The Steep Nekhoroshev s Theorem and optimal stability exponents

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1 The Steep Nekhoroshev s Theorem and optimal stability exponents M. Guzzo Dipartimento di Matematica Università degli Studi di Padova Via Trieste Padova, Italy L. Chierchia Dipartimento di Matematica e Fisica Università degli Studi Roma Tre Largo San L. Murialdo Roma, Italy G. Benettin Dipartimento di Matematica Università degli Studi di Padova Via Trieste Padova, Italy June, 204 Abstract Revising Nekhoroshev s geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev s theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be /2nα α n 2 α i s being Nekhoroshev s steepness indices and n 3 the number of degrees of freedom. On the base of a heuristic argument, we conecture that the new stability exponent is optimal. Introduction and results A. Motivations. In N.N. Nekhoroshev published a fundamental theorem [22, 23] about the exponential stability i.e., stability of action variables over times exponentially long with the inverse of the perturbation size of nearly integrable, real analytic Hamiltonian systems with Hamiltonian given, in standard action angle coordinates, by HI,ϕ = hi+εfi,ϕ, I,ϕ U T n, where: U R n is an open region, T n = R n /2πZ n is the standard flat n dimensional torus and ε is a small parameter. The integrable limit hi is assumed to satisfy a geometric condition, called by Nekhoroshev steepness the definition is recalled in 3 below. Under such assumptions, Nekhoroshev s states his theorem as follows : Let H in be real analytic with h steep. Then, there exist positive constants a, b and ε 0 such that for any 0 ε < ε 0 the solution I t,ϕ t of the standard Hamilton equations Compare [22, p. 4 and p. 8]; see also[22, p. 30] for a more detailed and precise statement.

2 for HI, ϕ satisfies for any time t satisfying I t I 0 ε b t ε exp ε a. Furthermore, a and b can be taken as follows: a = 2 2ζ +3n+4, b = 3a 2α n 2 where ζ = [ α α 2... α n 3 nα n 2 +n 2+n 3 ] , and α i are the steepness indices of h. Usually, a and b are called the stability exponents. Clearly, the most relevant quantity in this theorem is the stability exponent a appearing in the exponential, which gives the dominant time scale for the stability of the action variables. The exponential stability exponent a depends only on the number n of degrees of freedom and on the values of the firstn 2steepness indicesα i,i n 2.Notice that, forany fixedn,the best exponents a,b in 2 are obtained in the special case α =... = α n =, corresponding to convex or quasi convex hi which is the simplest instance of steep function. Actually, for any values of the steepness indices α i, the parameter ζ defined in 3 grows faster than 2 nn /2. The hypotheses of Nekhoroshev s theorem, as pointed out by Nekhoroshev himself, are qualitatively optimal, and, in particular, non steep Hamiltonians are in general non exponentially stable [22, ]. Furthermore, Nekhoroshev proved that steepness is a generic in C category property [2]. Finally, several interesting problems e.g., in Celestial Mechanics, compare below are steep but do not satisfy simpler assumptions such as quasi convexity. For all these reasons it seems natural and important to try to optimize the exponential stability exponents, especially with respect to the number n of the degrees of freedom which, in applications, typically range from n = 3 restricted three-body problems up to several tens planetary problems; this has been done, up to now, under simplifying assumptions but not in the general steep case. This paper is devoted to the general case. Before stating our result, let us briefly review the main extensions, applications and improvements concerning Nekhoroshev s theorem. Various extensions have been discussed, so as to cover the degeneracies of the Hamilton function which are usually met in some important mechanical systems fast rotations of the Euler Poinsot rigid body[, 2, 3]; the planetary N body problem[22, 24, 9]; restricted three body problems [8], elliptic equilibria [0, 25, 5, 32]. Furthermore, steepness could be used, in non-convex systems, to study the long term stability in problems such as the 2 For any fixed sequence, =,2,..., by considering a sequence of steep Hamiltonians h n with n degrees of freedom and steepness indices α,...,α n, the sequence of corresponding parameters ζ n := ζ satisfies ζ n ζ n n α α n 2, and the sequence of stability exponents a n := a satisfies a n a n 6n α α n 2. 2

3 Lagrangian equilibrium points L4-L5 of the restricted three body problem [4], asteroids of the Main Belt [20, 29, 6] and the Solar System [33, 30]. As far as improvements of the theoretical stability bounds i.e., improvements on the stability exponent a, quite complete results have been achieved in the special case of convex and quasi convex functions h: the proof of the theorem has been significantly simplified see [, 5, 6] and the stability exponent improved up to a = 2n, [9], [7], [3]; see [7] for exponents which are intermediate between a = 2n and a = 2n : such exponents in the convex case are nearly optimal, compare [35]. These improvements have been obtained by exploiting specific geometric properties of the convex and quasi convex cases, which allow to use conservation of energy in order to obtain topological confinement of the actions [6]. In fact, in the convex case, the analysis of the geometry of resonances, that is, the geometry of the manifolds {I U : k I = 0}, with I = hi and k Z n \{0}, is greatly simplified, since the frequency map I I is a diffeomorphism; on the other hand, in the general steep case, the Hamilton function cannot be used anymore in order to obtain topological confinement, and the geometry of resonances is significantly more complicate, due to possible folds and other degeneracies of the frequency map. Furthermore, while new different proofs of Nekhoroshev s theorem have appeared compare [27], which is based on the method of simultaneous Diophantine approximations introduced in [7], no improvements on the original Nekhoroshev s stability exponents, in the general steep case, are yet available 3, and no conecture about their optimal values has been discussed up to now 4. In this paper, we revisit and extend Nekhoroshev s geometric analysis obtaining, in particular, for 5 n 3, the new stability exponents a = /2np and b = a/α n with p being the product of the first n 2 steepness indices 6. The new stability exponents represent an essential improvement with respect to Eq. 2; in particular, the dependence of a on the number of degrees of freedom improves from quadratic to linear. It is also remarkable that, for α =... = α n = quasi convex case, we obtain the optimal stability exponents proved in [9, 7, 3], without using the local inversion of the frequency map, nor the Hamiltonian as a Lyapunov function. On the base of a simple heuristic argument, we conecture that the new stability exponent is optimal: the details will be given in point E below, here we only mention that the root /p comes in by a natural and necessary iterative dimensional argument related to a power law scaling of the amplitudes of the resonance domains and the tangential non degeneracies given by the steepness property, while the constant in front of p in the simplest and less degenerate case is the 2n coming from the quasi convex case. Putting these two things together one sees that the exponent /2np is, roughly speaking, necessary. A precise and fully quantitative formulation of our result is given in the following paragraph. 3 In the paper [26] there is a statement concerning improved values for the stability exponents, however, the proof appears to have a serious gap and such values are not ustified; see [28]. 4 The need for, at least, a conecture about the optimal value of the stability exponents in the general steep case was stressed also in [8], page 8. 5 The cases n 2 are, in general, totally stable and therefore are not included in our analysis. 6 This result has been announced in [2]. 3

4 B. Statement of the result.ac functionhiissaidtobesteepinu R n withsteepness indices α,...,α n and strictly positive steepness coefficients C,...,C n and r, if inf I U I > 0 and, for any I U, for any dimensional linear subspace Λ R n orthogonal to I with n, one has 7 max 0 η ξ min π ΛI +u C ξ ξ 0,r], 3 u Λ: u =η where π Λ denotes the orthogonal proection over Λ. To deal properly with initial data near the boundary, we will use the following notation: for any η > 0 and any D R n, we let D η := {I D : BI,η D}, where BI,η = {I R n : I I < η} is the real Euclidean ball centered in I of radius η and BI,η its closure. Theorem. Let H in be real analytic with h steep in U with steepness indices α,...,α n and let p := n 2 k= α k, a := 2np, b := a α n. 4 Then, there exist positive constants ε 0,R 0,T,c > 0 such that for any 0 ε < ε 0 the solution I t,ϕ t of the Hamilton equations for HI,ϕ with initial data I 0,ϕ 0 with I 0 U 2R 0 ε b satisfies I t I 0 R 0 ε b 5 for any time t satisfying: t T c exp ε ε a. 6 C. Quantitative formulation. Next, we provide explicit estimates for the parameters ε 0,R 0,T,c appearing in Theorem. To do this, we need to introduce some notations. Given extension parameters η,σ > 0 and any set D, we let the extended complex domains be defined by: D η = {I C n : I I η} and T n σ = {ϕ Cn /2πZ n : Imϕ i σ}. I D For any real action angle function ui,ϕ analytic in D η T n σ, with Fourier harmonics u k I, we denote its Fourier norm u η,σ = u k Dη e k σ, k Z n where. Dη denotes the sup norm in D η ; if it needs to be specified, we shall also use the heavier notation u D;η,σ. 7 For any vector u C n we denote by u := i ui 2 its hermitean norm and by u = i u. 4

5 Let H be real analytic in U T n with h steep in U with steepness indices α,...,α n and steepness coefficients C,...,C n and r. Without loss of generality, we can take the extension parameter in action space to be equal to the steepness coefficient r and we can find positive constants s,, and M such that: hi is real analytic on an open set which contains U r ; fi,ϕ is real analytic on an open set which contains U r T n s; For any I U, we have: and, for any I,I 2 U r, we have: Now, for n 2, let p := n 2 k= and define the parameters I I I 2 M I I 2. α k, q := np, β := +, 7 C Mr, 8 κ := M β E := max max 4Mκ 6 q n 2 α, 4 C 2 2 Then, in Theorem, one can take 2 ε := np 5 E 2np M f r,s 6 2 Mr b 8 2 ε 0 := ε min, n n c := ε a s 6, with µ 0 := max b, r 4nκ n b 2 2EC n. 9 a, s a, 6 R 0 := r n µ 0 ε b 24 2Mr, , κ n r 2 2EC n s T := 24 2M6E. 0 a ε f r,s α n D. On the proof. The proof of Nekhoroshev s theorem, in its various settings, can be split into: a geometric part, devoted to the analysis of distribution of small divisors in action space; an analytic part, devoted to the construction of normal forms; a stability argument yielding the confinement of the actions. 5

6 While the analytic part is obtained by adapting averaging methods to an analytic setting, the heart of Nekhoroshev s theorem resides in its geometric part. The geometric part of the steep case presented in [22, 23] still needed a deep revisitation, which is performed here and leads, in particular, to substantially improved stability exponents. The proof of Theorem will be obtained by deeply revisiting the geometric part of [22, 23]. The essential improvement are the following. First, we extend Pöschel s Geometric Lemmasee[3] to allow for a more general power law scaling of the amplitudes of the resonance domains. In this way, we allow for a definition of the resonance domains which depends on the Euclidean volume of a minimal cell of the lattice generating the resonance, and is compatible with steepness indices α i >. In contrast with the convex case, the analog of Pöschel s Geometric Lemma is here far to accomplish the geometric part of the theorem. In fact, motions with initial conditions characterized by a given resonance, may move along preferential planes of the action space, called fast drift planes. In particular, one needs to extend in the action space with fast drift planes the resonant domains obtained by a pull back from the frequency space: eventual degeneracies of the frequency map, which are typical of the steep non convex case, may produce topologically complicate sets. Nevertheless, a regularity of the distribution of these extended resonant sets must be proved: this is needed in order to grant the non overlapping of resonant domains of the same multiplicity. In [22, 23], the non overlapping is granted simply by construction of the resonance domains, but the price paid was an overestimate of resonant domains with the consequence of a strong n 2 scaling of the stability exponent 2. Here, we do not grant the overlapping by construction but, with a careful analysis of the topology of these sets, we obtain a better balance between optimal definition of resonant domains and their non overlapping. Finally, our geometric construction is fully compatible with the usual analytic part and stability argument, such as the so called resonance trap of [22, 23], and its improved version introduced in [5]. E. Heuristic discussion. The aim of this section is to provide some evidence that the result we obtained, for general steep Hamiltonians, is natural and cannot be substantially improved; in particular, the stability exponents a and b cannot be taken better. The reader is here assumed to have some familiarity with the traditional geometric construction entering Nekhoroshev theorem in the convex case. The essence of Nekhoroshev theorem is to take care of the resonant motions, namely motions taking place in a neighborhood of the resonant set k I = 0, k Z n, k K := Kǫ, with suitable Kǫ > 0. The reduction to a finite set of resonances, up to a cut off K, is possible thanks to the assumed analyticity of the perturbation: indeed the disregarded harmonics produce a drift of the actions with exponentially small speed of order exp σk; choosing eventually the cut off to be an inverse power of ǫ, precisely K ǫ a, the drift is negligible up to exponentially large times. Nonresonant motions are easily shown to be confined by constructing a nonresonant normal form in which all harmonics up to the cut off are averaged out. The challenge of Nekhoroshev theorem is to show that resonant motions are also confined, within the same exponentially large times, if steepness is assumed. 6

7 The nonresonant construction is possible in a nonresonant domain, namely a domain which proects, in the frequency space, to a set where all small divisors satisfy an inequality of the form k λ K ǫ k Z n \0 : k K. The product K ǫ at the r.h.s. needs perhaps an explanation. As is well known, by an elementary perturbation step no matter how it is performed a factor ǫ is gained in front of the perturbation, but a factor proportional to the square of the small divisor gets lost: so, ǫ is a kind of hard core of the resonant region, and the extra factor K in provides the effective gain of the step. Thanks to this extra factor, standard analytic techniques lead, with a number of steps proportional to K, to the wanted exponential estimates. Consider now a resonance of multiplicity, i.e. a resonance related to a lattice Λ generated by n independent vectors k,...,k, such that k i K for any i. The case of complete resonance, = n, corresponds to 0 and is not interesting. The exact resonance, in the frequency space, is the linear manifold of codimension satisfying k = 0 k Λ ; the corresponding set in the action space, in lack of invertibility of the frequency map, could be complicated: however in this discussion, oriented to show that our result cannot be substantially improved, we shall assume I is locally invertible and in the action space too the exact resonance is a manifold of codimension. The influence of the resonance is assumed to extend, still in the frequency space, up to a distance λ depending in an essential way on the multiplicity, larger for larger. Fine tuning requires that the size attributed to the resonant region also depends on the invidual lattice Λ, through the minimal volume of its cells, and not only on its dimension. This is important in the proof, but is not essential in this heuristic discussion, and will be disregarded. Understanding the correct dependence of λ on ǫ is a delicate matter. Suppose we are inside a resonance Λ, but far from other resonances. Then, standard resonant averaging shows that the motion, up to a negligible diffusion, is flattened on the so called plane of fast drift I + Λ, namely the plane parallel to Λ containing the initial datum I. Thespeed of fast drift is relatively large, namely of order ǫ. Unless during such motion new resonances are met, steepness directly provides confinement. Indeed, the essence of steepness is that, moving away from I along the plane of fast drift, or any other linear manifold, the frequency I must change, following essentially a power law in the distance from I. Consequently, if no new resonances enter the game, resonant motions, no matter how complicated, cannot extend over a long distance, remaining inside the resonant region. The maximal distance I the actions can travel inside a resonance, staying close to the fast drift plane, depends very much on the steepness indices. For convex Hamiltonians, the fast drift plane intersects transversely the resonance manifold, and I ; in the general steep case instead the fast drift plane may be tangent to the local manifold, and it turns out that I and are related by I /. 2 A fast motion of course might exit the resonant region by loosing a resonance relation: in such a case however it enters a less resonant region, and the argument repeats till, 7

8 I0 It Figure : Illustrating the possible mechanism of fast diffusion through crossing of fast drift planes, if resonances are not well separated. Dashed curves: the resonant manifolds. Gray neighbours: the corresponding resonant regions. Lines: the planes of fast drift. Irregular curve: a possible motion of actions from I0 to It, in a time of order /ǫ. possibly, the nonresonant region is reached, where fast motions cannot further extend this is the so called trapping argument introduced by Nekhoroshev; only in the convex case trapping can be replaced by energy conservation [6]. The actions can instead travel unboundedly large distances, if resonances of the same multiplicity are not well separated, and during the motion on the plane of fast drift I + Λ a new resonance is met. The possible scenario see figure is that the fast drift plane approaches a resonant manifold of larger multiplicity, there crosses some fast drift plane associated to a different resonance, and starts following it. The phenomenon could repeat: the new fast drift plane might cross, after approaching a convenient resonance of larger multiplicity, a third fast drift plane, and so on and so forth from resonance to resonance, till I gets large. In lack of steepness, a similar diffusion with speed ǫ inside the web of resonances is possible and, remarkably, is compatible with the presence, in the phase space, of a large set of KAM tori. A similar fast diffusion inside the web of resonances should not be confused with the Arnol d diffusion, which is a much slower quite different phenomenon. Such a possible scenario has been described already by Nekhoroshev, who provided examples of resonances acting as channels of superconductivity where large variations of the actions in a time /ǫ do occur [22, 23]. Numerical results, in turn, show the phenomenon of fast diffusion rather clearly [3, 34]: if the transversality between resonances and planes of fast drift is too weak, transitions from a resonance to a different resonance of the same multiplicity, through a resonance of higher multiplicity, are indeed observed, the higher multiplicity resonance behaving, so to speak, as a highways crossing. In such conditions, a sequence of similar transitions is numerically observed to produce a variation I of the actions of order one in a polynomial time scale. In conclusion: the above scenario is realistic, the problem of crossing of fast drift planes is real, resonances of the same multiplicity need to be well separated, as a necessary condition to prevent fast diffusion and assure stability of the actions over exponentially long times. So, let us now discuss which is the minimal separation between resonances, which is necessary in order to exclude the above crossing and prove long term stability of the actions. We shall reason as follows: we assume that, in the convex case, the Nekhoroshev Pöschel s geometric construction, leading to a = b = /2n, is optimal, and discuss 8

9 which is the unavoidable worsening one should expect in the general steep case. The expression 4 of the stability exponents a and b will naturally follow. The claim is that our values of a and b cannot be improved, unless the corresponding estimates in the convex case are also improved. The separation of resonances of the same multiplicity is obtained, both in the convex and in the general steep case, by appropriately tuning the sizes λ,...,λ n of the resonant regions of different multiplicity. In the convex case the appropriate scaling is λ + Kλ, λ n K, with K ǫ /2n. The latter relation assures indeed that the largest resonant zones, of multiplicity n, are still as small as ǫ b, with b = /2n; the factor K in the former one takes into account the fact that, by raising K, the angle between different resonant manifolds possibly gets small. In the general steep case the above scaling is replaced by λ + K A λ /, A, 3 still with λ n K. The exponent / of λ in 3 is unavoidable, for the possible tangencybetweenresonantmanifoldsandfast driftplanes.theexponenta isaquantity that for the moment we leave undetermined. Having established that, on the one side, λ must satisfy, while on the other side we need λ n K, the value of K cannot be too large: more precisely, from the above scaling law 3 it follows that the dependence of K on ǫ cannot be better than K ǫ a, a = 2Aα α n 2, with A = +A n 2 + A n 3 α n 2 + A n 4 α n 3 α n A α 2 α n 2 + α α n 2. 4 In the convex case α = = α n =, A = A n 2 =, the optimal value A = n is recovered. In the general steep case one should obviously conecture that such a value cannot improve, that is A n. Computing A, with rigorous estimates everywhere, is not a trivial task: remarkably, however, relevant compensations occur in the sum 4, and for all possible values of the steepness indices we are able to recover the value A = n, as in the convex case. Correspondingly, we get a = /2nα α n 2. Concerning b, its expression b = a/α n directly follows from λ n K the largest oscillation clearly occurs for = n, using 2 to relate I and. So, the values of a and b appear to be the optimal generalizations of the convex case. 8 F. The paper is organized as follows. The main part i.e., the geometric analysis is presented in 2: in 2. we introduce several auxiliary parameters needed to measure various covering sets, small divisors, cut-offs in Fourier space, time scales, etc. and 8 As pointed out, e.g., in [3] see [7] for a more refined discussion, in the convex case it is always possible to improve one exponent by worsening the other. The same argument repeats identical in the general steep case. 9

10 point out the relevant relations among them relations, which, although based on simple calculus, are proven, for completeness in Appendix B. In 2.2, merging and extending the geometric analysis of [22] and [3], we introduce a covering in action space formed by a suitable scale of resonant and non resonant regions. Section 2.3 is the heart of the paper, where the relevant analytic properties of the resonant and non resonant regions are proven; the section is divided into three lemmata: the first is about geometric estimates concerning resonant domains; the second deals with small divisor estimates and the third one is a non overlapping result for resonant regions corresponding to resonances of the same dimension. In 3 we recall briefly Pöschel s normal form theory [3] and show how it can be used in our setting. In the final section 4 we put all pieces together and prove Theorem with the constants listed in C above. In Appendix A we briefly review the notion of angles between linear spaces and, as mentioned above, Appendix B is an elementary check of the main relations among the auxiliary parameters. Acknowledgement. We are indebted with L. Niederman for kindly providing Ref [28] prior to publication. 2 Geometry of resonances 2. Auxiliary parameters In the proof of the Theorem several auxiliary parameters will occur; in this section we define such parameters and point out some compatibility relations expressed as in inequalities, which will be needed in the following we recall that ε,r 0,q are defined in 7 and 0. ε a K := ε 5 R := 2R 0 ε b 6 ρ := R 2n, 7 := q n := 0, q n :=, a n :=, a := q q + n 2. 9 Notice that the q s are strictly decreasing since a, indeed: a n 2 = nα n 2 + ; a = np + +, n

11 Let Λ be any maximal K lattice over Z n of dimension 9 n, Λ its volume, and set: Finally, we set λ := AK q, where A := 6E 2 r := κ λ C 22 δ Λ := λ Λ r 0 := λ 2MK T 0 := sr 0 e K s 6, 5ε f r,s 23 r Λ := δ Λ M 24 γ Λ := EK a δ Λ 25 R Λ := γ Λ 4MK, 26 TΛ := es 24 r Λ 27 e K s 6, T := min ε f T Λ 28 r,s Λ:dimΛ= T exp := min i=0,...,n T i. 29 It is then easy to check see Appendix B that under the assumption of Theorem, namely, 0 ε < ε 0, for any maximal K lattice of dimension n unless otherwise specified one has: A max max {,...,n } E a + 2 2a + 4, E a +2 a 30 Ks 6 3 ρ r Λ min 2,R Λ, 32 ρ δ Λ min r 4, 2r ρ r Λ,, n 2 33 δλ α KMκ C 4 γ Λ, n 2 34 R Λ r, 35 λ r 0 ε f r,s min 2 8 K, γ Λ r Λ 2 9 K, γ Λ R Λ K max r i ρ 37 0 i n 9 We recall that a maximal K lattice Λ is a lattice which admits a basis of vectors k Z n with k := n i= k i K, and it is not properly contained in any other lattice of the same dimension; the volume Λ of the lattice Λ is defined as the euclidean volume of the parallelepiped spanned by a basis for Λ; see [3]. Notice that for any K lattice of dimension, one has Λ K.

12 r 0 r 38 n r R 2 =0 2.2 Resonant and non resonant domains Fix I 0 U R and consider the set: 39 R r 2, 40 T exp T Ks exp. 4 ε 6 B := BI 0,R U. In order to prove the stability of all motions with initial actions I 0, we need to cover the domain B with open domains where suitable normal forms adapted to the local resonance properties may be constructed. We here introduce resonant zones and resonant blocks as in [3], but, since we do not require any local inversion for the frequency map I as it is typical of steepness [22, 23], see also [4], these domains are directly defined in the action space, without using any pull back from a frequencies space. Then, we define suitable extensions, in the spirit of the original construction of [22] see also [5]; see figure 2. We first define the resonant zones and blocks depending on the parameter K, representing a cut off for the resonance order, and also on the parameters 0 < λ <... < λ n < defined above. As in [3], we consider only the resonances defined by k I = 0 with k in some maximal K lattice Λ Z n. We define the resonant zone Z Λ := {I B : π Λ I < δ Λ }, 42 where Λ denotes the real vector space spanned by the lattice Λ, and the resonant block B Λ := Z Λ \Z +, = dimλ, 43 where: Z i := {Λ : dimλ =i}z Λ. We also define Z 0 := B and the non resonant block B 0 by B 0 := Z 0 \Z. We remark that, since I > δ Λ for any I B, the completely resonant zone Z Z n is empty and so is Z n. This implies B Λ = Z Λ, Λ s.t. dimλ = n. 44 Furthermore, if one defines B := {Λ :dimλ =}B Λ, 2

13 Z 0 B B 0 Z 0 B fast drift fast drift B 0 Convex, transversal B 0 fast drift Z 0 Steep, possibly tangent B 0 B Z 0 B Steep, not invertible Resonant manifolds are possibly tangent Figure 2: Sketch of the resonant sets: two resonances related to the lattices Λ and Λ of the same multiplicity are sketched, together with the resonance related to the lattice Λ Λ of multiplicity +, identified at the crossing of the two previous resonances, and some fast drift lines. Precisely, dashed curves represent the resonant manifolds of multiplicity ; dark gray areas denote resonant blocks of multiplicity ; light gray areas represent resonant zones of multiplicity +; by λ we denote some dimensional fast-drift lines. Different situations are illustrated: in the top-left panel we plot the situation which is typical of the quasi convex cases, with the fast drift planes which are transverse to the resonance; in the other panels we illustrate some situations which one may encounter in the more generic steep cases, where the fast drift planes may be tangent to the resonant manifolds, and also self-intersections bottom-left panel and tangencies of resonances bottom right panel are possible. In all the represented cases, the fast drift lines λ = I+ < Λ > Z Λ or λ = I+ < Λ > Z Λ through points I B Λ Λ do not intersect other resonant blocks of multiplicity +. one sees immediately that B = Z \Z +, so that, for any n, we have: B = B 0 B... B Z, 45 and, in particular, B = B 0 B... B n. 46 3

14 Next, following Nekhoroshev, we introduce discs D ρ Λ,η I := I I+ Λ BI,η Z Λ B ρ I Z Λ B ρ, 47 where I + Λ called by Nekhoroshev, fast drift plane denotes the plane through I parallel to Λ, C I denotes the connected component of a set C which contains I and η is any positive number less or equal than ρ. The extended resonant blocks are then defined by 0 : B ρ Λ,r Λ := D ρ Λ,r Λ I Z Λ B ρ, 48 I B Λ B ρ and the extended non resonant block by: B ρ 0 := B 0 B ρ. We remark that the set B ρ is not empty since ρ < R, and for any lattice Λ with dimλ = n, we have, by 44, 48 and footnote 0, B ρ Λ,r Λ = B Λ B ρ, dimλ = n Geometric properties of the resonant domains Geometric estimates for resonant domains For any maximal K lattice Λ, we need to estimate the diameter of the intersection of the fast drift planes I + Λ with the resonant zones: Lemma 2. For any I B Λ B ρ and I C ρ Λ,r Λ I we have: Proof We divide the proof of this lemma in three steps. Step. Let δ, ρ > 0 be such that I I κ δλ C r. 50 ρ δ min r, 2, 5 and define Z Λ δ = {I B : π Λ I < δ}. 52 Let us also denote by the linear space generated by I; by the linear space orthogonal to I and by Λ = π Λ the linear space obtained by proecting every vector u of Λ on. The first step will consist in proving that: 0 Notice that, if I B Λ, then I D ρ Λ,η I so that BΛ B ρ Bρ Λ,r Λ. 4

15 I For any I Z Λ δ B ρ and any I I + Λ Z Λ δ B ρ one has: Fix I 2 +Mr I I < 4 δ C. 53 I + Λ Z Λ δ B ρ I, with I I if I = I there is nothing to prove. Then, there exists a curve ut Λ such that u0 = 0, u = I I, and for any t, I. I +ut I + Λ Z Λ δ B ρ In particular, π Λ I +ut < δ. The proof of 53 will be based on the following claims i vii. i Λ is a vector space of dimension. Proof of i: Clearly, if u,...,u is a basis for Λ, then any vector in Λ can be represented as a linear combination of π u,..., π u Λ. We prove that the vectors π u,...,π u Λ are linearly independent, so that dimλ =. First, we remark that the only vector u of Λ satisfying: π u = 0 is u = 0. In fact, if there exists u 0 such that u Λ and π u = 0, then I Λ, and therefore we have: I = π Λ I < δ 2, which is not possible since for any I B we assumed I >. Now, let us consider c,...,c suchthat: i= c iπ u i = 0. Then,π i c iu i = 0, andtherefore i c iu i = 0. But, since the u i are linearly independent, it follows c,...,c = 0. ii For any u Λ, we have π Λ u = π u. Proof of ii: We first compute: π u = π Λ π u+π Λ π u. 54 Since π u Λ, we have π Λ π u = 0, so that 54 becomes: π u = π Λ π u. 55 But, π Λ u = π Λ π u+π u = π Λ π u+π Λ π u and since Λ, we have π Λ π u = 0, and therefore: π Λ u = π Λ π u. 56 From equations 55 and 56 we get ii. iii The angle 2 between Λ and Λ is equal to the angle between I and Λ, in formulae: Λ Λ = I Λ. 57 Proof of iii: By ii we have: Λ Λ = max u π Λ u = max u π u = u Λ,u 0 u Λ,u 0 Λ, and using x of Appendix A, we obtain Λ Λ = Λ = Λ. Notice that the set I + Λ Z Λ δ B ρ I is open in the relative topology of I + Λ and therefore is arc connected in I + Λ. 2 The notion of angle between linear spaces is briefly reviewed in Appendix A. 5

16 2 δ iv For any t, one has π Λ I +ut <. Proof of iv: We start with π Λ I +ut = I +ut 2 π Λ I +ut 2 = I +ut cosi +ut Λ 2 = I +ut sini +ut Λ sini +ut Λ and then we produce an upper bound estimate of the angle I +ut Λ. By using property ix of Appendix A, we first obtain: I +ut Λ I +ut Λ + Λ Λ. 58 Now, recalling that Λ and Λ have the same dimension claim i above, we see that by properties x and xi of Appendix A, Λ Λ = Λ Λ = Λ Λ = I Λ. From 58, we therefore obtain: Then, since: I +ut Λ I +ut Λ +I Λ. 59 sin I +ut Λ = π Λ I +ut I +ut sin I Λ = π Λ I I < δ < δ, 60 and δ/ / 2, both angles are strictly smaller than π/4, their sum is strictly smaller than π/2, and since sinx is monotone in [0, π/2], from 59 and standard trigonometry, we obtain: sini +ut Λ sini +ut Λ +I Λ sini +ut Λ + sini Λ < 2 δ. We therefore obtain: π Λ I +ut sini +ut Λ < v π ut < δ ut. Proof of v: Since ut Λ and I Z Λ δ, we have: 2 δ. vi I +π ut B. π ut = I ut I = π Λ I ut I < δ ut. 6

17 Proof of vi: Since I,I +ut B ρ, we have ut 2r and, using 40, we obtain ut r. Then, fromv and 5, we have: π ut < δ δ ut r ρ. Therefore, I +π ut B. vii ξ := π I I 0,r]. Proof of vii: Let us first assume ξ = 0, that is I I so that I I = I I I I. Since I I Λ and I I, this would imply also I Λ, and therefore: I = π Λ I < δ 2, which is not possible since for any I B we have I >. Therefore we have ξ > 0. Then, we have ξ = π I I I I = u r. Now, we are ready to complete the proof of 53. Since 0 < ξ r, let 0 η ξ the η which realizes the maximum in the definition of the steepness index of dimension, that is: min π Λ I +u > C ξ. 6 u Λ : u =η The curve π ut Λ oins I and I +π I I, and therefore [0,ξ] t [0,] π ut, so that there exists t [0,] such that π ut = η. From 6 it follows: π Λ I +π ut > C ξ. But using claims iv and v we also obtain: π Λ I +π ut π Λ I +ut +M π ut < 2 δ +M δ ut so that and therefore Using again v, we obtain: < 2 +Mr δ C ξ < 2 +Mr 2 +Mr π I I = ξ < δ, δ C. I I π I I + π I I 7

18 < 2 +Mr δ + δ C I I 2 +Mr δ + I I, 62 C 2 that is: I I < 2 +Mr 2 δ 2 +Mr < 4 C δ C. This finishes the proof of 53. Step 2. Next, we prove that: For any I Z Λ B ρ and any I D ρ Λ,r Λ I, we have: 2 +Mr I I δ Λ +Mr Λ α r Λ C Fix I D ρ Λ,r Λ I. Since D ρ Λ,r Λ I is open and connected, there exists a curve I +ut D ρ Λ,r Λ I, t [0,], such that I + r0 = I, I + u = I. Since D ρ Λ,r Λ I Z Λ, we have: π Λ I + ut < δ Λ for any t [0,], and also π Λ ut r Λ. In fact, since I +ut D ρ Λ,r Λ I Ĩ I+ Λ BĨ,r Λ, there exists a curve u t Λ such that ut u t r Λ, and therefore π Λ ut = π Λ ut u t ut u t r Λ. Then, we define u t := π Λ ut, so that u 0 = π Λ u0 = 0, I +u t I + Λ, and u t ut = π Λ ut ut = π Λ ut r Λ. Therefore, on the one hand we have I +u t B ρ+r Λ, on the other hand: π Λ I +u t π Λ I +ut + π Λ I +u t I +ut Therefore, for any t [0,], we have: I +u t We use, now, 53 step with π Λ I +ut + I +u t I +ut π Λ I +ut +M u t ut δ Λ +Mr Λ. I + Λ Z Λ δ Λ +Mr Λ B ρ r Λ I. δ := δ Λ +Mr Λ and ρ := ρ r Λ. In fact, I Z Λ Z Λ δ; I B ρ B ρ; from 33 it follows: δ min r ρ, 2. 8

19 Therefore, we have: 2 +Mr u t 4 for any t [0,]. In particular we have: δ 2 +Mr δ Λ +Mr Λ α = 4, C C 2 +Mr I I = u u u + u δ Λ +Mr Λ α r Λ +4. C Step 3. We may conclude the proof of the lemma. From 32 and 33 we obtain δ Λ +Mr Λ = 2δ Λ ρ 2r r ρ r Λ, so that applying 63 and using again 32, we have: 2 +Mr I I r Λ +4 δ Λ +Mr Λ δ Λ C M Mr 2 δ Λ C. Then, since and recall 33 δ Λ / <, we have δ Λ / δ Λ /, from which the first inequality in 50 follows at once; the second inequality follows from the fact that δ Λ λ and from the definition of r. Small divisor estimates Werecall[3]thataset B B isγ K nonresonantmoduloλifwehave k I γ for any k Z n \Λ such that k K;we will say that B B is γ non resonant if k I γ for any k Z n \{0} such that k K. The following result is a generalization of the Geometric Lemma in [3]. Lemma 2.2 i For any maximal K lattice Λ, the resonant block B Λ is γ Λ K non resonant modulo Λ, while the non resonant block B 0 is λ K non resonant. ii If = n, the extended block B ρ Λ,r Λ is γ Λ K non resonant modulo Λ; if n 2, the extended block B ρ Λ,r Λ is γ Λ /2 K non resonant modulo Λ. Proof of i: Let us first consider I B 0, so that I / Z. For any k Z n, with k K, let us denote by k the vector which generates the maximal one dimensional K lattice containing k. Since I / Z we have: π k I λ k λ k, and consequently k I = k π k I λ. Therefore, B 0 is λ K non resonant. Now, consider a maximal K lattice Λ, with := dimλ {,...,n } and let I B Λ. As in [3], let k / Λ with k K and denote by Λ + the maximal K lattice generated by Λ and k since Λ is maximal, dimλ + = +. For the purpose of this proof, let us denote π := π Λ, π := π Λ = Id π, π + := π Λ+, 9

20 where Id denotes the identity map. Since ππ + = π, it is easy to check that π k π + I πi +πk πi = k I. Thus, since the vectors π k and π + I πi = π π + I are proportional, and Λ + Λ π k, we obtain k I π k π + I πi πk πi = π k π + I πi πk πi Λ + π+ I Λ 2 πi 2 πk πi. Using πk k k K, πi < δ Λ, πi δ Λ+ we obtain: k I Λ + λ 2 + Λ Λ + 2 λ2 Λ 2 Kδ Λ. Using again Λ + Λ K, and K K a, we obtain: k I λ 2 + Λ K2 λ 2 Kλ λ+ 2 δ Λ K λ 2 K δ Λ AK 2a K 2 K δ Λ AK 2a K 2a K a = δ Λ K a A 2a, 64 so that, by 30, we finally get: k I δ Λ K a A 2a E a K a δ Λ = γ Λ. Proof of ii: If = n, the conclusion follows directly from lemma 2.2-i and 49. Let us therefore consider, for any =,...,n 2, I B ρ Λ,r Λ and I B Λ B ρ such that I C ρ Λ,r Λ I. By 33 and 63 we get 2 +Mr I I r Λ +4 δ Λ +Mr Λ. C Using also lemma 2.2-i, for any k Z n \Λ with k K, we have 2 +Mr k I k I KM I I γ Λ KM r Λ +4 But, since, by 32 and 25, KMr Λ γ Λ /4 from 34 there follows 2 +Mr δ Λ +Mr Λ α 4KM 2 +Mr δ Λ α 4KM 2 C C which, together with 65 yields k I γ Λ /2. Non overlapping of extended blocks and zones 20 δ Λ +Mr Λ. 65 C KMκ δλ C γ Λ 4, 66

21 Lemma 2.3 For any maximal K lattices Λ Λ of the same dimension =,...,n, we have B ρ Λ,r Λ Z Λ =. Proof. Let Λ Λ be maximal K lattices of the same dimension n and consider I B ρ Λ,r Λ : we have to prove that I / Z Λ, i.e., π Λ I δ Λ. 67 We divide the proof in two steps: the case n 2 and the case = n. Step. n 2. The argument follows from the following claims i vi. i For any η > 0, there exists I B Λ B ρ such that I I κ δλ C +η. Proof of i: SinceI B ρ Λ,r Λ, there exists I B ρ Λ,r Λ such that I I < η; by definition of B ρ Λ,r Λ there exists I B Λ B ρ such that I D ρ Λ,r Λ I. Then, i immediately follows from 50. ii π Λ I E a K a δ Λ. Proof of ii: Since Λ Λ, there exists k Λ such that k / Λ and k K. Therefore we have π Λ I k I / k and since I B Λ, ii follows from Lemma 2.2. iii π Λ I 2 Ea K a δ Λ. Proof of iii: Choose η γ Λ 4KM. Then, by using 34, i and ii, we obtain π Λ I π Λ I M I I E a K a δ Λ Mη Mκ δλ C E a K a δ Λ γ Λ 2K = 2 Ea K a δ Λ. Now, observe that, if we have 2 Ea K a δ Λ δ Λ, then 67 follows at once. Therefore, let us henceforth assume that 2 Ea K a δ Λ < δ Λ i.e. iv π Λ I A a K a δ Λ 2δ Λ. Λ Λ < 2 E a K a. 68 Proof of iv: Since Λ Λ, we consider k Λ such that k / Λ and k K, and we denote by Λ the maximal K lattice of dimension + which contains Λ and k. For the purpose of the proof of iv let us denote: First, since I B Λ, we have π := π Λ, π := π Λ, π := π Λ. π I π Id πi π πi π Id πi δ Λ. 69 Then, since I B Λ, I / Z Λ and we have π I δ Λ. 70 2

22 Let us consider the vector ν = π Λ k. We remark that ν Λ \0. In fact, on the one hand k / Λ, so that ν 0; on the other hand, since ν = k π Λ k is the sum of k Λ and of π k Λ, we have also ν Λ. Therefore, since ν is orthogonal to Λ, we have: π Λ = π Λ +π ν. 7 Moreover, we have: In fact, on the one hand we have ν k ν Λ Λ. 72 ν k ν = π Λ k k π Λ k = π Λ k, on the other hand we have From 7, we obtain: π Λ k Λ Λ. π Id πi 2 = π π Id πi 2 = π Id πi 2 π ν π Id πi 2 = π Id πi 2 ν π Id πi 2 ν We notice that: In fact, first we have π Id πi π Id πi π I π πi δ Λ δ Λ λ Λ A a K a Λ, Λ then, using 72, 68, we obtain Λ Λ = Λ Λ Λ Λ k Λ Λ < 2 k E a K a 2K E a K a, and therefore we have: π Id πi > λ Λ A a K a Finally, since K, a, and using also 30, we have π Id πi > λ Λ A a K 2K E a λ K Λ 2+ 2 E a > 0. 2K E a K a. = λ K Λ A a 2 E a 22

23 Therefore, from 73, 74, we have: π Id πi = π Id πi ν π Id πi 2 ν 2 π Id πi 2, and, since we obtain: π Id πi k = Id πi k = 0, π Id πi π Id πi min u k, u = ν u2 ν We remark that the maximum of ν u = π k ν u, for u k and u =, is obtained for u parallel to π k ν, that is for u = π k ν/ π k ν. Therefore, we have: and correspondingly: min u k, u = ν u2 ν 2 = max ν u = π k u k, u = ν = π k ν 2 ν 2 = ν k ν k. Therefore, from 75 and 72 we obtain ν 2 π k ν 2 ν 2 = π k ν ν π Id πi π Id πi ν k ν k π Id πi Λ Λ k. Then, since: π Id πi π I π πi δ Λ δ Λ, we obtain: π Id πi Λ δ Λ δ Λ Λ k λ + Λ k δ Λ Λ Λ k Aa K a δ Λ δ Λ, 76 and using 69 we obtain iv. We now are ready to finish the proof of 67 in the case n 2. From inequalities iv and i, we obtain: π I A a K a δλ α δ Λ 2δ Λ M κ +η. 77 C Using 34 and choosing η γ Λ 4KM, we obtain: π I A a K a δ Λ 2δ Λ 2 Ea K a δ Λ. 78 Since we are assuming 68 and since K a, we obtain π I A a K a δ Λ 2δ Λ 2 Ea K a δ Λ > A a K a 4 δ Λ E a K a δ Λ δ Λ > A a 4 E a δ Λ, 23

24 which, by 30, yields 67. Step 2. We now consider maximal K lattices Λ Λ of the same dimension = n. Since B ρ Λ,r Λ = B Λ B ρ, we have I B Λ and π I E a K a δ Λ. 79 In fact, since Λ Λ, there exists k Λ such that k / Λ and k K. Therefore we have π I k I / k and since I B Λ, by lemma 2.2 we have π I k I k E a K a δ Λ. We also have: First, since I B Λ, we have π I A a K a δ Λ 2δ Λ. 80 π I π Id πi π πi π Id πi δ Λ. 8 Then, since Λ Λ, we consider k Λ such that k / Λ and k K. In particular, since I B Λ, we have I. 82 Let us consider the vector ν = π Λ k. Since ν is orthogonal to Λ, we have: Moreover, we have: Id = π +π ν. 83 ν k π Λ k = ν Λ. 84 In fact, since the K lattice Λ,k is generated by Λ and k is properly contained in Z n, we have Λ π Λ k Λ,k. From 83, we obtain: πid πi 2 = Id πi 2 π ν Id πi 2, and since: and: we have: π ν Id πi = ν Id πi, ν Id πi δ Λ λ n > 0, π Id πi = Id πi ν Id πi2 ν 2 Id πi 2. 24

25 Then, since Id πi k = 0, we have π Id πi Id πi min u k, u = ν u2 ν We remark that the maximum of ν u = π k ν u, for u k and u =, is obtained for u parallel to π k ν, that is for u = π k ν/ π k ν. Therefore, we have: max ν u = π k u k, u = ν and correspondingly: min u k, u = ν u2 ν 2 = = π k ν 2 ν 2 = ν k ν k. ν 2 π k ν 2 ν 2 = π k ν ν Therefore, from 85, we obtain: and from 84 we obtain also: π ν k Id πi Id πi ν k, π Id πi Id πi Λ k. Then, since: Id πi I πi δ Λ, we obtain: π Id πi δ Λ Λ k Aa K a δ Λ δ Λ, 86 and using 8 we obtain 80. If E a K a δ Λ 2δ Λ, using 79, there is nothing more to prove. Therefore, we assume: Then, using 80, we obtain: E a K a δ Λ < 2δ Λ. π I A a K a δ Λ δ Λ > A a K a δ Λ Since K a, we have: and using 30 we obtain 67. π I > A a 2 E a δ Λ, 25 2 E a K a δ Λ. 87

26 3 Normal forms and dynamics in resonant blocks The geometric construction of 2 together with normal form theory allows to have some control of the dynamics in the extended blocks. We shall use normal form theory in the version given by Pöschel in [3]; see, in particular, the Normal Form Lemma at p. 92 of [3] which we shall use with parameters p = q = 2; notice that the constant M used in [3] is an upper bound on the derivative of I, which is used only as Lipschitz constant, so that our notation is consistent with that used in [3]. In fact, the following lemma holds Lemma 3. i Let I t,ϕ t be the solution of the Hamilton equations with initial condition 3 Ī0,ϕ 0 B ρ 0 Tn. Then, I t Ī0 r 0 88 for all times 4 t T 0. ii Let Λ be a maximal K lattice of dimension {,...,n }, and let I t,ϕ t be the solution of the Hamilton equations with initial data Ī0,ϕ 0 B Λ B +ρ T n. Let τ e be the possibly infinite exit time from 5 B ρ Λ,r Λ. Then, if τ e T Λ, we have I t Ī0 r for any time t < T Λ ; otherwise, there exists 0 i such that I τe B i B ρ. Proof of i: The non resonant block B 0 is λ non resonant see Lemma 2.2. Let us consider as extension vector r 0,s. Because of the definition of r 0, 38, 3 and the first inequality in 36, we can apply the normal form lemma in [3] in B 0. It then follows at once 88 for all times t T 0 with T 0 as in 28. Proof of ii: Let us first assume that τ e T Λ and consider the extension vector r Λ,s. By Lemma 2.2 ii, the domain B ρ Λ,r Λ is γ Λ /2 K non resonant modulo Λ. Thus, since r Λ r by 35, and because of 3, the definition of R Λ and the third inequality in 36, we can apply the Normal Form Lemma in [3] with p = q = 2, in B ρ Λ,r Λ. Thus, there exists a canonical transformation: conugating H to its resonant normal form: φ : B ρ Λ,r Λ r Λ2 T n s 6 B ρ Λ,r Λ rλ T n s I,ϕ I,ϕ = φi,ϕ 89 H Λ = H φ = h+εg +εf 90 with g a real analytic functions having the Fourier expansion g = k Λg k expik ϕ, 9 3 I.e. I t t=0 = Ī0: we are using here a slight abuse of notation in order not to confuse the point I0 in the statement of Theorem with the arbitrary point Ī0 used here. 4 Recall the definition of T 0 in I.e., τ e is such that I t B ρ Λ,r Λ for t < τ e and I τe / B ρ Λ,r Λ. 26

27 and the remainder f satisfying the exponential bound: f B ρ Λ,r Λ ; r Λ 2, s 6 e K s 6 f r,s. 92 Also, for any I,ϕ B ρ Λ,r Λ r Λ2 T n, by the third inequality in 36, one has: I I 8K γ Λ ε f r,s 2 6r Λ, so that φ B ρ Λ,r Λ T n B ρ Λ,r Λ r Λ 2 6 Tn. Finally, using the second inequality in 36 we have also: I I 8K γ Λ ε f r,s 2 6r Λ. Therefore, since I t B ρ Λ,r Λ for any t < τ e, we may define I t,ϕ t = φ I t,ϕ t, and using the specific form of Hamiltonian 90, we have π Λ I t I 0 ε t 0 f ϕ I t,ϕ t dt ε t sup B ρ Λ,r Λ r Λ 2 6 T n By Cauchy estimate see Lemma B.3 of [3] and by 92, we have: sup B ρ Λ,r Λ r Λ 2 6 T n so that, for any t < T Λ, we have: f ϕ 6 es f B ρ Λ,r Λ ; r Λ 2, s 6 6 es e K s 6 f r,s, π Λ I t 6ε I 0 es t e K 6 f s r,s 4 r Λ. As a consequence, the motion I t has the representation: f ϕ. I t = Ī0 +vt+dt 93 with vt Λ with v0 = 0, and dt < 3 4 r Λ: indeed, we can write I t = Ī0 +I t I t +I t I 0 +I 0 Ī0, and take vt = π Λ I t I 0 and dt = I t I t+π Λ I t I 0 +I 0 Ī0. Therefore, I t B ρ Λ,r Λ Z Λ B ρ and because of the representation 93 the distance between I t and the space Ī0 + Λ is smaller than 3 4 r Λ. Furthermore, I t is connected to Ī 0 in the set I Ī0+ Λ BI, 3 4 r Λ Z Λ B ρ so that I t C ρ C Λ, 3 4 ΛĪ0 ρ r Λ,r Λ Ī0. Thus, by Lemma 2. we have I t Ī0 r for any t < T Λ, as claimed. Let us now assume that the exit time τ e satisfies: 0 < τ e < T Λ. Since for any time t < τ e, we have I t C ρ Ī0, we have also: I Λ, 3 4 r τe C ρ As a consequence again Λ Λ, 3 4 ΛĪ0. r Lemma 2., we have I t Ī0 r < ρ for any t τ e and since Ī0 B +ρ, we also have I τe B ρ

28 Since I t C ρ Λ, 3 4 r ΛĪ0, the distance between I t and Ī0 + Λ is strictly smaller than 3 4 r Λ, and the distance between I τe and Ī0 + Λ is smaller or equal than 3 4 r Λ. Finally, since I t C ρ Λ, 3 4 r ΛĪ0, we have I t Z Λ, that is: π Λ I t < δ Λ. As a consequence, since I τe / B ρ Λ,r Λ, the only possibility is: π Λ I τe = δ Λ. But this means that I τe / Z Λ. On the other hand, by Lemma 2.3, I τe cannot belong to any Z Λ for any maximal K lattice Λ Λ of the same dimension ; therefore I τe / Z, whence, by 45, there must exist an i 0,..., such that I τe B i, which, together with 94, concludes the proof of the lemma. 4 The resonance trap argument and conclusion of the proof We are now in position to conclude the proof of the theorem, proving 5 and 6. In view of 46, the are are two alternatives 6 : a either I 0 B 0 B nρ; b ori 0 B Λ B +ρforsomemaximalk lattice ofdimension {,...,n }. In case a, by Lemma 3. i, by 37, the definition of T 0 and T exp 28, 29 and 4, the theorem is proved. In case b, by Lemma 3. ii, there are two alternatives: b either I t I 0 r 37 ρ, for t T exp T Λ b2 or there exist a t such that I t I 0 ρ for all t t and I t B i B ρ for some i {0,..., }. In case b, by 28, 29 and 4, and recalling that ρ = R 0 ε b /n, the theorem is proved. In case b2 we iterate the above scheme. Hence, after 0 k n steps we see that the action traectory I t ends up either in a trapping resonant region B Λ B i where it gets stuck for exponentially long times or it will end up in B 0 where also gets stuck for exponentially long time. Since in such k steps I t moves at most by kρ we see that in the possible fast drift we have I t I 0 kρ n ρ to which we have to add the displacement in the trapping region which is again at most ρ. Thus, for times t T exp we have I t I 0 nρ = R 0 ε b as claimed. We remark that, in the case b above, I t may visit several blocks in the time T exp ; let us denote by their minimal multiplicity, and t < T exp be such that I t B Λ B + ρ with dimλ =. Then, we have I 0 B Λ and, therefore, I t B Λ for all t T exp. In fact, since the geometry of resonances of the Hamilton function H is identical to the geometry of resonances of H, if we consider the solution I t,ϕ t of the Hamilton equations of H with I 0 = I t, and apply Lemma 3., we obtain that I 0 = I t B Λ B Λ. 6 Recall that I 0 is the center of B U, which is a sphere of radius r = 2R 0ε b = 2nρ. 28

29 A Angles between linear spaces In this appendix n 2, u,v,w,z... denote vectors in R n and L,L 2,L,... linear vector subspaces of R n of dimension m {,...,n }; π L denotes the orthogonal proection onto the linear space L and Arccos : [,] [0,π] denotes the principal branch of the inverse real cosine. Definition A. Let. The angle between u and v is defined as { u v Arccos, if u,v 0 u v = u v π 2. otherwise. Definition A.2 The angle between L and L 2 is defined as L L 2 := max u L \{0} u π L 2 u. We, next, list a few elementary properties of angles between linear spaces, whose simple proof is left to the reader for the proof of items x and xi, see, also, [Nekhoroshev79, p. 45]. i. u v [0,π] and u v [0,π/2] if and only if u v 0; L L 2 [0,π/2]. ii. L L 2 = π 2 if and only if7 L L 2 {0}. L L 2 < π 2 if and only if {u L : π L2 u = 0} = {0}. iii. u π L u = Arccos π L u u, u 0. iv. u π L u+u π L u = π 2, u 0. v. u π L u = min v L\{0} u v. vi. cosl L 2 = min max u L \{0} v L 2\{0} u v u v. vii. For any u and v one has 8 u v = v u. viii. If u 0 v, u v coincides with the Euclidean length of the shortest geodesic equivalently, shortest curve on the unit sphere S n := {ξ R n : ξ = } having as end points the proections of u and v on S n. ix. u v u w+w v. Also: L L 2 L L 3 +L 3 L 2. x. L L 2 = L 2 L. xi. If diml = diml 2, then L L 2 = L 2 L. B Parameter Relations For completeness, in this appendix, we prove the elementary inequalities Recall the definitions of the parameters given in First, we observe that from these definitions and the hypothesis 0 ε ε 0, it follows easily: E 4 ; A := 6E 24, K := ε ε δ Λ λ ; ρ = rµ 0 K /αn ; r = 2nρ ; a ε ε 0 a ; Λ K ; q + = 2a n ; q 2 n 2 ; q = a B. B.2. B.3 7 L := {u R n : u v = 0, v L}. 8 But, in general, L L 2 L 2 L : for example, if n = 3, L = {0,t,t : t R} and L 2 = {x 3 = 0}, then L L 2 = π/4, while L 2 L = π/2. 29