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A C Nekhoroshev theorem

  • We prove a C version of the Nekhoroshev's estimate on the stability times of the actions in close to integrable Hamiltonian systems. The proof we give is a variant of the original Nekhoroshev's proof and it consists in first conjugating, globally in the phase space, and up to a small remainder, the system to a normal form. Then we perform the geometric part of the proof in the normalized variables. As a result, we obtain a proof which is simpler than the usual ones.

    Citation: Dario Bambusi, Beatrice Langella. A C Nekhoroshev theorem[J]. Mathematics in Engineering, 2021, 3(2): 1-17. doi: 10.3934/mine.2021019

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  • We prove a C version of the Nekhoroshev's estimate on the stability times of the actions in close to integrable Hamiltonian systems. The proof we give is a variant of the original Nekhoroshev's proof and it consists in first conjugating, globally in the phase space, and up to a small remainder, the system to a normal form. Then we perform the geometric part of the proof in the normalized variables. As a result, we obtain a proof which is simpler than the usual ones.


    In this paper we prove a C version of Nekhoroshev's Theorem for the stability times in a close to integrable Hamiltonian system. The {main point is that our proof relies on an analytic and a geometrical construction which, although on the same line as the original Nekhoroshev's ones, are much simpler. We also obtain} an intermediate result that we think could have some interest in itself (see Theorem 1.2).

    To be definite and in order to avoid as much as possible technical complications, we study here a system of the form

    H(p,q)=H0(p)+εV(p,q),H0(p)=dj=1p2j2, (1.1)

    with VC(Td×Rd). However, our technique is applicable also to the general case of perturbations of steep integrable systems, and with a perturbation which is not globally bounded in the momenta p.

    The result we get is the following version of Nekhoroshev's Theorem.

    Theorem 1.1. Assume that VC(Td×Rd) is globally bounded and fix a positive b<12. Then, for any positive M, there exists CM,εM such that, if 0<ε<εM then, corresponding to any initial datum, one has

    |p(t)p(0)|CMεb , (1.2)
    |t|CMεM . (1.3)

    We recall that in the analytic (or Gevrey) case the time of stability, cf. Eq (1.3), is known to be exponentially long.

    Theorem 1.1 is not new, for example it is a direct consequence of Theorem 2.1 of [9]*. However, as far as we know our proof is new and the value of the exponent b that we get is better than those present in literature.

    *Actually such a theorem is also stated in [18], where reference is made to a slightly different statement present in [17]

    In the analytic case any b<1/2 is allowed at the price of worsening the estimate on the times, see for example [23]

    Essentially two methods of proof of Nekhoroshev's theorem are known: the original one [3,5,12,14,19,20,23], and Lochak's one [15,16] (see also [17,18] and [4,1] for infinite dimensional generalizations). We emphasize that Lochak's proof is much simpler than Nekhoroshev's one, but, at least in its original form, applies only to perturbations of quasiconvex integrable systems. The paper [9] is a generalization of Lochak's proof to the case of finite smoothness. Lochak's method was also extended to the steep case by using also ideas from the original proof by Nekhoroshev [21], see also [8,10,22].

    Our proof is a variant of Nekhoroshev's original one which consists of two steps: the analytic part and the geometric part. Classically, in the analytic part one shows that in a region of the phase space where only some resonances are present one can conjugate, up to a small remainder, the system to a system in resonant normal form. In the geometrical part one collects all the information and shows that, if the regions are suitably constructed, then for any initial datum there exists a region in which it remains for the considered times, and this leads to Nekhoroshev's estimate.

    In the classical approach the analysis of the geometrical part is slightly complicated by the fact that it has to be performed in the original coordinates, so that one has to take into account the effects of the coordinate transformation used to conjugate the system to normal form. The novelty of the present paper is that we use a canonical transformation which is globally defined and globally conjugates the system to a normal form which is different in each region of the phase space, depending on the resonances which are present in each region (a similar technique has been used for the first time in a probabilistic context in [7,11]). This is obtained by splitting each Fourier coefficient of V, namely ˆVk(p), into a part localized in the region |ω(p)k|<εδ (with a suitable δ) and a part localized in the nonresonant region. The part localized in the nonresonant region is then removed through the normalizing canonical transformation. Technically the localization is obtained simply by multiplying by a smooth cutoff function.

    Then the geometrical part consists in making a decomposition of the phase space in regions which are invariant for the dynamics of the normalized system. This leads in particular to the conclusion that, in the dynamics of the normalized system, estimate (1.2) is valid for all times. This is the content of the following theorem, which, as far as we know, is new.

    Theorem 1.2. Fix a positive b<12, then, for any positive M, there exists CM,εM such that, if 0<ε<εM, then there exists a canonical transformation (p,q)=T(˜p,˜q) and a (normal form) Hamiltonian HZ, with the following properties

    1). |p˜p|CMεb

    2). HTHZC2(Rd×Td)CMεM,

    3). Along the solutions of the Hamilton equations of HZ, one has

    |˜p(t)˜p(0)|CMεb ,tR .

    When adding the remainder, one gets the limitation (1.3) on the times.

    Finally we remark that Lochak's proof applies to system (2.5), but we think that our approach to the geometric part of the proof is the main interest of the present paper, since it is suitable for generalizations to the steep case.

    This paper originates from our research on the spectrum of Sturm Liouville operators in general tori [6], which lead to a quantum version of Nekhoroshev's theorem. When we were still lost on how to construct a quantum analogue of the geometric part, we had several very enlightening discussions with Antonio Giorgilli on the classical Nekhoroshev's theorem. At the end we realized that the quantum method we constructed had a classical counterpart which is the content of the present paper. It is a pleasure to dedicate this paper to Antonio Giorgilli in the occasion of his 70th birthday.

    One of us, Dario Bambusi, would like to thank Antonio who introduced him to science and in particular to the study of Hamiltonian dynamics: His presence has always been fundamental and I would be a different person if I had not met him. Thank you!

    In this subsection we present the tools we will use in order to deal with the C context.

    Having fixed a parameter 0<δ<12 and an interval U=[0,ε0) with some positive ε0, we give the following definition.

    Definition 2.1. A family of functions {fε}εU, fεC(Rd×Td) will be said to be a symbol of order m if for all α,βNd there exists a positive constant Cα,β such that

    supεUsuppRd,qTd|αpβqfε(p,q)εmε|α|δ|Cα,β. (2.1)

    In this case we will denote fεPm,δ. We will often omit the index ε.

    Remark 2.2. The constants Cα,β encode the smoothness properties of f. This is particularly clear for ε independent functions. Indeed in this case the function is of class Gevrey s if and only if there exist constants R,C s.t.

    Cα,βC(α!β!)sR|α|+|β| .

    We think that keeping track of the dependence of Cα,β on α and β should allow to obtain exponentially long stability times for the Gevrey case. Similar ideas have been developed in [2].

    Remark 2.3. It is immediate to see that fPm,δ if and only if for all integers N1 and N2 there exists a positive constant CmN1,N2 such that

    supεUsuppRd, kZd,αNd, |α|=N1|αpˆfk(ε,p)||k|N2ε(m|α|δ)CmN1,N2, (2.2)

    where

    ˆfk(ε,p)=1(2π)dTdfε(p,q)eikq,εU,pRd

    are the Fourier coefficients of f.

    Remark 2.4. The space Pm,δ endowed by the family of seminorms given by the constants CmN1,N2=CmN1,N2(f) of equation (2.2) is a Fréchet space.

    Remark 2.5. A direct computation shows that, if fPm1,δ and gPm2,δ, then

    1). f+gPmin{m1,m2},δ

    2). fgPm1+m2,δ

    3). the Poisson bracket, {f,g}Pm1+m2δ,δ.

    In the following, given a C function g, we will denote by Xg its Hamiltonian vector field and by Φtg the flow it generates (which in our framework will always be globally defined).

    In order to state the analytic Lemma, we start by defining what we mean by normal form of order N. From now of we fix the number N controlling the number of steps in the normal form procedure.

    Furthermore, we will denote

    a:=12δ ;

    we fix a positive (small) 0<β<1 and we define

    K=K(ε):=[1εβ]+1 (2.3)

    with the square bracket denoting the integer part. Eventually we will link β, δ, b, M and N.

    Definition 2.6. A family of functions Zε:Rd×TdR will be said to be in normal form if

    Zε(p,q)=|k|KˆZk(ε,p)eikq

    with

    ˆZk(ε,p)0|pk|εδ,kZd{0} , (2.4)

    Namely the k-th Fourier coefficient is supported in the resonant region |pk|εδ.

    Lemma 2.7. (Normal Form Lemma) Consider the system

    Hε:=H0(p)+Pε(p,q) (2.5)

    with H0 as in (1.1) and PεP1,δ, then there exists a canonical transformation T such that

    HεT=H0+Nj=1Zj+R(N) , (2.6)

    with ZjP1+a(j1),δ in normal form and R(N) s.t.

    supεUsuppRd,qTd|αpβqR(N)(p,q)ε1+Na|ε|α|δCα,β, (2.7)
    α,βNd×Ndwith |α|+|β|1 . (2.8)

    Furthermore, given a symbol fPm,δ, define Rf:=fTf, then one has

    suppRd,qTd|Rf(ε,p,q)|Cεm+12δ . (2.9)

    In the case f=pj, j=1,...,d, one has

    suppRd,qTd|Rpj(ε,p,q)|Cε1δ . (2.10)

    Definition 2.8. A function fulfilling Eqs (2.7) and (2.8) will be said to be a remainder of order N, or simply a remainder.

    The proof of Lemma 2.7 consists of a few steps: first we give a decomposition of an arbitrary symbol in a normal form part, a nonresonant part and a remainder, then we remove the nonresonant part of the perturbation and then we iterate. The canonical transformation used to remove the nonresonant part will be constructed using the Lie transform method, namely by using the time one flow of an auxiliary Hamiltonian. This requires the study of the Lie transform in our C context. We will also have to solve the cohomological equation in order to construct the auxiliary Hamiltonian. Finally we state and prove the iterative Lemma which is the last step of the proof of the Normal Form Lemma.

    Let us consider an even C function η:RR+ such that η(t)1 if |t|12 and η(t)0 if |t|1. For all kZd such that 0|k|K, we define the cut-off function

    χk(p)=η(pkεδ),pRd, (2.11)

    which is thus supported in |pk|εδ. Consider a smooth family of functions fεPm,δ, fε(p,q)=kZdˆfk(ε,p)eikq, we perform for fε the following decomposition:

    f(p,q)=f(res)(p,q)+f(nr)(p,q)+fK(p,q), (2.12)

    where

    f(res)(p,q)=0<|k|Kˆfk(ε,p)χk(p)eikq+ˆf0(ε,p),f(nr)(p,q)=0<|k|Kˆfk(ε,p)(1χk(p))eikq,fK(p,q)=|k|>Kˆfk(ε,p)eikq. (2.13)

    Remark 2.9. If fPm,δ then f(res),f(nr)Pm,δ. Furthermore f(res) is in normal form.

    Remark 2.10. Since compactly supported analytic functions do not exist, this step would be impossible in an analytic context. On the contrary a Gevrey cutoff would be possible, so in principle this method is suitable also to deal with the Gevrey case.

    Lemma 2.11. Let fP1,δ, then fKP1+Na,δ, so in particular it is a remainder in the sense that it fulfills Eqs (2.7) and (2.8).

    Proof. This is related to the fact that the Fourier coefficients of a C function decrease faster than any power of |k|1. Formally we have to bound the following seminorms

    C1+NaN1,N2(fK)=supεsupp,|k|>K,|α|=N1|αpˆfk(ε,p)|k|N2ε|α|δε1+Na|supεsupp,|k|>K,|α|=N1|αpˆfk(ε,p)|k|N2+N3ε|α|δε1+NaKN3| (2.14)

    and, choosing N3>Na/β, one has KN3εNa>1 and thus

    |(2.14)|supεsupp,|k|>K,|α|=N1|αpˆfk(ε,p)|k|N2+N3ε|α|δε|=C1N1,N2+N3(f)

    which is the thesis.

    Definition 2.12. Given a function gPm,δ, with m0, the time one flow Φ1gΦtg|t=1 will be called Lie transform generated by g.

    Given a function fPm1,δ, we study fΦ1g. To this end define the sequence f(l), by

    f(0):=f ,f(l):={f(l1);g}dldtl|t=0fΦtg ,l1 , (2.15)

    and remark that f(l)Pm1+l(mδ),δ if gPm,δ. We have the following lemma.

    Lemma 2.13. Let gPm2,δ and fPm1,δ, with m21δ and m11 then one has

    fΦ1g=Nl=0f(l)l!+R(N)Lie , (2.16)

    with R(N)Lie a remainder, in the sense that it fulfills Eqs (2.7) and (2.8).

    Proof. Use the formula for the remainder of the Taylor series (in time); this gives

    fΦ1g=Nl=0f(l)l!+1N!10(1+s)Nf(N+1)Φsgds .

    Of course the integral term is R(N)Lie. To estimate its supremum it is immediate. To estimate its first differential remark that

    d(f(N+1)Φsg)=df(N+1)(Φsg)dΦsg .

    Then, from the very definition of the flow one has that its differential fulfills

    ddtdΦtg=dXg(Φtg)dΦtg ,

    which is estimated by

    ddtdΦtgεm2δdΦtg ,

    where we used the fact that g is a symbol. From this it follows that, provided ε is small enough one has dΦtg2 for |t|1.

    From this and from the fact that f(l+1) is a symbol the thesis immediately follows.

    Concerning the cohomological equation we have the following simple lemma

    Lemma 2.14. Let fPm,δ and consider the cohomological equation

    {H0;g}+f(nr)=0 . (2.17)

    It admits a solution gPmδ,δ.

    Proof. Expanding in Fourier series, the cohomological equation takes the form

    jiH0pj0<|k|Kikjˆgk(p,ε)eikq=0<|k|Kˆfk(p,ε)(1χk(p,ε))eikq ,

    whose solution is

    ˆgk(p,ε)=ˆfk(p,ε)(1χk(p,ε))ipk .

    Since 1χk is supported in the region |pk|εδ/2, the result follows.

    In this subsection we prove the following lemma.

    Lemma 2.15. Let <N be an integer, and let H() be of the form

    H()=H0+Z()+f+R(N) , (2.18)

    with Z()P1,δ in normal form, fPm,δ with

    m:=1+a

    and R(N) a remainder.

    Then there exists a symbol g+1Pmδ,δ which generates a Lie transform Φ1g+1 with the property that H(+1):=H()Φ1g+1 fulfills the assumption of the lemma with +1 in place of .

    Proof. Decompose f as in (2.12) and let g+1 be the solution of the cohomological Eq (2.17) with f(nr) in place of f(nr) and compute

    H()Φ1g+1=H0+{H0;g+1} (2.19)
    +H0Φ1g+1(H0+{H0;g+1}) (2.20)
    +f(nr)+f(res)+f,K (2.21)
    +fΦ1g+1f (2.22)
    +Z()Φ1g+1Z() (2.23)
    +Z()+R(N)Φ1g+1 . (2.24)

    Exploiting Lemma 2.13, one can decompose the different lines as

    (2.22)=Nl=1f,(l)+˜R(N)1=:f1+1+˜R(N)1(2.23)=Nl=1Z()(l)+˜R(N)2=:f2+1+˜R(N)2

    with f1+1P2m2δ,δ, f2+1Pm+12δ,δ and ˜R(N)j remainders (actually of order higher than εNa+1).

    Concerning (2.20), just remark that the sequence H0,(l) defining the Lie transform of H0 (cf. (2.15)), can be generated computing H0,(1) from the cohomological equation, giving H0,(1)={H0;g+1}=f(nr)Pm,δ. In this way one gets H0,(l)P2(mδ),δ and also

    (2.20)=Nl=2H0,(l)+˜R(N)3=:f3+1+˜R(N)3 .

    It follows that, defining f+1:=f1+1+f2+1+f3+1,

    R(N)+1:=f,K+˜R(N)1+˜R(N)2+˜R(N)3+R(N)Φ1g+1 ,Z(+1):=Z()+f(res) 

    one has the thesis.

    In this section we define a partition of the action space Rd into blocks which are left invariant by the flow of a Hamiltonian which is in normal form, namely

    HZ(p,q):=H0(p)+Z(p,q), (3.1)

    with Z in normal form. As usual this partition will be labeled by the sub moduli of Zd identifying the resonances present in each region.

    Definition 3.1. (Module and related notations.) A subetaoup MZd will be called a module if ZdspanRM=M. Given a module M, we will denote MR the linear subspace of Rd generated by M. Furthermore, given a vector pRd we will denote by pM its orthogonal projection on MR.

    Remark 3.2. From the Definition 2.6 of normal form it immediately follows that, if a point pRd is such that

    |pk|εδkZd{0}s.t.|k|K,

    then

    {p,HZ}=0, (3.2)

    hence, in this region the action p is conserved along the motion of HZ.

    The first block we define is

    E0={pZd| |pk|εδ2βks.t.0<|k|K}, (3.3)

    where the correction 2β to the exponent has been inserted in order to separate E0 from the regions where some resonances are present.

    In order to define the other blocks, we introduce the following parameters:

    δ1=δ2β, (3.4)
    δs+1=δsβss=1,d1, (3.5)
    C1=1, (3.6)
    Cs+1=3s2sCs+1s=1,d1. (3.7)

    The next definition we give is meant to identify the points p which are in resonance with vectors of a given module MZd:

    Definition 3.3 (Resonant zones). For any module MZd of dimension s1 and for any (ordered) set {k1,,ks} of linearly independent vectors in M such that |kj|K for all j=1,,s, we define

    Zk1,,ks={pZd| |pkj|<Cjεδjj=1,,d}

    and

    Z(s)M={k1,,ks}lin.ind.inMZk1,,ks.

    Remark that the definition of Zk1,...,ks depends on the order in which the vectors kj are chosen. Thus the definition of resonant zone slightly differs from the analogous definition of resonant zone given in [13]. This is due to the fact that in the present construction we are interested in exhibiting a partition, and not only a covering, of the action space Rd. In particular we have the following remark.

    Remark 3.4. The resonant regions Z(s)M are not reciprocally disjoint; on the contrary, given an arbitrary module M of dimension s2, for any s<s, the following inclusion holds

    Z(s)MM:dimM=sZ(s)M.

    We now define the set composed by the points which are resonant with the vectors in a module M, but are non-resonant with the vectors kM.

    Definition 3.5 (Resonant blocks). Let M be a module of dimension s, we define the resonant block

    B(s)M=Z(s)M(s>sdimM=sZ(s)M).

    We prove below that the resonant blocks {B(s)M}M,s are reciprocally disjoint; nevertheless, they are not suitable for the geometric part, since they are not left invariant by the dynamics associated to a normal form Hamiltonian. For such a reason, we need the following definition:

    Definition 3.6 (Extended blocks and fast drift blocks). For any module M of dimension s, we define

    ˜E(s)M={B(s)M+MR}Z(s)M

    and the extended blocks

    E(s)M=˜E(s)M(s<sdimM=sE(s)M),

    where A+B is the Minkowski sum between sets, namely A+B={a+b | aA,bB}. Moreover, for all pE(s)M we define the fast drift blocks

    Π(s)M(p)={p+MR}Z(s)M .

    With the above definitions, the following result holds true:

    Theorem 3.7. 1). The blocks E0{E(s)M}s,M are a partition of Rd.

    2). Each block is invariant for the dynamics of a system in normal form.

    3). Along such a dynamics, for any initial datum one has

    |p(t)p(0)|3d2d1Cdεδ((d1)(d+1)+2)β. (3.8)

    Corollary 3.8. Theorem 1.2 holds.

    Proof. Choosing δ=14+b2, β=(12b)12(d2+1) and N=[M1a]+1 one immediately gets the result.

    The proof of Theorem 3.7 will occupy the rest of this subsection. We start by stating a few geometric results.

    Lemma 3.9. (Lemma 5.7 of [13]) Let 1sd, and let k1,...,ks be linearly independent vectors in Rd satisfying |kj|K for some positive K and for 1js. Denote by Vol(k1,...,ks) the s-dimensional volume of the parallelepiped with sides k1,...,ks; let moreover wSpan(k1,...,ks) be any vector, and let

    α:=maxj|wkj| ,

    then one has

    wsKs1αVol(k1,...,ks) . (3.9)

    For the proof see [13].

    Lemma 3.10. (Extended blocks are close to blocks.) For all p˜E(s)M there exists a point pB(s)M such that

    |pp|2sCsKs1εδs. (3.10)

    Proof. By the very definition of ˜E(s)M, if p˜E(s)M then there exists a point pB(s)M such that ppM. In particular one has that

    pp=pMpM. (3.11)

    Moreover, since p˜E(s)MZ(s)M, there exist k1,,ks linearly independent vectors in M, with |kj|K, such that

    |pMkj|=|pkj|Cjεδj,j=1,,s.

    Hence, remarking that for s independent vectors with integer components Vol(k1,...,ks)1, Lemma 3.9 implies

    |pM|sKs1Csεδs.

    Analogously, since pB(s)MZ(s)M,

    |pM|sKs1Csεδs.

    Thus (3.11) gives

    |pp||pM|+|pM|2sKs1Csεδs.

    Lemma 3.10 enables us to deduce the following result.

    Lemma 3.11. (Non overlapping of resonances) Consider two arbitrary resonance moduli M and M respectively of dimensions s and s. If ss and MM, then for all pE(s)M one has that

    dist(Π(s)M(p), Z(s)M)>sCsKs1εδs, (3.12)

    where dist(A,B)=inf{|ab| | aA, bB} denotes the distance between two sets.

    Proof. By contradiction, assume that (3.12) is not true. Then, by Lemma 3.10, one also has

    dist(B(s)M, Z(s)M)3sCsKs1εδs,

    It follows that there exist pB(s)M(p) and pZ(s)M such that

    |pp|3sCsKs1εδs.

    Since pZ(s)M and MM, there exists hM such that |h|K and |ph|Csεδs. Compute now

    |ph||pp||h|+|ph|3sKs1CsεδsK+Csεδs,

    thus, recalling that K=[εβ]+1, one has that

    |ph|<(3s2s+1)Csεδsβs.

    Due to definitions (3.4), it follows that

    |ph|<Cs+1εδs+1.

    Hence pZ(s+1)Mh, where Mh is the resonance module generated by M{h}, which is impossible, since pB(s)M implies that it is not in any higher dimensional resonance zones.

    Lemma 3.12. Fix an arbitrary module M of dimension s, for all pE(s)M,

    diam (Π(s)M(p))2sKs1Csεδs.

    Proof. Arguing as in the proof of Lemma 3.10, if p and p are two points in Π(s)M(p), then

    |pM|, |pM|sCsKs1εδs.

    Hence, recalling that ppM, we deduce that

    |pp|2sCsKs1εδs

    Remark 3.13. Recall that K=[εβ]+1; then Lemma 3.12 implies that for any module M and for all pE(s)M

    diam(Π(s)M(p))3d2d1Cdεδ((d1)(d+1)+2)β.

    We are now in position to prove Theorem 3.7. Remark that its proof has also as a consequence the fact that, for any resonance modulus M,

    E(s)M(s<sMME(s)M)¯Z(s)M,

    which shows that it is possible to move from the extended block E(s)M only losing resonances (that is, entering a block E(s)M with MM), or remaining inside the same resonant zone Z(s)M. The latter option will be excluded by the dynamics, which ensures that the motion entirely evolves along planes parallel to M.

    Proof of Theorem 3.7. Since each point pRd belongs either to E0 or to Z(s)M for some M and s, it immediately follows from the definition of the extended blocks that E0{E(s)M}M,s is a covering of Rd. If E(s)M and E(s)M are such that ss, then the two blocks are disjoint by their very definition; if s=s and MM, then their intersection is empty by Lemma 3.11, recalling that for all M and s one has E(s)MZ(s)M. This proves Item 1.

    We now prove the invariance of the extended blocks {E(s)M}s,M along the flow ΦtHZ, arguing by induction on their dimension s.

    Inductive base: s=0.} As already observed in Remark 3.2, if p(0)E0, then p(t)p(0) tR, hence the invariance of the block E0 immediately follows.

    Inductive step: Fix M of dimension s1 and a point pE(s)M. We are now going to prove that

    p(t)Π(s)M(p) ,tR. (3.13)

    Suppose by contradiction that there exists a finite time ˉt>0 such that

    p(t)Π(s)M(p)t<ˉt,andp(ˉt)Π(s)M(p).

    Then for such ˉt one has that

    p(ˉt){p+MR}. (3.14)

    Indeed, for any normalized vector λRd such that λM, consider the quantity I(t)=p(t)λ. For all t such that 0t<ˉt one has

    ˙I(t)={I(t), HZ(p(t),q(t))}=0<|k|KiˆZk(ε,p(t))(kλ)eikq(t)=0,

    due to the fact that ˆZk(ε,p(t))=0 if kM, but kλ=0 if kM. Hence

    p(ˉt)λ=limtˉt p(t)λ=pλ,

    from which (3.14) follows, given the arbitrariety of the vector λM.

    Recall now the definition of Π(s)M(p){p+MR}Z(s)M; since by eq. (3.14) p(ˉt){p+MR}, it must be that

    p(ˉt)Z(s)M. (3.15)

    Since E0{E(s)M}s,M is a partition of Rd, there exists M such that p(t)E(s)M with s=dimM (possibly with s=0, if p(ˉt)E0). We analyze all the possible configurations.

    1).p(ˉt)E(s)M with s=s: then, since by its definition p(ˉt)Z(s)M, it must be MM. Thus

    p(ˉt)Z(s)MZ(s)M,

    which is empty by Lemma 3.11. Hence this case is contradictory.

    2).p(ˉt)E(s)M with s>s. This leads again to a contradiction, since due to Remark 3.4 one would have

    p(ˉt)E(s)MZ(s)M(Mofdim.sMMZ(s)M)Z(s)M,

    but p(ˉt)Z(s)M implies that neither p(ˉt)Z(s)M, nor p(ˉt)Z(s)M for any M of dimension s, with MM, due to Lemma 3.11.

    3).p(ˉt)E(s)M, with s<s. Due to the induction assumption, the blocks E(s)M of dimension s<s are invariant under the dynamics of HZ, thus no orbit can enter or exit from it

    Hence none of the above situations is possible, contradicting the assumption that ˉt<. Since the same occurs for negative times, we can conclude that

    p(t)Π(s)M(p)tR.

    Estimate (3.8) then follows from Remark 3.13. Moreover, since p(t)Π(s)M(p)˜E(s)M and by inductive hypothesis each block E(s)M with s<s has been proven to be invariant under the flow of HZ for all real times, it must be

    p(t)˜E(s)M(ssdimM=sE(s)M)=E(s)M ,tR.

    Here we come to study the dynamics of the Hamiltonian HT.

    In the following, we will denote (¯p(t),¯q(t))=ΦtHT(p,q), with HT as in Lemma 2.7. Furthermore, in order to be able to study the dynamics of a point starting in an extended block, say E(s)M, we consider the following sets:

    (Π(s)M(p))εδ={pRd | dist(p,Π(s)M(p))<εδ}.

    The result we obtain is the following:

    Proposition 3.14. For all N there exists a positive threshold εN such that, if εεN, then pRd

    |¯p(t)p|εδ((d1)(d+1)+2)βts.t.|t|εNa. (3.16)

    Proof. Fix pRd, then, for any time t such that |t|εNa, one has that either ¯p(t)E0, or ¯p(t)Π(st)Mt(pt), for some fast drift block identified by a suitable MtZd of dimension st1 and some (not unique) ptE(st)Mt. Let t0[εNa,εNa] be such that Mt0 is of minimal dimension, namely such that

    st0=dimMt0=min|t|εNadimMt.

    Of course, if there exists a time t0 such that p(t0)E0, then Mt0={0} and Π(st0)Mt0(pt0)={pt0} and (Π(st0)Mt0(pt0))εδBεδ(pt0), namely the ball of center pt0 and radius εδ.

    We are going to prove that

    p(t)(Π(st0)Mt0(pt0))εδ|t|εNa (3.17)

    This is obtained arguing essentially as in the proof of Theorem 3.7.

    Let ˉt>0 be the exit time of ¯p(t) from (Π(st0)Mt0(pt0))εδ, namely the time s.t. t with 0t<ˉt

    ¯p(t+t0)(Π(st0)Mt0(pt0))εδ,and¯p(ˉt+t0)(Π(st0)Mt0(pt0))εδ.

    We prove that |ˉt+t0|>εNa, from which (3.17) follows. Indeed, suppose by contradiction that |ˉt+t0|εNa. Then for any normalized vector λRd with λMt0, we consider the quantity

    I(t)=¯p(t+t0)λ

    due to Lemma 3.11, for 0t<ˉt,

    |˙I(t)|=|{I(t), HT}|=|{I(t), R(t)}|KNε1+Naδ (3.18)

    where KN is a constant bounding the r.h.s. of (2.7). Since |t+t0|<εNa,|I(t)I(0)|2KNε1δ, thus, passing to the limit tˉt, we obtain

    |I(ˉt)I(0)|2KNε1δ,

    which is strictly less than εδ2 if ε is small enough.

    If Mt0={0}, this enables us to conclude that

    dist(¯p(ˉt+t0),pt0)<εδ2,

    which contradicts the definition of ˉt as the time of exit from B(pt0,εδ).

    Assume now st01, then (3.18) implies

    dist(¯p(ˉt+t0),Πst0Mt0(pt0))<εδ2. (3.19)

    Since

    Π(st0)Mt0(pt0)={pt0+MR}Z(st0)Mt0;

    now, by the definition of ˉt,¯p(ˉt+t0)(Π(st0)Mt0(pt0))εδ, equation (3.19) implies that in particular ¯p(ˉt+t0)Z(st0)Mt0. Then one argues as in the proof of Theorem 3.7 to deduce that, by Lemma 8, the point ¯p(ˉt+t0) cannot belong to any block E(s)M with sst0. Thus it must be

    ¯p(ˉt+t0)E(s)Mforsomes<st0,

    which contradicts the minimality hypothesis on st0. Hence, arguing analogously for negative times, we can deduce that (3.17) holds. Finally, recall that by Remark 3.13 this implies that

    |¯p(t)p|3d2d1Cdεδ((d1)(d+1)+2)β.

    Combining the estimate in Proposition 3.14 and estimate (2.10) on the size of the deformation induced on the action variables by the canonical transformation T, we are finally able to deduce that for all NN and tR such that |t|εNa there exists a positive constant KN such

    |p(t)p(0)||p(t)¯p(t)|+|¯p(t)¯p(0)|+|¯p(0)p(0)|KNε1δ+3d2d1Cdεδ((d1)(d+1)+2)β+KNε1δ(2KN+3d2d1Cd)εδ((d1)(d+1)+2)β,

    which concludes the proof of Theorem 1.1.

    We acknowledge the support of GNFM.

    The authors declare no conflict of interest.



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