Notice that $Df_i(x)\tilde E_j^i (x) =\tilde E_j^{i+1} (f(x))$.
If the spaces $E_j(x)$ depend continuously on $x\in\Lambda_{\varep,\ell}$
we see that $\tilde E_j^i$ will depend continuously also and the modulus
of continuity can be estimated from the modulus of continuity of the
coordinate changes.
In particular, we can find operators $\pi_i(x)$ such that
$$\eqalign{
&\pi_i (x)\tilde E_j^i (x) =\tilde E_j^i (0)\cr
&\|\pi_i (x) -Id\|\le\omega (|x|)\cr}$$
with $\omega(t)$ decreasing $\omega (t)\to 0$ as $t\to 0$.
We claim that, for $\delta$ small enough, the pseudo-orbit given by
$$\bfx =\cases{0&if $i\ne kN$\cr
\tilde x_N&if $i= kM$\cr}$$
and the operator $M$ defined by $(M\eta)_i = M_{i-1}\eta_{i-1}$ with
$M_{i-1} = Df_{i-1}(0)$ if $i\ne kN$;
$M_{i-1}\pi_i (\tilde x_N) Df_{i-1} (0)$ if $i=kN$.
Notice that we do not have upper bounds for $M$ but that Poincar\'e's
recurrence theorem implies there is a sequence of $N$'s going to
infinity. Hence, we will have to prove the estimates of \clm(fixedpoint)
for $N$ sufficiently large.
We will assume without loss of generality that $N$ is large enough that
$$\eqalign{
&\ell\exp ((\mu^+ -\varep)N) <1\cr
&\ell\exp ((-\mu^- -2\varep)N) <1\cr}$$
where $\mu^+$ is the smallest positive Lyapunov exponent and $\mu^-$
is the negative Lyapunov exponent of smallest absolute value.
(Heuristically, we are waiting long enough that the non-uniform
hyperbolicity has had time to start acting.)
Clearly, $\|\Tau_f (\bfx) -\bfx\| = |\tilde x_N|$ which can be
estimated by $K \delta$, where, as before,
$K$ is a constant that only depends on the
supremum of the derivatives of the coordinate mappings.
$$\eqalign{
\| D\Tau_f -M\| & =\| Df_{N-1} (0) -\pi_N (\tilde x_N) Df_{N-1}(0)\|\cr
&=\| Df_{N-1} (0)\|\omega (|\tilde x_N|)\cr}$$
which also tends to zero with $\delta$.
To estimate $(M-Id)^{-1}$ we study the equations $(M-Id)\eta =\bfw$ in
a way quite similar to the proof of part b) of \clm(spectrum).
Notice that, by construction $M$ preserves the decomposition
$\chi =\bigoplus_i\chi_i$ with
$\chi_i =\{\bfv\mid v_k\in\tilde E_i^k\}$.
Proceeding as in the proof of \clm(spectrum) b), we find that the equation
$$M\eta -\eta = \bfw$$
admits the solutions (analogous to \equ(solutions)) given by:
$$\eqalign{
&\pi_{i+1}^{[<]}\eta_{i+1} = -\pi_{i+1}^{[<]} w_{i+1}
+\sum_{j=0} M_i\cdots M_{i-j}\pi_{i-j}^{[<]} w_{i-j}\cr
&\pi_{i+1}^{[>]}\eta_{i+1} =\sum_{j=0}^\infty M_i^{-1}\cdots
M_{i+j}^{-1}\pi_{i+1+j}^{[>]} w_{i+1+j}\cr}
\EQ(newsolutions)$$
To bound the products in the sums in \equ(solutions) we use the inequalities
in the definition of the set $\Lambda_{\varep,\ell}$.
We see that
$$\eqalign{
&M_{kN} M_{kN+1}\cdots M_{(k+1)N} =\cr
&Df_{kN}(0)\cdots Df_{k+N}(0) +
Df_{kN}(0)\cdots Df_{(k+1)N-1} (0)\pi_{N-1}(\tilde x_N)
Df_{(k+1)N} (0)\cr}$$
Hence the norm is bounded by
$\ell\exp ((\mu^- +\varep)(N-1))\ell\exp (\varep N)\omega (\delta)
+\ell\exp ((\mu^- +\varep) N)$.
We see that, by making $N$ sufficiently large and $\delta$ sufficiently
small, we can ensure that this is bounded away from 1.
Hence, the first series in \equ(newsolutions) converges uniformly and
we can bound the result by a constant times $\|\bfw\|$.
An analogous argument works for the second sum.
Applying \clm(fixedpoint) we conclude that there is a fixed point of
$\Tau_f$ close to zero, that is an orbit that shadows the $\varep$
pseudo-orbit $\bfx$.
We claim that this orbit has to be periodic of period $N$.
To prove that, we recall that the fixed point of $\Tau_f$ was obtained
iterating the operator $\Phi = -(M-Id)^{-1} (\Tau_f - Id) +Id$.
Notice that $\Tau$ maps sequences of period $N$ into sequences of period $N$
and using the formulas for the inverse of $M-Id$, so does $(M-Id)^{-1}$.
Hence $\Phi$ maps periodic sequences into periodic sequences and, since
the starting sequence is periodic with period $N$, so should be the
fixed point.
To prove the claim about the Lyapunov exponents of the orbit, we use the
characterization of Lyapunov exponents in terms of the spectrum of
$D\Tau_f$.
Notice that, by making $N$ sufficiently large and $\delta$ sufficiently
small, we can guarantee that the fixed point of $\Tau_f$ would be
as close as desired to zero.
Since $\|D\Tau_f (\bfx) - D\Tau_f (0)\|\le\| \bfx\|\,\| D^2\Tau_f\|
\le\|\bfx\|\,\| D^2 f\|$
(notice that this bound does not depend on $N$) we have that the spectrum
of $D\Tau_f (\bfx)$ would be arbitrarily clsoe --- in the sense of sets ---
to the spectrum of $D\Tau_f(0)$. By making $\delta$ sufficiently small
we can approximate $\spec (D\Tau_f(0))$ by $\spec(M)$.
The argument to show that $\spec(M)$ is close to being circles around
the Lyapunov exponents is very similar to the arguments we have already
used. It suffices to show that if $z$ is away from the Lyapunov
exponents we can solve the equation
$$M\bfv - z\bfv = \bfw
\EQ(resolvent)$$
If $\pi_i^{[<|z|]}\pi^{[>|z|]}$ denote the projection onto
$\bigoplus_{e^{\lambda_i}< |z|} E_i,\bigoplus_{e^{\lambda_i} > |z|}
E_i$ respectively.
By using the analogue of \equ(solutions) and the estimates analogue to
those used to bound \equ(newsolutions) we can show that, provided that
$N$ is sufficiently large, $\delta$ sufficiently small, \equ(resolvent)
has a solution if $|z|$ is not a Lyapunov exponent.
This finishes the proof of \clm(Lyapunov).
\QED
\SECTION Discussion
The above results justify Greene's criterion for the families for which there is
a sharp transition between the parameter values for which the Aubry Mather
set is an analytic circle and those parameter values for which it is
a Cantor set with non-zero Lyapunov exponent. (We refer to those
cases collectively as the neutral cases.)
We have not obtained any results in the cases that
the Aubry-Mather set is a non-smooth circle or for those in which it
is a Cantor set with zero Lyapunov exponent.
Our results do not exclude that, on an open interval of parameters,
the neutral cases occur or that the hyperbolic cases coexist with
smooth tori in all the scales.
For the standard map and the golden mean rotation number there is a very convincing renormalization group picture [McK1], [McK2]
which suggests that at precisely one value of the parameter, the
Aubry-Mather set of golden mean rotation number
is a non-smooth invariant circle and that, for smaller
values of the parameter, the Aubry-Mather set is
an smooth circle whereas for values
bigger that the critical value, it is a hyperbolic set.
This picture, due to McKay, is based on a detalied study of the
dynamics of a renormalization operator acting on the space of maps.
The operator has an attractive fixed point --called trivial since it can be
computed explicitely -- and a non trivial one which McKay computed quite
convincingly. This non-trivial fixed point has a stable manifold of
codimension $1$ and a one-dimensional
unstable manifold, one of whose sides ends on the trivial fixed point.
If a map, under iteration of the renormalization group, converges to the
trivial fixed point, the Aubry Mather Cantor set of
golden mean rotation is an smooth circle. (This part of the
picture and some generalizations has been justified rigorously in [Ha].)
If a map under repeated renormalization
converges to the non-trivial fixed point, the Aubry Mather set of
golden mean rotation is a not very smooth invariant circle.
If it approaches the unstable manifold on
the side oposite to the trivial fixed point, the Aubry-Mather Cantor
set of golden mean rotation is a hyperbolic Cantor set.
The remarkably simple behaviour of the standard map family can be justified
with the help of this picture by realizing that the
curve described by the standard map family crosses
transversally the stable manifold of the non-trivial fixed point and is
very close to the unstable manifold of the renormalization map. Hence, by the
$\lambda$-lemma, succesive iterations of the renormalization group
will make it converge to the unstable manifold.
Notice also that for families sufficiently close
to the standard map the $\lambda$-lemma also applies
and they will also converge to the
unstable manifold. If we keep in mind that the
effect of the renormalization group on a map is to make the space scale
smaller and the time scale longer, the high iterations
of the renormalization group capture
phenomena that happen on small scales and for long times.
The convergence onto the unstable manifold has the consequence
that, for all families, sufficciently close that the
$\lambda$-lemma applies, the long term behavior
at small scales is the same.
Unfortunately this very satisfactory picture has only
a local nature. The papers [W1], [W2], [KMcK] and our
forthcomming paper [FL] present evidence that the
the dynamics of the renormalization group operator
has more complicated features than just a saddle point.
These effects can be observed in standard like families in which we
substitute for the $\sin$ in the standard family a
trigonometric polynomial of two coefficients.
In that case, it seems quite possible that there is not
a sharp transition between the smooth behavior and the
hyperbolic one. For those systems, Greene's criterion
seems to work less efficiently than in the
case of the standard map.
Nevertheless, the discussion
of our proofs suggests that Greene's criterion can be used very effectively and
very safely as a negative criterion for the
non-exextence of smooth invariant circles. When the
residues are reasonably big for a periodic orbit of high period
we can be quite confident that, for this parameter value, there
is no smooth invariant circle. Unfortunately,
when the renormalization group picure does not
holdi, the values computed
with different orbits do not stack up as predictably as in the
case of the standard mapping and it does not seem possible to
extrapolate (See [W1], [W2], [KMcK], [FL]).
Let us also mention that, even if by now it is quite well
stablished rigorously that convergence to a trivial fixed point of the
renormalization group operator implies that there is
an smooth invariant circle (See e.g [H], [SK] for rigorous proofs)
the converse is much more difficult.
We point out that there are ways of constructing non-smooth invariant circles
which seem to have quite non-Diophantine rotation numbers.
The method is described in [Ma3] and attributed there to
Yoccoz.
The idea is to exploit the a-priori Lipschitz bounds
porvided by Birkhoff theory to conclude that from a sequence
of invariant circles we can extract a subsequence
converging in $C^0$. Since a K.~A.~M. circle
implies the existence of nearby K.~A.~M. circles with
slightly worse Diophantinbe properties, it is
possible to construct invariant circles with non-Diophantine
rotation number. In the renormalization group languange this suggests that
there are invariant circles on the basin of attraction of the
the trivial fixed point. Since these circles do not
seem to converge under renormalization, it is not clear that
Greene's criterion would apply to them.
\SECTION Acknowledgements
The work of R.~de la Llave has been partially supported by
National Science Foundation grants. He also would like to acknowledge
stimulating discussions with R. McKay that took place while
both were enjoying the hospitality of S. Wiggins.
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