SUPEREXPONENTIAL STABILITY OF KAM TORIALESSANDRO MORBIDELLICNRS, Observatoire de la Cote d'AzurBP 229, 06304 { NICE Cedex 4, France.ANTONIO GIORGILLIDipartimento di Matematica dell' Universit�a di Milanoand Gruppo Nazionale di Fisica Matematica del CNR,Via Saldini 50, 20133 Milano, Italy.Abstract. We study the dynamics in the neighbourhood of an invariant torus of a nearlyintegrable system. We provide an upper bound to the di�usion speed, which turns out tobe of superexponentially small size exp(exp(�1=%)), % being the distance from the invarianttorus. We also discuss the connection of this result with the existence of many invariant toriclose to the considered one.1. Introduction and resultsWe consider the problem of Arnold di�usion in nearly integrable Hamiltonian systems,with the aim of producing bounds on the di�usion speed in the spirit of Nekhoroshev'stheory. We concentrate our attention in the neighbourhood of the invariant tori, theexistence of which is guaranteed by KAM theory. We show that a clever use of theknown results of both KAM theory and Nekhoroshev's theory leads to strong conse-quences, although of local character.The key remark is that in the neighbourhood of an invariant KAM torus it isnatural to introduce the distance from the torus as a new perturbation parameter.Thanks to this change of perspective we can apply a Birkho� procedure, thus reducingthe perturbation to an exponentially small size in 1=%, % being the distance from theinvariant torus. The new action variables introduced by the Birkho� normalizationare good adiabatic invariants, since they remove most of the quasiperiodic oscillationof the old action variables due to perturbation, namely, the so called deformation. Atthis point we change again our perspective, and investigate the dynamics with theglobal approach of Arnold on the one hand, and of Nekhoroshev on the other hand.All the results follow from a direct application of known theorems. In particular, a

2 A. Morbidelli and A. Giorgillirelevant consequence is that the speed of the Arnold di�usion (if any) turns out to beof superexponentially small size exp(exp(�1=%)). This is our main contribution.In view of this result, we can consider an invariant KAM torus as the head of astructure which dominates the dynamics in its neighbourhood. It appears that sucha structure has a typical radius which depends on the size of the perturbation, anddecreases to zero as the perturbation increases towards the critical size correspondingto the destruction of the torus. The neighbourhood of the torus turns out to containmany other invariant tori, constituting a set the relative volume of which tends to 1when approaching the head torus. More precisely, the relative volume of the comple-ment of the set of invariant tori turns out to be as small as exp(�1=%). This representsa local improvement with respect to the estimates more or less explicitly contained inmany previous statements: see for instance ref. [13]. The new aspect that we point outis that the dynamics is strongly a�ected by the existence of such a structure, inasmuchas the chaotic di�usion of orbits starting in the gaps between tori is thus forced torequire a very long time. According to the known Nekhoroshev's estimates, such atime is exponentially large with the inverse of the perturbation. In our case, as wealready remarked, the perturbation is exponentially small in 1=%. This determines thesuperexponential estimate for the di�usion time.The picture resulting from the discussion above contrasts with the quitewidespread opinion, especially among physicists, that the existence of invariant tori insystems with more than two degrees of freedom is not so relevant (see for instance [10]).Such an opinion is supported by the known fact that the tori do not isolate separatedregions in phase space, thus allowing for the so called Arnold's di�usion. More recently,some authors have pointed out that KAM tori are very sticky, by applying locally theNekhoroshev theory to the neighbourhood of an invariant torus: see for instance [16].Their approach is actually equivalent to the application of the Birkho� normalizationprocedure using the distance from the invariant torus as a perturbation parameter.However, we go beyond this level. We show in fact that although the KAM tori arenot isolating, they form nevertheless a kind of impenetrable structure that the orbitscannot escape (nor enter, of course) for an exceedingly long time, very large even withrespect to the known Nekhoroshev's estimates. Thus, the behaviour of the system inthe region containing invariant KAM tori can be said to be e�ectively integrable.We come now to a formal statement of our result. We consider a canonical systemof di�erential equations with an Hamiltonian of the form(1) H(p; q; ") = H0(p) + "H1(p; q; ") ;where p 2 G � Rn, G being an open set, and q 2 Tn are the action{angle variables,and " a small parameter. The Hamiltonian is assumed to be a real analytic functionof all its variables. The frequencies of the unperturbed system will be denoted by

Superexponential stability of KAM tori 3!(p) = @H0@p , and the Hessian matrix of H0 with respect to p will be denoted by C(p).Also, we shall denote by j � j a norm on functions over their domain of analyticity, forinstance the usual supremum norm, and by k � k a norm for vectors in Rn or Cn, forinstance the Euclidean norm.We pick up a point p� such that the corresponding frequency !� := !(p�) satis�esthe diophantine condition(2) jk � !�j � jkj�� for all k 2 Zn ; k 6= 0 ;for some > 0 and � > n� 1; here, we de�ne jkj = jk1j+ : : :+ jknj. We also assumethat the Hamiltonian admits an analytic bounded extension to a domain(3) D�(p�) = B�(p�)�Tn� ;for some positive �. Here, B�(p�) is the open ball of radius � and center p� in Cn, andTn� = fq 2 Cn : jIm (q)j < �g. By the analyticity of the Hamiltonian, such a � exists.With this setting we can state ourTheorem: Consider the Hamiltonian (1) in the domain D�(p�) de�ned by (3), with!� � !(p�) satisfying (2), and assume that the following conditions hold with positiveconstants ", d < 1 and m:(a) jH1j < " in D�(p�);(b) C(p) satis�es the nondegeneracy condition dkvk < kC vk < d�1kvk;(c) the matrix C = C(p�) satis�es jCv � vj > mjv � vj for all v 2 Rn with v ? !�.Then there exists a positive "� such that for every " < "� the following statementholds true: there is a positive %� depending on " such that for every % < %� there is ananalytic canonical transformation mapping (J; ) 2 B%(0)�Tn to (p; q) 2 B�(p�)�Tnwith the properties:(i) J = 0 is an invariant torus carrying a quasiperiodic ow with frequencies !�.(ii) The domain D%(0) contains an in�nity of invariant tori the relative volume ofwhich tends to 1 as % ! 0, being estimated by 1 � exp(�1=%); the structure ofthese invariant tori is close to that of the tori J = const, in the sense that everyinvariant torus lies in a neighbourhood of some torus J = const, the radius of theneighbourhood being estimated by exp[�(1=%)].(iii) For every initial datum J0 2 B%(0) one has that jJ(t) � J0j is estimated byexp��(1=%)1=(�+1)� for jtj � T , whereT ' exp�exp(1=%)1=(�+1)� :Let us add a comment concerning the local quasi{convexity hypothesis (c). Weuse this hypothesis in connection with Nekhoroshev's theorem. It should be remarkedthat the original formulation of Nekhoroshev requires the less stringent condition ofsteepness, which however is more di�cult to handle. Later formulations of the theorem

4 A. Morbidelli and A. Giorgillimake use of the easier condition of convexity, i.e., jC(p)v � vj > mv � v for all p 2 Gand for all v 2 Rn. However, as stressed by Nekhoroshev himself, it is enough torequire that the latter inequality holds for all v ? !(p). This is the so called conditionof quasi{convexity. The fact that we restrict our attention to the neighbourhood ofthe invariant torus p� allows us to further relax the quasi{convexity condition to theform (c).2. Scheme of the proofThe proof of our theorem relies on a composition of known results, namely of KAMtheorem[9] and Nekhoroshev's theorem[14][15], both in a local and a global formulation.More precisely, we need two preliminary steps in order to prove the statement (i), andtwo independent steps in order to prove the statements (ii) and (iii) respectively. The�rst step reduces the Hamiltonian to Kolmogorov's normal form, thus ensuring theexistence of an invariant torus (the head torus). The second step is the construction of aBirkho� normal form up to a �nite order in the neighbourhood of the head torus, withexponential estimates of Nekhoroshev's type. The mapping (J; )! (p; q) referred toin the statement of the theorem is actually the composition of the mapping leadingto Kolmogorov's normal form and of that leading to Birkho�'s normal form. Thethird step is the application of KAM theorem in the global version due to Arnold[1].The fourth and last step is the application of Nekhoroshev's theorem. We performthe four steps above in separate subsections. We omit all the unnecessary technicaldetails, making a direct use of the known theorems in some available form, withoutattempting to �nd optimal estimates.2.1 Use of Kolmogorov's normal formWe follow the formulation of Kolmogorov's theorem provided in [3]. The hypotheses(a) and (b) allow to apply the theorem. Accordingly, there exists a positive "� suchthat for every " < "� there exists a positive �0 < � and an analytic canonical trans-formation (p0; q0)! (p; q), mapping D�0(0) into D�(p�), which gives the Hamiltonianthe Kolmogorov's normal form(4) H 0(p0; q0) =Xi !ip0i + 12Xi;j C 0i;jp0ip0j + f(p0; q0)with f at least quadratic in p0. In particular, the real matrix C 0ij satis�es the conditions(b) and (c) with new constants m0 < m and d0 < d, while the quadratic part of f haszero average over the angles q0.

Superexponential stability of KAM tori 52.2 Use of the local formulation of Nekhoroshev's theoremWe expand now the perturbation f in (4) in power series about the origin (i.e., theinvariant torus). We write the Hamiltonian as(5) H(p0; q0) =Xi !ip0i +H2(p0; q0) +H3(p0; q0) + : : :where Hs(p0; q0) is for s � 2 a homogeneous polynomial of degree s in p0. As f isanalytic in D�0(0), the expansion is convergent, and the norm of Hs decreases at leastgeometrically with s, being of order �0s. Remark that the Hamiltonian (5) resemblesthat of a system of perturbed harmonic oscillators, as considered for instance in [8].Thus, we can apply the local formulation of Nekhoroshev's theorem. To this end, asusual, we perform a Birkho� normalization up to some �nite order r, thus giving theHamiltonian the form(6) H(J; ) =Xi !iJi + Z(r)(J) +R(r)(J; ) ;which is analytic inD%(0) for some positive % < �0, depending on r. Here, Z(r)(J) is thenormalized part of the new Hamiltonian, while R(r)(J; ) is the still non normalizedremainder. In particular the quadratic part of Z(r) is nothing but (1=2)Pi;j C 0ijJiJj ,i.e., it coincides with the average quadratic part of the Kolmogorov's normal form (4).Thus, the validity of the conditions (b) and (c) is preserved by the Birkho�'s normal-ization procedure.Standard estimates (see for instance [5], [7], [8], [17]) allow to prove that in D%(0)the size of the remainder is of order (r!)�+1%r. An optimal choice of r as a function of%, i.e., r ' (1=%)1=(�+1), allows one to prove that there exists a positive % such thatfor % < % one has(7) ���R(r+1)��� < A jf j exp"��%%�1=(�+1)# ;with a constant A depending on the number n of degrees of freedom.2.3 Use of Arnold's formulation of KAM theoremWe rewrite the Hamiltonian (6) in the form H(J; ) = H0(J) +H1(J; ), where(8) H0(J) =Xi !iJi + Z(r)(J) ; H1(J; ) = R(r)(J; ) :To this Hamiltonian we apply the statement of the main theorem in [1]. The ana-lyticity hypothesis is clearly satis�ed, as well as the condition jH1j < M for someM . Indeed, in view of the result of the previous section, the remainder R(r)(p; q) is

6 A. Morbidelli and A. Giorgillian analytic function in a domain D% for some positive %, and is bounded there byA jf j exp h� (%=%)1=(�+1)i. We stress that, choosing % small enough, the size of H1can be made arbitrarily small, exponentially with 1=%. Concerning the nondegeneracycondition det jC0(J)j 6= 0 ; with C0i;j = @2H0@Ji@Jj ;which must hold for all J in the domain where the theorem is applied, we remark thatit holds in some neighbourhood of the torus J = 0. Indeed, it holds for J = 0 sinceone clearly has C0(0) = C 0ij , where C 0ij is just the matrix appearing in (4), and so, bycontinuity, it holds in a neighbourhood of J = 0. Thus, a straightforward applicationof Arnold's theorem shows that there exists a positive %1 < % such that for % < %1 therelative volume of the KAM tori in the neighbourhood of J = 0 is large, and tends to1 as % ! 0. According to Neishtadt[13], the volume of the complement of the set ofinvariant tori is estimated by p", where " is the size of the perturbation. In our casewe replace " by exp(�1=%). This proves the statement (ii).2.4 Use of Nekhoroshev's theoremAs in the last section, we rewrite the Hamiltonian as H(J; ) = H0(J)+H1(J; ), withH0 and H1 given by (8). We use the formulation of Nekhoroshev's theorem given in[4]. Again, the analyticity and nondegeneracy conditions are satis�ed, as was alreadyremarked. The condition on the smallness of the perturbation is satis�ed too, due tothe exponential decrease of the remainder. Instead of the convexity condition requiredin [4],(9) kC0(p)v � vk � mv � v for all v 2 Rnwe use the quasi{convexity, i.e., we add the condition v ? !(p) as remarked in theintroduction. In view of the hypothesis (c), such a condition is clearly satis�ed in someneighbourhood of p = 0. Thus, we apply Nekhoroshev's theorem, according to which,denoting by " the size of the perturbation, for " smaller than a positive " one has(10) jJ(t)� J(0)j < P"1=c for jtj � T �1"�1=2 exp"�1"�1=c# ;with constants P, T and c ' n2. (We stress that the estimates are not optimal: forbetter estimates see for instance [11], [17].) In our case the size of the perturbation is" = A jf j exp h� (%=%)1=(�+1)i. We conclude that there exists a positive %2 such thatfor every % < %2 the upper bound to the di�usion time is of order exp [exp(1=%)],as claimed. This proves the statement (ii). The constant %� in the statement of thetheorem must be identi�ed with the minimum between %1 and %2.

Superexponential stability of KAM tori 73. DiscussionWe discuss here some points that we consider relevant for interpretation and use ofour result. We discuss in particular three points. (i) The possible optimality of ourestimate, and the relation with some recent results of Chierchia and Gallavotti[6] onthe existence of di�usion. (ii) The possible interpretation of the known phenomenonof the existence of a quite sharp threshold separating order from chaos in nearlyintegrable systems. (iii) The applicability of the same approach to di�erent situations,for instance, the case of an elliptic equilibrium.Concerning the �rst point, namely the optimality of the results, a remark is inorder. There is a widespread opinion that Nekhoroshev's theorem is in some senseoptimal, although evidence of this fact has never been produced. Thus, the followingquestion naturally comes to mind: the Nekhoroshev's procedure has been already usedin sect. 2.2; how can we go further in optimization using again the Nekhoroshev'sapproach (as is done in sect. 2.4)?To clarify this point, let us consider again the Hamiltonian (1). Attempting totransform that system into an integrable one up to terms of order "2 is clearly im-possible, as remarked long ago by Poincar�e, unless some restriction is imposed eitheron the domain, or on the Fourier terms to be removed, or both. Indeed, the existenceof resonances actually causes a change of the actions (in resonant zones) with speedof order ". Nekhoroshev's theory, in its most general formulation, states essentiallythat the main e�ect of the perturbations reduces to an oscillation along a directionof fast drift, just due to the resonance. Such an oscillation is necessarily bounded asfar as the resonant zones do not overlap: actually, the strongest condition in the socalled geometric part of the Nekhoroshev's theorem is precisely the non overlappingof resonances. Superimposed to the oscillation there could be a very slow di�usion,but the di�usion speed is exponentially slow with the inverse of the perturbation. Inbrief, a perturbation of order " causes an oscillation of order, e.g., "1=2, but a di�usionof order exp(�1=") only.Let us now come back to our problem. The procedure illustrated in sect. 2.2 is anattempt to prove that in the neighbourhood of a torus the system is still integrable,in Birkho�'s spirit. Such a procedure is successful up to a certain �nite order, becausethe diophantine condition on the frequencies ensures that in the neighbourhood of thetorus there are no resonances of low order. This can be understood on the basis of thefollowing heuristic argument. An elementary estimate shows that inside a ball of radius% there are no resonances of order lower than %�1=(�+1). On the other hand, due tothe analyticity of the Hamiltonian, the coe�cient of a resonance of order s is of orderexp(�s). Thus, in a ball of radius %, Birkho�'s procedure stops when one encountersa resonance of order %�1=(�+1), causing a perturbation of order exp(�%�1=(�+1)). Thisis indeed the exponential remainder given by the local Nekhoroshev's estimate in the

8 A. Morbidelli and A. Giorgillineighbourhood of radius % of the torus.At this point comes the new idea that we introduce in the present work: we addthe geometric part of the Nekhoroshev's theorem. This is justi�ed by the followingstandard argument. It is known that the number of resonances of order s increasesas sn, where n is the number of degrees of freedom; on the other hand, the size of aresonant region is of order exp(�s), being essentially controlled by the coe�cient ofthe resonant term. Thus, approaching the torus, resonances do accumulate geometri-cally with their order, but each of them controls a region of an exponentially smallvolume. One thus concludes that the total relative volume of the resonant regions issmall, which precludes overlapping. This fact, on the one hand, leaves enough spacefree from resonances to allow the existence of a set of invariant tori of large relativemeasure. On the other hand, the general conclusion of Nekhoroshev's theorem thata perturbation of order " causes a di�usion at most of order exp(�1=") applies heretoo. The superexponential estimate in this case follows from " ' exp(�1=%).We stress that the superexponential estimate cannot be directly obtained fromKolmogorov's normal form, without the Birkho� normalization of the second step.Heuristically, this can be justi�ed as follows. Nekhoroshev's theory, including the geo-metric part, is developed making always reference to the unperturbed action variables.The main contribution to the change in time of such quantities is due to the so calleddeformation, which in the case of Kolmogorov's normal form is estimated by somepower of %. On the other hand, when the distance % from the invariant torus decreasesto zero a deformation bounded only by a power of % would forbid a consistent construc-tion of the geography of resonances, as required by the geometric part of Nekhoroshev'stheorem. The good action variables introduced by Birkho�'s normalization procedureallow us to overcome this di�culty.The natural question now is: is our new result optimal? We believe that the answeris yes. Indeed, the recent paper by Chierchia and Gallavotti[6], although not applicableto our case in a straightforward manner, suggests that a resonance of multiplicity oneand size " gives macroscopic di�usion along the resonance line in a time proportionalto exp(1="). Since in the vicinity of a KAM torus the size of the strongest resonancesis exp(�1=%), one has to expect that di�usion, if any, actually requires a time of orderexp(exp(1=%)).We come now to the point (ii), namely the problem of the existence of thresholdsfor transition from order to chaos. Here, our argument is completely heuristic. Theclassical exponential bound, as given by local Nekhoroshev{like results, suggests thatdi�usion takes place in a rather smooth way, although quite rapidly, when the per-turbation parameter is increased. Conversely, the numerical simulations shows thatdi�usion takes place in a very sharp fashion, hardly compatible with the Nekhoro-shev's exponential. Here we understand that such a sharp change from order to chaoscorresponds to a transition from resonance non{overlapping to resonance overlapping,

Superexponential stability of KAM tori 9which completely changes the structure of orbits in phase space and activates di�erentmechanisms of chaotic di�usion, the di�usion speed passing from exponentially slowwith respect to resonance strength to directly proportional to resonance strength.Concerning the point (iii), we discuss on the one hand the application to the caseof an elliptic equilibrium, and, on the other hand, a possible extension to our previouswork [12].We �rst consider the case of an elliptic equilibrium point with the frequencies ofthe harmonic part of the Hamiltonian satisfying a diophantine condition (2). The mainremark is that such an equilibrium is nothing but a degenerate invariant torus. Thus,one can follow the same steps as above, just skipping the construction of Kolmogorov'snormal form. The application of Arnold's theorem then leads to the existence of a setof invariant tori, the relative volume of which tends to 1 as 1� exp(�1=%) in the limit%! 0, (% being in this case the distance from the equilibrium point); this improves alittle, from a quantitative viewpoint, the statement in [2], appendix 8. The applicationof Nekhoroshev's theorem is instead new: if the convexity condition (c) of the theoremis satis�ed, then one obtains a superexponential estimate of the di�usion time, insteadof the usual exponential one.We come now to our previous result in [12]. We considered there an Hamiltonianof the form (1), and proved that the di�usion speed can be made arbitrarily smallprovided one starts close enough to an invariant torus. However, we could not recovera superexponential estimate because we did not actually use in a complete fashion thepowerfulness of the geometric part of the Nekhoroshev's theorem. A straightforwardapplication of this idea leads to a quantitative reformulation of that result, statingthat for " su�ciently small there exists a domain, D" say, which contains open ballsof radius ", characterized by a di�usion speed of order exp�� exp(1=")�. The domainD" can be identi�ed with the nonresonant region of the geometric construction byNekhoroshev.Acknowledgements. We are grateful to D. Bambusi, G. Benettin, F. Fass�oand L. Galgani (in alphabetical order) for very useful discussions and suggestionsduring the preparation of this manuscript.References[1] V. I. Arnold: Proof of a theorem of A. N. kolmogorov on the invariance ofquasi{periodic motions under small perturbations of the Hamiltonian. Russ.Math. Surv., 18, 9 (1963).

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