INSTRUCTIONS: ... contains a total of 90007 characters. 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124 | 1159 ASCII characters 125 } 22 ASCII characters 126 ~ INFORMATION: BODY: % The original version of this macros is due to J.P Eckmann % % \magnification \magstep1 \vsize=22 truecm \hsize=16 truecm \hoffset=0.8 truecm \normalbaselineskip=5.25mm \baselineskip=5.25mm \parskip=10pt \immediate\openout1=key \font\titlefont=cmbx10 scaled\magstep1 \font\authorfont=cmcsc10 \font\footfont=cmr7 \font\sectionfont=cmbx10 scaled\magstep1 \font\subsectionfont=cmbx10 \font\small=cmr7 \font\smaller=cmr5 %%%%%constant subscript positions%%%%% \fontdimen16\tensy=2.7pt \fontdimen17\tensy=2.7pt \fontdimen14\tensy=2.7pt %%%%%%%%%%%%%%%%%%%%%% %%% macros %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \def\dowrite #1{\immediate\write16 {#1} \immediate\write1 {#1} } %\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi} \newcount\EQNcount \EQNcount=1 \newcount\SECTIONcount \SECTIONcount=0 \newcount\APPENDIXcount \APPENDIXcount=0 \newcount\CLAIMcount \CLAIMcount=1 \newcount\SUBSECTIONcount 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``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE, \else\period \fi\fi \if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi \if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if T\isyrJPE \ \else\period \fi\fi\fi \if T\istoappearJPE (To appear)\period \fi \if T\ispreprintJPE Preprint\period \fi \if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi \if T\isvolJPE \unhbox\volboxJPE\unskip, \fi \if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi \if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% SOME SMALL TRICKS%%%%%%%%%% \def \breakline{\vskip 0em} \def \script{\bf} \def \norm{\vert \vert} \def \endnorm{\vert \vert} \def\Tr{\mathop{\rm Tr}\nolimits} \def\Im{\mathop{\rm Im}\nolimits} \def\im{\mathop{\rm im}\nolimits} \def\dist{\mathop{\rm dist}\nolimits} \def\Dom{\mathop{\rm Dom}\nolimits} \def\spec{\mathop{\rm spec}\nolimits} \def\varep{\varepsilon} \def\torus{{\bf T}} \def\OO{{\cal O}} \def\tildeu{{\tilde u}} \def\tildev{{\tilde v}} \def\barH{ {\bar H}} \def\tildeH{ {\tilde H}} \def\barK{ {\bar K}} \def\tildeK{ {\tilde K}} \magnification=\magstep1 \overfullrule=0em %\voffset=1truein\hoffset=1truein \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. 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