\documentstyle{amsppt} \magnification=1200 \def\vru{\vrule height0.15truecm width0.2truecm depth0.15truecm} \tolerance=1000 \hfuzz=1pt \vsize=21.truecm \hsize=15.8 truecm \hoffset=0.4truecm \normalbaselineskip=5.25truemm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=19pt \parskip=0.1pt plus1pt \font\titlefont=cmbx10 scaled\magstep1 \font\ninerm=cmr9 \vskip2.0truecm %\NoBlackBoxes %%%%%%%%%%this is a macro equivalent to \nologo%%%%%%%%%%%%%%%%%%%%%%% \catcode\@=11 \def\logo@{\null} \catcode\@=12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\titlefont Fixed point theorems in the Arnol'd model about instability} \centerline{\titlefont of the action-variables in phase-space} \vskip2.0truecm \centerline{\bf Paolo Perfetti} \centerline{Dipartimento di matematica II Universit{\accent"12 a} di Roma} \centerline{via della Ricerca Scientifica 00133 Roma, Italy} \vskip1.0truecm \noindent {\bf Abstract}\ {\ninerm We consider the hamiltonian $H={1\over2}(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in{\Bbb R}^2$ (\lq\lq Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds {\it (whiskers)} of invariant, \lq\lq a priori unstable tori", for any vector-frequency $(\omega,1)\in{\Bbb R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the {\tt Assertion B} pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method togheter with the \lq\lq conical metric" ( see the footnote ** of pag. 583 of [A]). } \vskip0.5truecm \noindent {\bf \mathhexbox2780.Introduction}\ By means of contraction techniques we prove the existence of {\it whiskered tori} (i.e. invariant tori that possess stable and unstable manifolds) for the dynamical system whose hamiltonian is (\lq\lq Arnol'd model") $$H(I_1,I_2,\varphi_1,\varphi_2,t)={1\over2}(I_1^2+I_2^2)+\varepsilon (\cos\varphi_1-1)+ \varepsilon\mu(\cos\varphi_1-1)(\sin\varphi_2+\cos t).$$ \noindent The hamiltonian $H$ is defined over the \lq\lq extended phase-space" $\tilde{\Cal M}={\Cal M}\times{\Bbb T}= {\Bbb R}^2\times{\Bbb T}^3;$ $(I_1,I_2)\in{\Bbb R}^2,$ $(\varphi_1,\varphi_2,t)\in{\Bbb T}^3$ $({\Bbb T}={\Bbb R}/2\pi{\Bbb Z}),$ $\varepsilon$ and $\mu$ are real parameters. Physically $H$ represents a pendulum $({1\over2}I_1^2+\varepsilon(\cos\varphi_1-1))$ coupled to a rotator $({1\over2}I_2^2)$ through a force periodically depending on $\varphi_1,$ $\varphi_2$ and $t.$ The canonical equations of motion are \left\{ \eqalign{ &\dot\theta=1,\qquad\qquad\dot\varphi_1=I_1,\qquad\qquad\dot\varphi_2=I_2\cr &\dot I_1=\varepsilon\sin\varphi_1(1+\mu(\sin\varphi_2+\cos \theta)) \qquad\qquad\qquad \dot I_2=-\varepsilon\mu\cos\varphi_2(\cos\varphi_1-1)\cr }\right. \eqno(0.0) \noindent and we denote by $\phi^\mu_t\colon\tilde M\to\tilde M$ the associated hamiltonian flow. \noindent For $\varepsilon=0$ the system is integrable. For $0<\varepsilon\le\varepsilon_o$ ($\varepsilon_o$ small enough), as well known by KAM theorem, most (in the sense of Lebesgue measure) of the tori are continued. The measure of the initial data excluded by the theorem is $O(\sqrt\varepsilon)$ (although they are dense). Now for $\varepsilon\ne0$ and $\mu=0,$ following [A], we consider the motions that \lq\lq develop along the separatrix of the pendulum". We therefore write the equations $$H^1(I_1,\varphi_1)= {1\over2}I_1^2+\varepsilon(\cos\varphi_1-1)=0,\qquad\quad H^2(I_2,\varphi_2)= {1\over2}I_2^2={1\over2}\omega^2$$ and consider their orbits. A standard calculation gives \eqalign{ &\theta(t)=\theta^o+t,\qquad\qquad I_2(t)\equiv\omega,\qquad\quad I_1(t)\doteq \pm\beta(t)=\pm{2\sqrt\varepsilon \over \cosh(\sqrt\varepsilon t)}\cr &\varphi_2(t)=\varphi_2^o+\omega t,\qquad \varphi_1(t)\doteq\pm\alpha(t)=\pi\pm2\arctan\sinh(\sqrt\varepsilon t). \cr } \eqno(0.1) $\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o):\tilde{\Cal M}\to \tilde{\Cal M}$ is the flow generated by the hamiltonian $H^0=H^1+H^2$ when the initial datum, for what concerns the variables $(I_1,\varphi_1),$ is $(\pm2\sqrt\varepsilon,\pi).$ \noindent The invariant set ${\Bbb T}_\omega=\{(I_1,\varphi_1,I_2,\varphi_2,\theta) \vert (I_1=0, \varphi_1=0, I_2=\omega, (\varphi_2,\theta)\in{\Bbb T}^2\}$ is geometrically an unstable torus (from which the definition \lq\lq a priori unstable" used in [CG]) and it is run linearly: $\theta^o\to \theta^o+t,$ $\varphi^o_2\to\varphi^o_2+\omega t.$ Adopting the definitions of [A] we call $Y^-_\omega$ the set of points such that, $dist(\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o), {\Bbb T}_\omega)\to0$ as $t\to\infty$ (it is called the {\it stable manifold} of ${\Bbb T}_\omega$). It is a three dimensional manifold in a five dimensional phase-space. The {\it unstable manifold} at ${\Bbb T}_\omega$, $Y^+_\omega,$ is defined as the set of points such that $dist(\phi_t^0(\pm2\sqrt\varepsilon,\pi,\omega,\varphi_2^o,\theta^o), {\Bbb T}_\omega)\to0$ as $t\to-\infty;$ because of the degeneracy of the model we have $Y^-_\omega\equiv Y^+_\omega$ and $Y^+_\omega\cap Y^-_{\omega^\prime}=\emptyset$ $\forall \omega\ne{\omega^\prime}.$ This last relation is connected to the impossibility of the trajectories (0.1) to make a \lq\lq large variation" in the $(I_1,I_2)$ variables; in fact $I_2$ is a constant while $\vert I_1\vert\le2\sqrt\varepsilon.$ \vskip0.7truecm \noindent Now set $\varepsilon\ne0$ and $\mu\ne0.$ The torus ${\Bbb T}_\omega$ continues to be invariant for any value of $\mu$ and in [A] ({\tt Theorem 3} pag.582) the following theorem (\lq\lq Arnol'd diffusion") is stated \vskip0.5truecm \noindent {\sl Assume $00$ there exists a $\mu_o(\varepsilon,A,B)$ such that if $0<\mu<\mu_o$ then $\vert I_2(t^o)-I_2(0)\vert\ge(B-A)$ where $t^o=t^o(\mu)$ }\footnote{$\mu_o\sim e^{-1/\sqrt\varepsilon}.$ Moreover it is easy to verify that $\mu_o\to0$ as $B\to\infty.$ } \vskip0.5truecm\noindent The proof is based on two fundamental steps: 1) {\tt Assertion B} of [A]: \lq\lq {\sl ${\Bbb T}_\omega$ is a whiskered transition torus for $\mu$ small enough}" 2) $Y^+_\omega\cap Y^-_{\omega^\prime}\ne\emptyset$ for $\omega^\prime$ close to $\omega.$ \vskip0.3truecm \noindent The proof of Part 1) is the object of this paper. Part 2) is proven in [A]. \noindent In order to prove 1) we shall need: i) the existence (continuation) for $\mu\ne0$ of the {\it stable manifold}, $Y^-_\omega,$ and {\it unstable manifold}, $Y^+_\omega,$ at ${\Bbb T}_\omega$ for any real $\omega$ (proved in {\bf \mathhexbox2782}; we have retained for the manifolds the same notation of the case $\mu=0$). This is what in the step 1) is represented by the word \lq\lq ${\Bbb T}_\omega$ is a whiskered torus for $\mu$ small enough" ii) \lq\lq The transition property" of ${\Bbb T}_\omega$ with $\omega\in{\Bbb R}\backslash{\Bbb Q}$ so that \lq\lq ${\Bbb T}_\omega$ is a whiskered transition torus" (see {\bf \mathhexbox2784}). \noindent Point ii) means: given an arbitrary point $p$ of the {\it stable manifold} at ${\Bbb T}_\omega$ let us consider an arbitrary neighborhood $U$ of $p$ and set $\Omega=\cup_{t\ge0}\phi_t(U)$ ($\phi_t$ is the flow of the system (0.0)). Now consider the {\it unstable manifold}, an arbitrary point $q$ of it and an arbitrary manifold ${\Cal N}$ that crosses $Y^+_\omega$ at $q.$ Moreover the manifold ${\Cal N}$ is such that $T_q{\Cal N}\oplus T_qY^+_\omega=T_q M$ (i.e.$T_q{\Cal N}$ and $T_qY^+_\omega$ span the whole tangent space; it is said that ${\Cal N}$ {\it complements} $Y^+_\omega$ at $q$); then $\Omega$ intersects ${\Cal N}.$ \noindent We do not prove ii) as it has been done in [Ma] and extended in [Cr] (they consider the hamiltonian functions previously considered in [Gr] and [Tr] respectively). The idea of the proof is based on a suitable generalization of the $\lambda-$lemma by Palis (see [PM] pag.80, [Wi] pag. 324, [Ru] pag.88) As said above we make use of contraction arguments in order to prove the existence of {\it stable} and {\it unstable} manifolds. We will work in the analytic class as the Cauchy theorem makes the estimates very easy. Frequently other less regular function spaces are introduced in connection with fixed-point theorems but they imply more involved estimates adding little to the comprehension of the problem. In recent years there have been many papers about instability of quasi-integrable dynamical system ([Xia1], [CG], [Xia2], [Ga2], [Xia3], [Ge1], [Ge2], [C1], [Moe], [RW1], [RW2], [C2], [Ni]). We just point out that the above papers use KAM theorem to prove the existence of the invariant tori or their stable and ustable manifolds. This forces to exclude, as usual, a dense set of initial data of small Lebesgue measure. The presence of holes in the space of the actions complicates a lot the construction of the transition chains (see [CG]). >From a non-perturbative point of view we cite the work of Bessi [Be]. Bessi uses variational techniques in order to prove the existence of trajectories connecting far away regions of phase-space and finds good estimates about the times of transition. \vskip0.3truecm \noindent The plan of the paper is the following: in {\bf\mathhexbox2781} we construct a suitable Poincar\'e map, say $M,$ for the system (0.0); in {\bf\mathhexbox2782} we construct the stable and unstable manifold, called ${\Cal Y}^-$ and ${\Cal Y}^+$ respectively, for $M$; in {\bf\mathhexbox2783} using the equations of ${\Cal Y}^-$ and ${\Cal Y}^+$ we linearize the motion on ${\Cal Y}^\pm$; finally in {\bf\mathhexbox2784} we show how [Ma] applies to our setting in order to prove the transition property (see ii) above). \vskip0.5truecm \noindent {\bf Acknowledgments}\ I thank Luigi Chierchia for many careful comments, advises and suggestions. \vskip0.6truecm \noindent {\bf \mathhexbox2781.Setup}\qquad In this section we do a suitable Poincar\'e map on the system (0.0). Therefore the relevant quantities that we shall consider: tori, stable and unstable manifolds are all relative to the map. With the transformation over the variables $(I_1,\varphi_1,I_2,\varphi_2,\theta)=U(q,p,I_2,\varphi_2,\theta)$ made of two canonical maps plus a rescaling the equations (0.0) become (more details are in Appendix 1) \left\{ \eqalign{ &\dot\theta=1,\qquad\quad\dot\varphi_2=I_2,\qquad\quad \dot I_2=-\varepsilon\mu\cos\varphi_2(\cos\tilde\varphi_1 (q,p)-1)\cr &\dot q=\sqrt\varepsilon q(1+F_J)-{\sqrt\varepsilon\over\sqrt2}\mu \sin\tilde\varphi_1(q,p)(\sin\varphi_2+\cos \theta) \cr &\dot p=-\sqrt\varepsilon p(1+F_J)+{\sqrt\varepsilon\over\sqrt2}\mu\sin\tilde \varphi_1 (q,p)(\sin\varphi_2+ \cos \theta)\cr }\right. \eqno(1.0) $F(J)=O(J^2)$ and $\tilde\varphi_1$ are analytic functions of $q$ and $p$ in the region $\vert q\vert +\vert p\vert\le O(1)$; $\tilde\varphi_1(0,0)=0.$ \noindent >From the equations (1.0) one immediately realizes that: \noindent 1) ${\Bbb T}_\omega=\{(q,p,I_2,\varphi_2,\theta)\vert q=0, p=0, I_2=\omega, (\varphi_2,\theta)\in{\Bbb T}^2\}$ is an invariant set for any value of the parameter $\mu$ and is linearly i.e. $\varphi_2(t)=\varphi_2^o+ \omega t,$ $\theta(t)=\theta^o+t.$ \noindent 2) if $\mu=0$ the $q$ and $p$ axis are invariant. $F(J)$ is a first integral (actually any function of $J$ is a prime integral) and for $\vert q\vert+\vert p\vert$ small enough it is $1+F_J>0.$ Then the points of the $p$ axis are exponentially shrunk to $0$ while the points of the $q$ axis are exponentially pushed away from the origin. In the new variables $(q,p,I_2,\varphi_2,\theta)$ the {\it stable manifold}, $Y^-_\omega$ is written as $\{(q,p,I_2,\varphi_2,\theta)\vert q=0,p\in(-\sigma,\sigma),I_2=\omega,(\varphi_2,\theta)\in{\Bbb T}^2\}$ while the {\it unstable manifold} $Y^+_\omega$ as $\{(q,p,I_2,\varphi_2,\theta)\vert q\in(-\sigma,\sigma), p=0,I_2=\omega,(\varphi_2,\theta)\in{\Bbb T}^2\}$ $(\sigma=O(1)).$ \vskip0.4truecm \noindent Now we do the following Poincar\'e-map: let $\phi^\mu_t(q^o,p^o,I^o_2,\varphi^o_2,0)\colon \tilde{\Cal M}\to \tilde{\Cal M}$ be the flow of the system (1.0) ($0$ stands for $\theta^o=0$ and we omit it for brevity). Denoting $r^o=(q^o,p^o,I^o_2,\varphi^o_2),$ for any integer $n$ we define $\phi^\mu_{2n\pi}(r^o,0)\vert_{{\Cal M}}\doteq\hat\phi^\mu_{2n\pi}(r^o) \colon B^2_\sigma\times{\Bbb R}\times{\Bbb T}\to B^2_\sigma\times{\Bbb R} \times{\Bbb T}$ ($B^2_\sigma$ is a two-dimensional sphere of radius $2\sigma^2$ in the $(q,p)$ space). The map $\hat\phi^\mu_{2\pi}$ is written symbolically as $(r^o)^\prime\doteq(x^\prime, y^\prime, I^\prime,\varphi^\prime)= \hat\phi^\mu_{2\pi}(x,y,I,\varphi)\doteq\hat\phi^\mu_{2\pi}(r^o)$ \left\{ \eqalign{ &\theta^\prime=\theta=0\cr &y^\prime=y e^{-2\pi\sqrt\varepsilon(1+F_J)} +\mu \sqrt\varepsilon f_1(x,y,I,\varphi,\mu) \cr &x^\prime=x e^{2\pi\sqrt\varepsilon(1+F_J)} -\mu \sqrt\varepsilon f_2(x,y,I,\varphi,\mu) \cr &\varphi^\prime=\varphi+2\pi I+\mu\varepsilon f_3(x,y,I,\varphi,\mu) \cr &I^\prime=I-\mu\varepsilon f_4(x,y,I,\varphi,\mu) \cr }\right. \eqno(1.1) where $f_1,\ f_2=O(\vert x\vert+\vert y\vert)\qquad f_3,\ f_4= O((\vert x\vert+\vert y\vert)^2).$ In fact we have \left\{ \eqalign{ &q(2\pi;r^o)=q^o e^{2\pi\sqrt\varepsilon(1+F_J)} -\mu{\sqrt\varepsilon\over\sqrt2}\int_0^{2\pi}ds\sin\tilde\varphi_1 (q,p)(\sin\varphi_2+\cos s)\cr &\qquad\qquad\qquad\qquad +\sqrt\varepsilon \int_0^{2\pi}ds\Bigl[\bigl(q(1+F_J)\bigr)(s;r^o,\mu)- \bigl(q(1+F_J)\bigr)(s;r^o,0)\Bigr]\cr &p(2\pi;r^o)=p^o e^{-2\pi\sqrt\varepsilon(1+F_J)} -\mu{\sqrt\varepsilon\over\sqrt2} \int_0^{2\pi}ds\sin\tilde\varphi_1 (q,p) (\sin\varphi_2+\cos s)-\cr &\qquad\qquad\qquad\qquad -\sqrt\varepsilon \int_0^{2\pi}ds \Bigl[\bigl(p(1+F_J)\bigr)(s;r^o,\mu)- \bigl(p(1+F_J)\bigr)(s;r^o,0)\Bigr] \cr &\varphi_2(2\pi;r^o)=\varphi_2^o+2\pi I^o_2+\mu\varepsilon \int_0^{2\pi} dt \int_0^t ds\cos\varphi_2(\cos\tilde\varphi_1(q,p)-1)\cr &I_2(2\pi;r^o)=I_2^o-\mu\varepsilon\int_0^{2\pi}ds \cos\varphi_2(\cos\tilde\varphi_1(q,p)-1)\cr }\right. \eqno(1.2) In analogy with the differential equations (0.0) we define: ${\Cal Y}^-_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert dist(\hat\phi_{2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to+\infty$ (the {\it stable manifold} at ${\Cal T}_\omega$) and ${\Cal Y}^+_\omega=\{r^o\in B^2_\sigma\times{\Bbb R}\times{\Bbb T}\vert dist(\hat\phi_{-2\pi n}^\mu(r^o),{\Cal T}_\omega)\to0$ as $n\to\infty$ (the {\it unstable manifold} at ${\Cal T}_\omega$). A different value of $\theta^o$ would generate a different Poincar\'e map. If we call ${\Cal Y}^-_\omega(\theta^o)$ the stable manifold, ${\Cal Y}^+_\omega(\theta^o)$ the unstable one and ${\Cal T}_\omega(\theta^o)$ the invariant torus of frequency $\omega$ for the Poincar\'e map at $\theta^o$ i.e. taking as surface map $\{(p,q)\in B^2_\sigma, I\in{\Bbb R}, (\varphi,\theta)\in{\Bbb T}^2 \vert \theta=\theta^o\},$ then we have $Y^-_\omega=\cup_{0\le \theta^o\le2\pi}({\Cal Y}^-_\omega)(\theta^o),$ $Y^+_\omega=\cup_{0\le \theta^o\le2\pi}({\Cal Y}^+_\omega)(\theta^o)$ and ${\Bbb T}_\omega=\cup_{0\le \theta^o\le2\pi}{\Cal T}_\omega(\theta^o)$ (see Appendix 1). We then reduce to the study of the existence of ${\Cal Y}^-_\omega$ and ${\Cal Y}^+_\omega;$ the existence of ${\Cal T}_\omega$ is trivial matter due to the peculiarity of the model. We have to keep in mind that after the integrations with respect to time, there remain functions only of the initial data $r^o.$ For any $I^o\in{\Bbb R}$ and for any $\mu$ the set ${\Cal T}_{I^o}=\{q=0, p=0, I=I^o, \varphi\in{\Bbb T}\}$ is invariant. Geometrically it is a circle contained in the hyperplane $\theta=0;$ on it the motion is linear $\varphi_2^{(n)}=\varphi_2^{(0)}+2\pi I^on,$ $n\in{\Bbb N};$ it is periodic if $I^o\in{\Bbb Q}$ and quasi-periodic if $I^o\in{\Bbb R}\backslash{\Bbb Q}.$ Of course we have ${\Bbb T}_{I^o}=\cup_{0\le \theta^o\le2\pi}{\Cal T}_{I^o}(\theta^o).$ When $\mu=0$ the following sets are also invariant: ${\Cal Y}^-_{I^o} =\{q=0, p\in(-\sigma,+\sigma), I=I^o, \varphi\in{\Bbb T}\}$ (the stable manifold) and ${\Cal Y}^+_{I^o} =\{q\in(-\sigma,+\sigma), p=0, I=I^o, \varphi\in{\Bbb T}\}$ (the unstable manifold); they are two dimensional in the four dimensional space ${\Bbb R}^3\times{\Bbb T}.$ \vskip0.5truecm \noindent {\bf \mathhexbox2782.Existence of perturbed stable and unstable manifolds} \vskip0.3truecm \noindent Now we investigate the point i) of the introduction. >From now on we shall consider only the map (1.1). What follows is closely related to the Hadamard's theorem about the existence of stable and unstable manifolds of fully hyperbolic systems and in general to the theory of invariant sets for flows and maps defined on a manifold (possibly ${\Bbb R}^N$). There is an abundant literature on this argument and we mention only few references [Ke], [Fe1], [Fe2], [Fe3], [La2], [La1], [Ga3], [Ze], [HPS], [Wi], [PM], [Ir], [Ru]. \noindent Following Hadamard's approach and using a suggestion by Arnol'd (see footnote ** of pag.583 of [A]) we look for the stable manifold at ${\Cal T}_\omega$ as ${\Cal Y}^-_\omega=\{(x,y,I,\varphi)\vert x=\mu yA^s(y,\varphi;\omega,\mu), y\in(-\sigma,+\sigma), I=\omega+\mu\sqrt\varepsilon y{\Cal I}^s(y,\varphi;\omega,\mu), \varphi\in{\Bbb T}\}$ while ${\Cal Y}^+_\omega=\{(x,y,I,\varphi)\vert x\in(-\sigma,+\sigma), y=\mu xA^u(x,\varphi;\omega,\mu), I=\omega+ \mu\sqrt\varepsilon y{\Cal I}^u(x,\varphi;\omega,\mu), \varphi\in{\Bbb T}\}.$ In order to simplify the estimates (leaving all the essential difficulties) we shall consider the space of the holomorphic functions ${\Cal A}(\Delta_{y,\varphi}^{\sigma,\xi})$ (for ${\Cal Y}^-_\omega$) given by the analytic functions over $(-\sigma,\sigma)\times{\Bbb T}$ that admit an holomorphic extension to the complex domain $\Delta_{y,\varphi}^{\sigma,\xi} =\{(y,\varphi)\in{\Bbb C}^2\vert\ \vert y\vert<\sigma, \vert Im\varphi\vert<\xi<1\}$ and are continuous on $\overline\Delta_{y,\varphi}^{\sigma,\xi}.$ With the norm $\Vert g\Vert= \sup_{\overline\Delta_{y\varphi}^{\sigma,\xi}}\vert g(y,\varphi)\vert,$ ${\Cal A}(\Delta_{y,\varphi}^{\sigma,\xi})$ is a Banach space. Of course for ${\Cal Y}^+_\omega$ the definition of the space of functions is exactly the same except that there is $x$ in place of $y$ and $u$ in place of $s.$ Now we prove the following \vskip0.5truecm \noindent {\bf Theorem 2.0}{\sl\ For any $\omega\in{\Bbb R}\backslash\{0\}$ and for any $\vert\mu\vert\le\mu_o$ there exist two unique functions $(A^s,{\Cal I}^s)\colon[-{\sigma\over2},{\sigma\over2}]\times{\Bbb T} \to{\Bbb R}\times{\Bbb R}$ such that the set ${\Cal Y}^-_\omega=\{(x,y,I,\varphi)=(\mu yA^s(y,\varphi;\omega,\mu,\varepsilon),y, \omega+\mu\sqrt\varepsilon y{\Cal I}^s(y,\varphi;\omega,\mu,\varepsilon) ,\varphi)\vert y\in[-{\sigma\over2},{\sigma\over2}],\ \varphi\in{\Bbb T}\}$ is invariant respect to the Poincar\'e map (1.1). $A^s$ and ${\Cal I}^s$ are analytic in $(y,\varphi),$ in $\omega\in{\Bbb R}\backslash\{0\}$ and $\mu.$ } \vskip0.4truecm \noindent Remarks\quad i) in the functions $A^s$ and ${\Cal I}^s,$ $\varphi$ and $y$ are the variables while the presence of $\omega,$ $\mu$ and $\varepsilon$ is of parametric type\quad ii) clearly the theorem could be formulated choosing less regular function spaces (see e.g. [La1]) \quad iii) $\sigma=O(1);$ $\mu_o$ is a function of $\sigma,$ $\varepsilon,$ $\xi,$ and of the Poincar\'e map; moreover $\mu_o\to0$ as $\sigma\to0,$ as $\varepsilon\to0$ and $\varepsilon\to\infty$ \qquad iv) in the next section we will prove that ${\Cal Y}^-_\omega$ is the stable manifold \vskip0.3truecm \noindent {\bf Proof.}\qquad Let's consider the closed set $\bar{\Cal A}\subset{\Cal A}(\Delta_{y\varphi}^{\sigma,\xi})$ whose functions have the norm bounded by a constatnt ${\Cal A}$ $\Vert g\Vert_{\sigma,\xi}= \sup_{\Delta_{y,\varphi}^{\sigma,\xi}}\vert g(y,\varphi)\vert\le A_o.$ Moreover we will consider the functions from $\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}$ into ${\Bbb C}$ where $\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}= \{(x,y,I,\varphi)\in{\Bbb C}^4\vert\ \vert x\vert<\sigma, \vert y\vert<\sigma, \vert I-\omega\vert <\rho, \vert Im\varphi\vert<\xi<1\}.$ Analogously to what happens in ${\Cal A}(\Delta_{y\varphi}^{\sigma,\xi})$ we define $\Vert h\Vert_{\sigma,\xi,\rho}= \sup_{\overline\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}} \vert h(x,y,I,\varphi)\vert$ for functions defined and analytic on $\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}$ and continuous on $\overline\Delta_{x,y,I,\varphi}^{\sigma,\xi,\rho}.$ The function $(A^s,{\Cal I}^s)$ will be found as fixed point of a contraction map from $\bar{\Cal A}\times\bar{\Cal A}$ into itself. Then the uniqueness and analiticity immediately follow. First of all we impose the invariance. A brief computation gives \eqalign{ &A={\sqrt\varepsilon\over ay}f_2(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)+ {1\over a}\bigl( {1\over a}+{\mu\sqrt\varepsilon\over y} f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)\bigl) A({y\over a}+\cr &+\mu\sqrt\varepsilon f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi),\cr &\qquad\qquad\qquad ,\varphi+2\pi\omega+ 2\pi\mu\sqrt\varepsilon y{\Cal I}+ \varepsilon\mu f_3(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi))\cr &{\Cal I}={\sqrt\varepsilon\over y} f_4(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)+\bigl( {1\over a}+{\mu\sqrt\varepsilon\over y} f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi)\bigl) {\Cal I}({y\over a}+\cr &+\mu\sqrt\varepsilon f_1(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi),\cr &\qquad\qquad\qquad ,\varphi+2\pi\omega+ 2\pi\mu\sqrt\varepsilon y{\Cal I}+\mu\varepsilon f_3(\mu yA,y,\omega+\mu\sqrt\varepsilon y{\Cal I},\varphi))\cr } \eqno(2.0) and $a=e^{2\pi\sqrt\varepsilon(1+F_J)}$. We indicate (2.0) as $M(A,{\Cal I})(y,\varphi)=\bigl(M_1(A,{\Cal I}),M_2(A,{\Cal I})\bigl) (y,\varphi)$ and try to solve it by means of contraction techniques: we look for two functions $(A^s,{\Cal I}^s)$ such that $M_1(A^s,{\Cal I}^s)=A^s, M_2(A^s,{\Cal I}^s)={\Cal I}^s.$ We show that there exists a real number $00$ and any $n\in {\Bbb N}^N$, one has: $$\Vert\partial_z^n g\Vert_{{\Cal D}'}\equiv\sup_{{\Cal D}'} \vert{\partial^{\vert n\vert} g\over \partial_{z_1}^{n_1}\cdot\cdot\cdot\partial_{z_N}^{n_N}}\vert\le \vert n\vert!\ \delta^{-\vert n\vert} \ \Vert g\Vert_{\Cal D}$$ \noindent The proof is a straightforward exercise based upon Cauchy's integral formula using the contour $\vert z^\prime_h-z_h\vert=\delta$, $h=1,...,N$ ($z\in {\Cal D}'$ fixed and $z^\prime\in {\Cal D}$ variable of integration)}. \vskip0.5truecm \noindent {\bf Appendix 2}\qquad Sketch of the proof of theorem 3.0. We rewrite the system (3.2) as \left\{ \eqalign{ &\eta^\prime=\eta e^{-2\pi\sqrt\varepsilon} +\mu\eta\bigl(h_1+h_2\bigr)(\eta,\varphi)\cr &\varphi^\prime=\varphi+2\pi\omega+2\pi\mu\sqrt\varepsilon{\Cal I}^s_\omega(\eta,\varphi)+ \mu\varepsilon F_3(\eta,\varphi)\cr }\right. \eqno(4.2) where \eqalign{ &h_1={1\over\eta\mu}\int_0^1d\lambda {d\over d\lambda} {1\over b(\eta,\varphi;\lambda\mu)}={1\over\eta} \int_0^1-d\lambda {1\over b^2}{d\over d\zeta} a[\zeta Y^2(\eta,\varphi;\zeta)A^s\Bigl( Y(\eta,\varphi;\zeta),\varphi\Bigr)]\vert_{\zeta=\lambda\mu}\cr } $\vert\eta\vert\le\sigma_1,$ $\vert Im\varphi\vert\le{\xi\over4};$ for the definition of $a[\cdot]$ see (3.4). Now $${d\over d\zeta}a[\zeta Y^2(\eta,\varphi;\zeta)A^s\bigl( Y(\eta,\varphi;\zeta),\varphi\bigr)]= \Bigl(Y^2A^s+2\zeta Y A^s{\partial Y\over\partial\zeta}+ \zeta Y^2{\partial Y\over\partial\zeta} {\partial A^s\over\partial Y}\Bigr){da\over dJ}$$ and in the domain $\vert\eta\vert\le\sigma_1,$ $\vert\mu\vert\le{1\over2}\mu_o,$ $\vert Im\varphi\vert\le{\xi\over4}$ we have $\Vert{1\over\eta}{d\over d\zeta}a[\cdot]\Vert \le{A_o e^{3\pi\sqrt\varepsilon}\over\sigma\mu_o}$ and then $\vert- {1\over\eta\mu} \int_0^1d\lambda b^{-2}(\eta,\varphi;\lambda\mu){d\over d\lambda} b(\eta,\varphi;\lambda\mu)\vert=\vert h_1(\varphi;\lambda\mu)\vert\le{4A_o\over\mu_o} e^{\pi\sqrt\varepsilon}.$ For what concerns $h_2$ we observe that $Y(\eta,\varphi)\to0$ as $\eta\to0$ and then $h(\eta,\varphi)\to0$ as $\eta\to0$ (see (3.4)). Using the maximum principle for holomorphic functions in the form of Schwarz's lemma we bound $\Vert{h\over\eta}\Vert_{\sigma_1,{\xi\over4}}\le {\pi A_o\over\sigma_1}\sqrt\varepsilon e^{3\pi\sqrt\varepsilon}.$ Now substitute $\eta=r+\mu R(r,\theta)$ and $\varphi=\theta+\mu r^2\Phi(r,\theta)$ into the system (4.2) and write a fixed point equation \eqalign{ &R(re^{-2\pi\sqrt\varepsilon},\theta+2\pi\omega)=e^{-2\pi\sqrt\varepsilon} R(r,\theta)+\bigl(r+\mu R(r,\theta)\bigr)\cdot\cr &\cdot\bigl(h_1+h_2\bigr) (r+\mu R(r,\theta),\theta+\mu r^2\Phi(r,\theta))\doteq V_1(R,\Phi)\cr } \eqalign{ &\Phi(r,\theta)=e^{-4\pi\sqrt\varepsilon} \Phi(re^{-2\pi\sqrt\varepsilon},\theta+2\pi\omega)- 2\pi{\sqrt\varepsilon\over r^2}{\Cal I}^s_\omega(\eta(r,\theta),\varphi(r\theta))-\cr &-{\varepsilon\over r^2}F_3(\eta(r,\theta),\varphi(r\theta)) \doteq V_2(R,\Phi).\cr } As in the previous cases the Cauchy estimates enable us to say that in the domain $\Delta^{\sigma_2,{\xi\over16}}_{\eta,\theta}$ and for $\vert\mu\vert\le\mu_1$ there exists a unique solution (fixed point) of the previous equations that we call $(R^s,\Phi^s).$ About their norm we have taken $\Vert R^s\Vert_{\sigma_2,{\xi\over16}}\ge{20A_o\sigma_1\over\mu_o\sigma_o} e^{3\pi\sqrt\varepsilon}(1-e^{-2\pi\sqrt\varepsilon})^{-1},$ $\Vert\Phi^s\Vert_{\sigma_2,{\xi\over16}}\ge {7A_o\sigma_o\sqrt\varepsilon\over\sigma_2^2} (1-e^{-4\pi\sqrt\varepsilon})^{-1}.$ \noindent It is important to note that $R^s(0,\varphi)\equiv0$ and being $R^s$ an analytic function it can be written as $r\hat R^s.$ Moreover for writing ${{\Cal I}^s_\omega\over r^2}$ meaningful we need to keep in mind the content of what written immediately before the Theorem 3.0. \vskip0.5truecm \centerline{\bf References} \vskip0.2truecm \noindent [A] V.I.ARNOL'D {\sl Instabilities of dynamical systems with several degrees of freedom} Sov.math.Dokl. {\bf 6} (1964) p.581-585 \noindent [Be] U.BESSI {\sl An Approach to Arnold Diffusion through the Calculus of Variations}, Nonlinear Analysis, T.M.A., vol.26 num.6 (1996) \noindent [C1] L.CHIERCHIA {\sl Arnol'd instability for nearly-integrable analytic hamiltonian systems}, in \lq\lq Local and Variational Methods in the Study of Hamiltonian Systems" Ambrosetti-Dell'Antonio eds.World Scientific Pub. 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