Nekhoroshev Estimates Quasi-convex Hamiltonian Systems · Quasi-convex Hamiltonian Systems Jürgen...

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nek-1 12.1.2002 20:40 Version 3.1, July 25, 1992 Math. Z. 213 (1993) 187–216 Nekhoroshev Estimates for Quasi-convex Hamiltonian Systems Jürgen Pöschel 0 Introduction This paper is concerned with the stability of motions in nearly integrable, time- independent hamiltonian systems of n degrees of freedom, with hamiltonian H = h ( I ) + f ( I ,θ) in n -dimensional action-angle variables I depending on a small parameter such that f . The equations of motions are ˙ I =−θ H, ˙ θ = I H in usual vector notation, where the dot denotes differentiation with respect to the time variable and indicates partial differentiation. For = 0 these equations reduce to the integrable equations ˙ I = 0, ˙ θ = ω( I ) def = I h . All motions are linear, and the actions in particular stay put for all times. For ar- bitrarily small = 0, however, it is a well known fact that the dynamics may be extremely complicated, and that the actions may vary unboundedly over unbounded

Transcript of Nekhoroshev Estimates Quasi-convex Hamiltonian Systems · Quasi-convex Hamiltonian Systems Jürgen...

Page 1: Nekhoroshev Estimates Quasi-convex Hamiltonian Systems · Quasi-convex Hamiltonian Systems Jürgen Pöschel 0 Introduction This paper is concerned with the stability of motions in

nek-1 12.1.2002 20:40

Version 3.1, July 25, 1992Math. Z. 213 (1993) 187–216

Nekhoroshev EstimatesforQuasi-convex Hamiltonian Systems

Jürgen Pöschel

0 Introduction

This paper is concerned with the stability of motions in nearly integrable, time-independent hamiltonian systems of n degrees of freedom, with hamiltonian

H = h(I ) + fε(I, θ)

in n -dimensional action-angle variables I, θ depending on a small parameter ε suchthat fε ∼ ε . The equations of motions are

I = −∂θ H, θ = ∂I H

in usual vector notation, where the dot denotes differentiation with respect to the timevariable and ∂ indicates partial differentiation.

For ε = 0 these equations reduce to the integrable equations

I = 0, θ = ω(I )def= ∂I h.

All motions are linear, and the actions in particular stay put for all times. For ar-bitrarily small ε = 0, however, it is a well known fact that the dynamics may beextremely complicated, and that the actions may vary unboundedly over unbounded

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2 Section 0: Introduction

time intervals even in nondegenerate systems. For example, the two degree of free-dom hamiltonian

H = 1

2(I 2

1 − I 22 ) + ε sin(θ1 − θ2)

admits the special solution I (t) = (−εt, εt) , θ(t) = − 12 (εt2, εt2) , for which we

have

‖I (t) − I (0)‖ = ‖I (t)‖ =√

2 εt

for all t [20].It was a remarkable achievement by Nekhoroshev to observe that nonethe-

less, generically, the variation of the actions of all orbits remains small over a finite,but exponentially long time interval. More precisely, he proved that for sufficientlysmall ε one has

‖I (t) − I (0)‖ ≤ R∗εb for |t | ≤ T∗ exp(ε−a

)for all orbits, provided the hamiltonian H is real analytic, and the unperturbedhamiltonian h meets certain generic transversality conditions known as steepness[19,20,21,11].

These bounds are referred to as the stability radius and the stability time of thesystem, respectively. Their most important constituents are their stability exponentsa and b . They depend on the number of degrees of freedom, n , and the so calledsteepness indices and tend to zero as n tends to infinity.

We will not enter into a discussion of steepness here, however. Rather, it isthe purpose of this paper to give simple and explicit expressions for these stabilitybounds for the important special cases of convex and quasi-convex systems, whichare the ‘steepest’ among all steep systems. That is, the unperturbed hamiltonian h isassumed to be convex on all of its domain, or on each of its level sets, respectively. Theformer arise typically in classical mechanics as the hamiltonians of nearly integrableconservative systems. The importance of the latter stems from the fact that time-dependent perturbations of convex systems turn into time-independent perturbationsof quasi-convex systems after passing to an extended phase space in the usual fashion.

Such systems admit the simplest and best stability estimates. In particular, wewill obtain

a = 1

2n.

Besides that we have b = a on all of phase space, and even b = 1/2 in an opensubdomain of large measure, which conforms with KAM-theory. In addition, the

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Section 0: Introduction 3

exponent a improves in small neighbourhoods of resonances, up to the value

a = 1

2

near periodic orbits.In his original paper [20] Nekhoroshev was mainly concerned with the generic

steep case, so his estimates are not particularly sharp. In the eighties, Benettin,Gallavotti, Galgani and Giorgilli investigated the convex case more carefully in orderto obtain bounds relevant for stability problems in Celestial and Statistical Mechanics,for example. In a number of papers, for instance [2,4,8,9], they refined the originalproof to finally obtain a stability exponent around

a = 1

n2 + n.

Their work also helped to make these results more widely known and accessible.Recently, Lochak [12,13] presented a novel approach to the convex case en-

abling him to obtain any exponent a with

a <1

2n + 1.

His analysis centers around periodic orbits, which represent the worst type of reso-nances in the system. He observes that on account of convexity suitable neighbour-hoods of suitable periodic orbits are stable for exponentially long times. Covering thewhole phase space by such neighbourhoods with the help of a standard result fromthe theory of diophantine approximations he then obtains the general estimate. In hiswords, “the important property that emerges is that, the more resonant the initial con-dition, the more stable the corresponding trajectory will be” in convex systems [13].

In this paper, however, we follow the traditional lines in order to obtain theresults indicated above. First, normal forms are constructed on various subdomainsof phase space known as ‘resonance blocks’ which are described as neighbourhoodsof ‘admissible resonances’. In the second step, these normal forms lead to stabilityestimates, which turn out to be rather straightforward in the convex case. Thirdly, ageometric argument is given how to cover the whole phase space by such resonanceblocks.

Of these three steps only the last, geometric one is essentially new. The idea is todescribe resonance zones not through small divisor conditions, because their geometryis somewhat complicated. Instead, they are given as tubular neighbourhoods of exactresonances, for which the required small divisor estimates are subsequently verified.

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4 Section 1: Set Up and Main Result

This leads to a simpler and essentially optimal covering of action space by resonanceblocks. In turn, this results in a substantial improvement of the stability exponent.

After a previous version of this paper was completed the author learned fromAnatolij Neishtadt a simple argument which allowed to sharpen the Normal FormLemma and the Resonant Stability Lemma below. The same remark enabled Lochak,in a forthcoming joint paper with Neishtadt [15], to obtain the stability exponenta = 1/2n along his lines as well. Yet another application of the same idea may befound in [5].

Before proceeding to the details I like to take the opportunity to thank SergejKuksin for numerous discussions during his stay in Bonn, which kept me working onthis subject. I am also indebted to Giancarlo Benettin, Amadeo Delshams, AntonioGiorgilli, Pierre Lochak and Anatolij Neishtadt (in alphabetical order) for their manyremarks and comments, which helped to improve this paper to its present state.Finally, I thank the Forschungsinstitut fur Mathematik at the ETH Zurich for itshospitality while this work was completed, and the Deutsche Forschungsgemeinschaftfor their financial support through a Heisenberg grant.

1 Set Up and Main Result

Set Up. We consider a nearly integrable hamiltonian

H = h(I ) + fε(I, θ),

that is real analytic in the action-angle variables

I ∈ P ⊆ Rn, θ ∈ Tn = Rn/2πZn

and depends on a small parameter ε in an arbitrary fashion. More specifically, thehamiltonian is assumed to be real analytic on a fixed complex neighbourhood ofP × Tn of the form

Vr0,s0 P = Vr0 P × Ws0Tn ⊆ Cn × Cn,

where for an arbitrary subset D of Rn , Vr D = I : ‖I − D‖ < r ⊆ Cn denotesthe complex neighbourhood of radius r around D with respect to the euclideannorm ‖ · ‖ , and Ws0T

n = θ : |θ | = maxi |Im θi | < s0 denotes the complex stripof width s0 around the torus Tn . The set P may be any point set. Later on, we willalso use the notation Ur D = Vr D ∩Rn to denote real neighbourhoods of a subset Dof Rn .

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Section 1: Set Up and Main Result 5

We assume n ≥ 2 in order to exclude the trivial case of one degree of freedomsystems, which are always integrable.

For the sake of convenience the perturbation parameter ε is assumed to bechosen in such a way that

fε P,r0,s0≤ ε

for all sufficiently small ε ≥ 0 in the following exponentially weighted norm. If uis analytic on Vr,s D with Fourier expansion

∑k uk(I ) eik · θ , then

u D,r,s = supI∈Vr D

∑k∈Zn

|uk(I )| e|k|s,

where |k| = |k1| + · · · + |kn| . The basic properties of these norms are collected inAppendix B. In particular, they are related to the usual sup-norms by

|u|D,r,s ≤ u D,r,s ≤ cothnσ |u|D,r,s+2σ

for all σ > 0.The hessian of the integrable hamiltonian h ,

Q(I ) = ∂2I h(I ),

is assumed to be uniformly bounded with respect to the operator norm induced by theeuclidean norm, also denoted by ‖ · ‖ :

supI∈Vr0 P

‖Q(I )‖ ≤ M < ∞.

For the symmetric operator Q this norm amounts to the spectral radius norm. Theseassumptions will be in effect throughout the following discussion.

Main Result. To state the main result we need to quantify the notion of con-vexity. Let l and m be positive numbers. The integrable hamiltonian h is said to bem -convex, if the inequality

〈Q(I )ξ, ξ〉 ≥ m‖ξ‖2, ξ ∈ Rn,

holds at every point I in Ur0 P . More generally, h is said to be l, m -quasi-convex,if at every point I in Ur0 P at least one of the inequalities

|〈ω(I ), ξ〉| > l‖ξ‖, 〈Q(I )ξ, ξ〉 ≥ m‖ξ‖2

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6 Section 1: Set Up and Main Result

holds for each ξ ∈ Rn . Thus, uniform m -convexity is required not only on thehyperplane orthogonal to ω(I ) but also outside a cone around ω(I ) whose angledecreases with ‖ω(I )‖/ l . This amounts to strict convexity when ‖ω(I )‖ ≤ l .

Theorem 1. Suppose h is l, m -quasi-convex, and

fε P,r0,s0≤ ε ≤ ε0 = mr2

0

210 A2n,

where r0 ≤ 4l/m and A = 11M/m . Then for every orbit with initial position(I0, θ0) in P × Tn one has

‖I (t) − I0‖ ≤ R0

ε0

)a

for |t | ≤ T0 exp( s0

6

(ε0

ε

)a)

except when ‖ω(I0)‖ ≤ mr0/8 in which case ‖I (t) − I0‖ ≤ r0 for all t . Theparameters are

a = 1

2n, R0 = r0

A, T0 = A2 s0

Ω0,

where Ω0 = sup‖I−I0‖≤R0‖ω(I )‖ .

In this statement the analyticity radius r0 plays the role of an external parameter,which may be adjusted in some suitable way. For example, replacing r0 by the smallerquantity r0 (ε/ε0)

(1−µ)/2 with 0 ≤ µ ≤ 1 one immediately obtains the followingmore general version of Theorem 1.

Theorem 1*. Under the same hypotheses as in Theorem 1 one has for everyorbit

‖I (t) − I0‖ ≤ R0

ε0

)µa+(1−µ)/2

for |t | ≤ T0 exp( s0

6

(ε0

ε

)µa)

for every µ in 0 ≤ µ ≤ 1 , except when ‖ω(I0)‖ ≤ mr0/8 · (ε/ε0)(1−µ)/2 , in which

case one has perpetual stability in a slightly larger ball.

Quasi-convex systems are iso-energetically nondegenerate, as one easily veri-fies. Hence, KAM-theory applies as well, at least for sufficiently small ε , providinga Cantor family of invariant tori, on which orbits are indeed perpetually stable. Intwo degree of freedom systems this results in perpetual stability of all other orbitsas well, due to their well known confinement between invariant tori. So in this case

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Section 2: Normal Forms 7

the Nekhoroshev estimate is of minor interest. Instabilities such as Arnold diffusionmay only occur when the number of degrees of freedom is at least three, and here theNekhoroshev estimate provides an upper bound for the speed of such processes.

The rest of this paper is almost entirely devoted to the proof of Theorem 1. Thefollowing three sections describe its three basic steps: the construction of normalforms, the stability estimates based on convexity, and the geometry of resonances.The subsequent section finishes the proof. In addition, it contains further results onthe exceptional stability of orbits very close to resonances and perturbations of linearintegrable systems.

2 Normal Forms

At a given point in phase space the relevant part of a perturbation of an inte-grable system consists of those Fourier terms that correspond to resonances, or nearresonances, among the frequencies of the unperturbed motion at that point. All otherterms may be transformed away to obtain a resonant normal form that allows a moredetailed analysis of the evolution of the system.

Let be a sublattice of Zn . We are considering subsets D of P where thefrequencies of the integrable system,

ω(I ) = ∂I h(I ),

satisfy nonresonance conditions up to a certain order except for those integer vectorslying in . To make this quantitative, a subset D ⊆ P is said to be α, K -nonreso-nant modulo , if at every point I in D ,

|k · ω(I )| ≥ α for all k ∈ ZnK \,

where ZnK = k : |k| ≤ K . For the time being, α and K will be regarded as

independent parameters, but eventually they will be chosen as suitable functions of ε .In the vicinity of such a set D the perturbed hamiltonian h + fε will be put

into a -resonant normal form h + g + f∗ up to f∗ . That means,

g =∑k∈

gk(I ) eik · θ

contains only -resonant terms, while f∗ is a general term. The aim is to make f∗exponentially small in K .

A special situation arises when is the trivial sublattice of Zn containingonly 0. In this case the set D is said to be completely α, K -nonresonant. In the

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8 Section 2: Normal Forms

corresponding normal form, g is independent of the angles θ and thus integrable.Naturally, the analysis of this case is simpler than in the presence of resonances. Thiswill become apparent in the next section.

The Lemma. To state the Normal Form Lemma a few notations are required.We write TK and P for the projections of Fourier series onto their partial sumscontaining only terms with k ∈ Zn

K and k ∈ , respectively. Moreover, we write pand q for two positive numbers related by p−1 + q−1 = 1. Later on, we let q = 9and p = 9/8, but this choice is quite arbitrary.

From now on we write s for s0 for brevity.

Normal Form Lemma. Suppose that D ⊆ P is α, K -nonresonant mod-ulo . If

ε ≤ 1

27q

αr

K, r ≤ min

pMK, r0

),

and Ks ≥ 6 , then there exists a real analytic, symplectic coordinate transformation: Vr∗,s∗ D → Vr,s D , where r∗ = r/2 and s∗ = s/6 , such that H = h + g + f∗is in -resonant normal form up to f∗ with

g − g0 r∗,s∗ ≤ 16cε2, f∗ r∗,s∗ ≤ e−Ks/6ε,

where g0 = PTK fε and c = 2q K/αr . Moreover,

‖ΠI − I‖ ≤ cεr

uniformly on Vr∗,s∗ D , where ΠI denotes the projection onto I -coordinates.

A more detailed estimate of is given at the end of this section. Incidentally, need not be a sublattice of Zn for the Normal Form Lemma to be true. Any subsetof Zn is allowed that is symmetric about the origin. This greater generality, however,will not be needed here.

Proof of the Lemma. We prove the Normal Form Lemma by a finite iterativeprocedure reminiscent of the KAM-scheme. At each step – described in the IterativeLemma below – a coordinate change is performed that takes the hamiltonian closerto normal form. Choosing the number of steps and their sizes carefully their productproduces a normal form with an exponentially small remainder.

A significant feature of the lemma is the dependence of its smallness conditionon K alone, rather than its square. This is accomplished by making the first step inthe iteration large, while all subsequent steps are uniformly small. Such a scheme

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Section 2: Normal Forms 9

first appeared in [18] in a related averaging problem, and the author is indebted toAnatolij Neishtadt for communicating this idea.

Alternatively, one could work with a suitable vectorfield norm rather than ahamiltonian one [5,6]. This approach avoids the inconvenience of different step sizesand is of particular use in other related problems, but it also sacrifices the compactnessof the hamiltonian formulation. Anyhow, the one big step would now be requiredbeforehand to pass from the hamiltonian to the vectorfield norm.

Yet another approach avoiding successive coordinate transformations is de-scribed for example in [3,8,10]. There, the Lie transform method is employed whichworks in a fixed coordinate system in a manner similar to the comparison of coeffi-cients in power series expansions.

From now on we write · r,s for · D,r,s and Vr,s for Vr,s D for brevity, sinceD is fixed. We also write | · |P , · P for the phase space norm defined by |(I, θ)|P =max (‖I‖, |θ |) and its induced operator norm, respectively.

Iterative Lemma. Suppose D ⊆ P is α, K -nonresonant modulo , andthe hamiltonian H = h + g + f is real analytic on Vr,s . If

f r,s <αρσ

2q,

with 2ρ < r ≤ α/pMK and 2σ < s , then there exists a real analytic, symplectictransformation : Vr−2ρ,s−2σ → Vr,s so that H = h + g+ + f+ with g+ − g =PTK f and

f+ r−2ρ,s−2σ ≤(

1 − f r,s

αρσ/2q

)−1 (g ∗ + f r,s

αρσ/q+ e−Kσ

)f r,s,

where

g ∗ =∑

j

1

2

rj − r + ρ+ σ

sj − s + σ

)g rj ,sj

,

if g = ∑j gj with terms gj bounded on Vrj ,sj ⊇ Vr,s . Moreover,

|W ( − id)|P ,1

2W (D − I )W −1

P≤ q

αρσf r,s

uniformly on Vr−2ρ,s−2σ , where W = diag(ρ−1 In, σ

−1 In)

and In denotes the n -dimensional unit matrix.

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10 Section 2: Normal Forms

The proof of the Iterative Lemma is a purely technical matter and is relegatedto Appendix A. Here we show how the Normal Form Lemma follows from it.

Proof of the Normal Form Lemma. For the very first step let ρ0 = r/8,σ0 = s/6 and r1 = r − 2ρ0 = 3

4r , s1 = s − 2σ0 = 23 s . Since we assume Ks ≥ 6,

we have

ε0def= ε ≤ 1

27q

αrs

Ks≤ αρ0σ0

16q.

So the Iterative Lemma applies. There exists a coordinate transformation 0: Vr1,s1 →Vr,s so that H 0 = h + g0 + f1 with g0 = PTK fε and

f1 r1,s1≤ 8

7

(qε0

αρ0σ0+ e−Kσ0

)ε0.

In general, the first summand is dominant, and assuming that e−Kσ0 ≤ qε0

αρ0σ0we

obtain

ε1def= f1 r1,s1

≤ 8

3

qε20

αρ0σ0= 27qε2

0

αrs≤ ε0

4.

Otherwise we perform the first step with twice the step size and already obtain thefinal result

f1 r1,s1≤ 8

7

(qε0

4αρ0σ0+ e−2Kσ0

)ε0 ≤ 8

7

(1

4+ 1

e

)e−Kσ0ε0 ≤ eKs/6ε.

Moreover, for 0 we have

|W0(0 − id)|P ,1

2W0(D0 − I )W −1

0 P≤ qε0

αρ0σ0

with W0 = diag(ρ−10 In, σ

−10 In) in both cases.

Next we apply the Iterative Lemma N times with fixed parameters

ρ = r

8N, σ = s

4N,

where the integer N is chosen so that e−Kσ is bounded by 1/8 but now is as largeas possible. Thus we choose N so that

N ≤ Ks

12 ln 2< N + 1.

In particular, Ks ≥ 8N . We can assume that N ≥ 1, since otherwise there is nothingmore to do.

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Section 2: Normal Forms 11

We will obtain a succession of transformations i : Vi+1 → Vi = Vri ,si for1 ≤ i ≤ N , where ri = r1 − 2(i − 1)ρ and similarly si , so that the product = 01· · ·N maps VN+1 = Vr/2,s/6 into Vr,s . Moreover, for the componentsof the successive hamiltonians Hi = h + gi−1 + fi on Vi with corresponding norms· i we will find the estimates

εidef= fi i ≤ 41−iε1, gi − gi−1 i ≤ εi

for 1 ≤ i ≤ N + 1 and 1 ≤ i ≤ N , respectively, as we now verify by induction.Indeed, the smallness condition of the Iterative Lemma is satisfied by Hi at

each step, since we have

εi ≤ ε1 = 27qε20

αrs≤ αrs

213q N 2= αρσ

28q,

which follows from

ε0 ≤ 1

27q

αrs

Ks≤ 1

210q

αrs

N.

To estimate fi+1 we write gi−1 = (gi−1 − gi−2) + · · · + (g1 − g0) + g0 and get

gi−1 ∗ ≤ 1

2(εi−1 + · · · + ε1) + 1

4

ρ0+ σ

σ0

)ε0

≤ ε1 + ε0

N

≤ 1

28

αρσ

q+ 1

25

αρσ

q

≤ 1

24

αρσ

q.

Moreover, fi i ≤ 1

28

αρσ

qand e−Kσ ≤ 1

8. Hence, we obtain

fi+1 i+1 ≤(

1 − 1

27

)−1 (1

24+ 1

28+ 1

8

)fi i ≤ εi

4≤ 4−iε1

as we wanted to show. We also have gi − gi−1 i = PTK fi i ≤ fi i as required,and in addition

|W (i − id)|P ,1

2W (Di − I )W −1

P≤ qεi

αρσ

uniformly on Vi+1 with W = diag(ρ−1 In, σ−1 In) .

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12 Section 3: Stability Estimates

Let H = h + g + f∗ be the final hamiltonian after N such iterations. Then wefind

f∗ r∗,s∗ = fN+1 N+1 ≤ 4−N−1ε0 ≤ 4−Ks/12 ln 2ε0 = e−Ks/6ε

and

g − g0 r∗,s∗ ≤ 4

3· ε1 ≤ 28qε2

0

αrs≤ 25q K

αr· ε2.

Finally, the estimates for the i and the usual telescoping arguments yield

|W0( − id)|P ,1

2W0(D − I )W −1

P≤ 2qε0

αρ0σ0≤ 25q K

αr· ε

uniformly on Vr∗,s∗ . This in particular contains the estimate

‖I − I‖ ≤ 2qε0

ασ0≤ 2q K

α· ε.

This proves the Normal Form Lemma.

3 Stability Estimates

Bounds on the variation of the action are obtained by analyzing the hamilton-ian system in normal form. This analysis is simplest in the completely nonresonantregime. Here, no further assumptions are needed. In particular, no convexity hypoth-esis is needed.

Proposition 1 (Nonresonant Stability Estimate). Suppose the domain D ⊆P is completely α, K -nonresonant. If

ε ≤ 1

27q

αr

K, r ≤ min

pMK, r0

),

then

‖I (t) − I0‖ ≤ r for |t | ≤ sr

5εeKs/6

for every orbit of the perturbed system with initial position in D × Tn .

Proof. First we assume that Ks ≥ 6. Let r = r/2, s = s/6. By the NormalForm Lemma there exists a real analytic symplectic transformation : Vr ,s D →

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Section 3: Stability Estimates 13

Vr,s D , ( I , θ ) → (I, θ) , which takes the given hamiltonian H into H = h+g+ f∗with ∂θ g = 0 and f∗ r ,s ≤ η = εe−Ks/6 . Moreover,

‖ I − I‖ ≤ ρ = r

26= r

25

uniformly on the smaller domain. Thus, −1 takes any given initial position in D×Tn

into some initial position in Uρ D × Tn .Fix such an initial position ( I 0, θ0) . The real ball B0 of radius r −ρ around I 0

is contained in Ur D . As long as the forward evolution of the action I stays inside B0

we have d I/dt = −∂θ f∗ and therefore

‖ I (t) − I 0‖ ≤∫ t

0‖∂θ f∗‖ dτ ≤ t sup

B0×Tn‖∂θ f∗‖ ≤ tη

es

by the Cauchy estimate of Lemma B.3. It follows that

‖ I (t) − I 0‖ ≤ r − ρ for 0 ≤ t ≤ sr

5εeKs/6 ≤ es(r − ρ)

η.

Going back to original coordinates we obtain

‖I (t) − I (0)‖ ≤ ‖I (t) − I (t)‖ + ‖ I (t) − I (0)‖ + ‖ I (0) − I (0)‖≤ ρ + (r − ρ) + ρ ≤ r

for the same time interval. The estimate for backward orbits is, of course, the same.For Ks ≤ 6 the same reasoning applies to the hamiltonian in original coordi-

nates, letting g = 0 and f∗ = fε .

Next we look at the hamiltonian in the vicinity of resonances. Here, we followan idea of Gallavotti [7], also used by Lochak [12,13], that due to the convexity ofthe unperturbed hamiltonian an orbit stays in a fixed coordinate neighbourhood of aresonance for an exponentially long time interval. Indeed, on account of the NormalForm Lemma, near a -resonant point I∗ the action essentially evolves in the plane

through I∗ spanned by the vectors in . The restriction of h to is a convexfunction with a critical point at I∗ , which in addition is nearly invariant because ofenergy conservation. Thus, the hamiltonian h is almost a Liapunov function for acertain amount of time.

To make this quantitative we introduce the following notion. Let be anontrivial sublattice of Zn characterizing a resonance, and

R = ω ∈ Rn : k · ω = 0 for all k ∈

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14 Section 3: Stability Estimates

the linear space of exact -resonances. A domain D ⊆ P is said to be δ -close toexact -resonances, if

‖ω(I ) − R‖ = minυ∈R

‖ω(I ) − υ‖ ≤ δ

for all I in D .We first consider the hamiltonian in normal form coordinates.

Lemma 1. Suppose D ⊆ P is δ -close to exact -resonances, and thehamiltonian H = h + g + f∗ is in -resonant normal form up to f∗ on Vr,s D withg r,s + f∗ r,s ≤ ζ . If h is m -convex and

δ ≤ mr

4, ζ ≤ mr2

32,

then for every orbit with initial action I0 in Uρ D , ρ = δ/4M , one has

‖I (t) − I0‖ ≤ r − ρ for |t | ≤ T∗ = s

12‖ω(I0)‖mr2

η,

where η = f∗ r,s .

Note that the lemma also applies to = Zn . In this case there is no propernormal form to speak of. The assumptions, however, imply that h is convex in Dand close to its absolute minimum, which assures stability.

The lemma also makes sense when f∗ vanishes yielding eternal stability. Thus,very near resonances in a convex integrable system, instability is not caused by theresonant terms of a perturbation but rather by the nonresonant ones which can not betransformed away.

Proof of the Lemma. Fix an initial position (I0, θ0) in Uρ D × Tn with ρ =δ/4M and consider its forward orbit. The real ball B0 of radius r − ρ around I0

is contained in Ur D , and there is a positive, possibly infinite time Te of first exit ofI (t) from B0 . Let T = min(Te, T∗) .

Following Lochak [12] we consider h = h(I (t))∣∣T

0 and I = I (t)∣∣T

0 .Expanding around I0 in the ball B0 we get

h = 〈ω0, I 〉 +∫ 1

0(1 − s) 〈Q(I (s))I, I 〉 ds,

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Section 3: Stability Estimates 15

where ω0 = ω(I0) , and I (s) = I0 + sI describes a straight line in B0 . Hence,

m

2‖I‖2 ≤ |h| + |〈ω0, I 〉|

by the convexity of Q .Let P denote the orthogonal projection onto the real space spanned by , and

let Q = I − P . By the choice of ρ and the mean value theorem, the domain Uρ Dis 5

4δ -close to exact -resonances. Therefore,

|〈ω0, PI 〉| ≤ ‖Pω0‖ ‖PI‖≤ 5

4δ · ‖I‖ ≤ 5

2mδ2 + m

6‖I‖2.

The vector QI is given by the integral along the orbit of ∂θ f∗ only, since g is in -resonant normal form. Thus,

|〈ω0, QI 〉| ≤∫ T

0|〈ω0, ∂θ f∗〉| dτ

≤ T ‖ω0‖ supB0×Tn

‖∂θ f∗‖ ≤ T ‖ω0‖ · 1

esf∗ r,s

by the Cauchy estimate of Lemma B.3. Finally, H = H(I (t), θ(t))∣∣T

0 vanishes byenergy conservation and therefore |h| ≤ |g| + | f∗| ≤ 2ζ.

Collecting terms and using the notation of the lemma and its hypotheses on δ ,ζ and T we thus arrive at

m

3‖I‖2 ≤ 2ζ + 5

2mδ2 + T ‖ω0‖ · 1

esf∗ r,s

≤ mr2

16+ 5mr2

32+ mr2

32

= mr2

4<

m

3(r − ρ)2,

since ρ = δ/4M ≤ r/16 by assumption and thus r ≤ 16/15 (r − ρ) . It follows that‖I‖ < r − ρ and consequently that T = T∗ . This proves the lemma for forwardorbits, but backward orbits are, of course, the same.

As a matter of fact, at any point I the action essentially evolves in the corre-sponding level set of h and thus in the hyperplane orthogonal to ω(I ) . It is only onthese hyperplanes where the convexity of h is really needed to make the preceding

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16 Section 3: Stability Estimates

argument work. This leads to the weaker notion of quasi-convexity, which was madequantitative in Section 1.

Lemma 2. The previous lemma remains valid for an l, m -quasi-convex hamil-tonian h , if r ≤ 2l/m , and if ‖ω(I0)‖ is replaced by

Ω = sup‖I−I0‖≤r−ρ

‖ω(I )‖.

Proof. We continue with the same set up and the same notation as before, butthis time assume that Te < T∗ to reach a contradiction. Thus we assume that ‖I‖ =r − ρ at a first time T = Te < T∗ . We show that this implies |〈ω(I (s)), I 〉| ≤l‖I‖ for all 0 ≤ s ≤ 1. Then we can again apply the convexity argument, arrivingat ‖I‖ < r − ρ , which yields the desired contradiction.

Indeed, proceeding as before we have

|〈ω(I (s)), PI 〉| ≤ 5

4δ‖I‖,

|〈ω(I (s)), QI 〉| ≤ T Ω · 1

esf∗ r,s,

with Ω = supI∈B0‖ω(I )‖ . In view of δ ≤ mr/4, the hypotheses on T∗ and

r ≤ 2(r − ρ) it follows that

|〈ω(I (s)), I 〉| ≤ mr

3‖I‖ + mr

16(r − ρ) ≤ mr

2‖I‖ ≤ l‖I‖

as we wanted to show.

We now transfer the contents of the last two lemmata to our hamiltonian inoriginal coordinates.

Proposition 2 (Resonant Stability Lemma). Suppose the domain D ⊆ Pis δ -close to exact -resonances, but α, K -nonresonant modulo . If h is l, m -quasi-convex and

ε ≤ p

q

mr2

27, δ ≤ mr

8, r ≤ min

pMK,

4l

m, r0

)

with q ≥ 3 , then for every orbit with initial position in D × Tn the estimate

‖I (t) − I0‖ ≤ r for |t | ≤ s

288Ω

mr2

εeKs/6

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Section 3: Stability Estimates 17

holds, where Ω = sup‖I−I0‖≤r ‖ω(I )‖ . In particular, in the fully resonant case = Zn , this estimate holds for all t .

Proof. We take the freedom to increase δ so that

δ = mr

8.

The hypotheses imply that ε satisfies the assumptions of the Normal Form Lemma.Letting r = r/2, s = s/6, and assuming first that Ks ≥ 6, there exists a transfor-mation : Vr ,s D → Vr,s D , ( I , θ ) → (I, θ) , that takes the original hamiltonian Hinto the -resonant normal form H = h + g + f∗ up to f∗ with

g r ,s + f∗ r ,s ≤ ζ = 2ε, f∗ r ,s ≤ η = e−Ks/6ε.

Moreover, with α ≥ pMKr ,

‖ I − I‖ ≤ 2q K ε

α≤ 2qε

pMr≤ mr

26 M= δ

8M

uniformly on the smaller domain by our choice of δ . Hence, −1 takes any initialposition in D ×Tn in the original coordinates to some initial position in Uρ D ×Tn ,ρ = δ/4M , in normal form coordinates.

The two previous lemmata now apply, since also

ζ ≤ mr2

27= mr2

32, δ = mr

8= mr

4,

and r ≤ 2l/m . Hence, for every orbit with initial position in Uρ D × Tn we have‖ I (t) − I 0‖ ≤ r − ρ for

|t | ≤ s

12Ω

mr2

η= s

288Ω

mr2

εeKs/6, Ω = sup

‖ I− I 0‖≤r−ρ

‖ω( I )‖.

This yields the bound ‖I (t) − I0‖ ≤ r as in the proof of Proposition 1 for the statedtime interval, since Ω ≤ Ω .

If, on the other hand, Ks < 6, then we simply apply those two lemmata to thegiven hamiltonian H with g = 0 and f∗ = fε . And if = Zn , then we may setg = fε and fε = 0 to obtain the last claim.

Incidentally, the last lemma and its proof also hold in the nonresonant case, withthe assumption of δ -closeness simply dropped. But Proposition 1 clearly provides abetter estimate in this situation.

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18 Section 4: Geometry of Resonances

4 Geometry of Resonances

In order to apply our stability estimates to all orbits with initial positions inP ×Tn we need to cover all resonances in P by properly chosen neighbourhoods of‘nonresonant resonances’. To this end, an analogous covering is first constructed infrequency space and then pulled back to P via the frequency map.

In principle such a covering is easily constructed. From a given neighbourhoodof all resonances of some order d , 1 ≤ d < n , a sufficiently large neighbourhood ofall resonances of order d + 1 is removed, so that blocks of ‘nonresonant resonancesof order d ’ remain. It is the quantitative aspect which is somewhat more subtle.

First we observe that it suffices to consider resonances characterized by maximalK -lattices in Zn , the family of which is denoted by MK . These are lattices thatare generated by vectors in Zn

K = k : |k| ≤ K and are not properly contained inany other lattice of the same dimension. For, if were not a K -lattice, then itseffective dimension relevant for the nonresonance conditions is smaller than its truedimension, since integer vectors of norm greater than K are never considered. Andif were not maximal, then we could enlarge it to a maximal lattice of the samedimension without affecting the associated resonance space R nor the nonresonanceconditions modulo .

In the following construction a key role is played by the volume of a nontrivialinteger lattice . This quantity, denoted by || , is defined as the volume of theparallelepiped, or fundamental domain, spanned by the vectors of any choice of basisfor .

This definition makes sense. If A is an n×d -matrix, then its d columns span aparallelepiped in n -space whose d -dimensional volume is

√det AtA . If A and A are

two n ×d -matrices whose column vectors form two bases for a given d -dimensionallattice , then there is a unimodular d × d -matrix U such that A = AU . Thus,

det AtA = det U tAtAU = det U det AtA det U = det AtA.

Hence, there is no ambiguity in defining || = √det AtA as the volume of a nontrivial

lattice , where A is any n × d -matrix whose columns form a basis for . – Forthe trivial lattice the volume is set equal to 1 for consistency.

Now, fix n positive parameters λ1 < λ2 < · · · < λn . With each nontrivialmaximal K -lattice of dimension d we associate its resonance zone

Z = ω : ‖ω − R‖ < δ , δ = λd

|| ,

which is the open neighbourhood of radius δ around the n − d -dimensional space

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Section 4: Geometry of Resonances 19

of exact -resonances R = ω : k · ω = 0 for all k ∈ . Removing a neigh-bourhood of all next order resonances we obtain the resonance blocks

B = Z\Z∗d+1,

where Z∗d = ⋃

dim =d Z for 1 ≤ d ≤ n and Z∗n+1 = ∅ . We also set

Z = Rn, B = Z\Z∗1 ,

where stands for the trivial lattice containing only 0. The block B comprises thecompletely nonresonant regime.

Clearly, we have

Rn = Z = B ∪ Z∗1

= B ∪ B∗1 ∪ Z∗

2

...

= B ∪ B∗1 ∪ · · · ∪ B∗

n ,

where B∗d = ⋃

dim =d B . So we obtain a covering of frequency space by resonanceblocks B with ∈ MK .

Geometric Lemma. Let E > 0 and A ≥ E + √2 . If

λd+1

λd≥ AK , 1 ≤ d < n,

then each block B for a nontrivial is α, K -nonresonant modulo with

α ≥ E K δ,

while B is completely α, K -nonresonant with α = λ1 .

Proof. In the following it suffices to assume ‖k‖ ≤ K rather than the strongerinequality |k| ≤ K .

First, let ω ∈ B . Fix k with 0 < ‖k‖ ≤ K , and let be the unique one-dimensional maximal lattice containing k . Then || ≤ ‖k‖ , hence δ ≥ λ1/‖k‖ ,and so

|k · ω| = ‖k‖ · ‖ω − R‖ ≥ ‖k‖ δ ≥ λ1,

since ω /∈ Z . This proves the lemma for B .

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20 Section 4: Geometry of Resonances

Now, let a maximal K -lattice of dimension d ≥ 1 and let ω ∈ B . We mayassume that d < n , since there is nothing to prove for = Zn . Fix k not in with‖k‖ ≤ K . Since is maximal, the integer vector k is linearly independent of , andthe maximal K -lattice + generated by and k together has dimension d +1 ≤ n .

Let P and P+ denote the orthogonal projections onto the real spaces spannedby and + , respectively, and let Q = I − P . We then have

k · ω = k · P+ω

= Qk · Q P+ω + Pk · P P+ω

= Qk ·(I − P)P+ω + Pk · Pω

= Qk ·(P+ω − Pω) + Pk · Pω,

since P P+ = P .Let V and V+ denote the volumes of and + , respectively, and let λ , λ+

be short for λd , λd+1 . Then

‖Qk‖ ≥ V+

V,

because the volume of + is not bigger than the volume of times the length of thecomponent of k orthogonal to , which is Qk . Moreover,

‖Pω‖ ≤ λ

V≤ Kλ

V+, ‖P+ω‖ ≥ λ+

V+≥ AKλ

V+,

since ω ∈ Z , but ω /∈ Z+ . Thus,

‖P+ω − Pω‖ =√

‖P+ω‖2 − ‖Pω‖2 ≥ Kλ

V+

√A2 − 1

by Pythagoras. Finally, the vectors Qk and P+ω − Pω are collinear, because theyboth lie on the line orthogonal to within the real space spanned by + . Hence theabsolute value of their scalar product equals the product of their lengths, and we obtain

|k · ω| ≥ ‖Qk‖ · ‖P+ω − Pω‖ − ‖Pk‖ · ‖Pω‖≥ Kλ

V

√A2 − 1 − Kλ

V≥ E K δ,

since√

A2 − 1 − 1 ≥ A − √2 ≥ E .

We now pull back this covering of frequency space to a covering of action space.The following formulation of the result includes the parameters m and M only forlater convenience. Here they are not needed.

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Section 4: Geometry of Resonances 21

Covering Lemma. Given positive parameters p, b, r0 and K ≥ 1 , fix a

constant A ≥ pM

bm+ √

2 and let

r = r0

An K n

Ad K d

|| , d = dim ,

and

α = pMKr, δ = bmr.

Then there exists a covering of P by resonance blocks D , ∈ MK , such that eachblock D is α, K -nonresonant modulo , but δ -close to exact -resonances fornontrivial .

Proof. Define the blocks B with respect to the parameters

λd = bmr0

An K nAd K d , 1 ≤ d ≤ n.

The Geometric Lemma applies with E = pM/bm . For nontrivial we haveδ = λd/ || = bmr , and each block B is αλ, K -nonresonant modulo withα = E K δ = pMKr . This also applies to the block B , as one readily verifies.Setting

D = I ∈ P : ω(I ) ∈ B

and discarding all sets which turn out to be empty we obtain the postulated coveringof P .

With this choice of parameters the covering of frequency space is optimal inthe following sense. On one hand, in the completely nonresonant block B we have

|k · ω| ≥ α ∼ 1

K n−1, 0 = |k| ≤ K ,

for all K , which can not be improved on account of Dirichlet’s theorem on diophantineapproximations. On the other hand, for any bounded region G the Lebesgue measureof Z∗

1 ∩ G is bounded by

∑dim =1

δ ∼ mr0

K n

∑1≤‖k‖≤K

K

‖k‖ ∼ mr0.

So this sum is of the order of 1, which could not be any better.

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22 Section 5: Proof of Theorem 1 and Further Results

5 Proof of Theorem 1 and Further Results

Proof of Theorem 1. We now fix the constants q = 9, p = 9/8 and, in theCovering Lemma, b = 1/8 and

A = 11M

m≥ pM

bm+

√2.

Then, for every r0 > 0 and every K ≥ 1, there exists a covering of P by suitableresonance blocks D , ∈ MK , with parameters α = pMKr and δ = mr/8,where

r = r0

An K n

Ad K d

|| , d = dim .

In particular, r0/An K n = r ≤ r ≤ rZn = r0 for all those .The stability estimates now apply to all blocks simultaneously, if we have

r0 ≤ 4l/m and

ε ≤ mr2

210= ε0

K 2n, ε0 = mr2

0

210 A2n.

Conversely, for ε ≤ ε0 we may choose K ≥ 1 so that equality is achieved and thestability estimates apply to every block with

K =(ε0

ε

)a, a = 1

2n.

For the stability radius we find the upper bounds

r ≤

r0 for = Zn,

r0/AK otherwise.

The first alternative applies if I0 falls into the block DZn which is tantamount to‖ω(I0)‖ ≤ δZn = mr0/8. Otherwise the second alternative applies, which yields thepostulated stability radius R0/K .

To bound the stability time from below in the resonant regime, we observe thatfor d ≥ 1, we have mr2

/ε ≥ 210 A2 and thus

s

288Ω

mr2

ε≥ A2 s

Ω0, Ω0 = sup

‖I−I0‖≤R0/K‖ω(I )‖

This yields the postulated stability time. Moreover, this time is infinite in theblock DZn , that is, for ‖ω(I0)‖ ≤ mr0/8.

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Section 5: Proof of Theorem 1 and Further Results 23

In the nonresonant regime D we have

r

ε= r0

An K n

K 2n

ε0= 210 An K n

mr0.

Since we can assume that Ω0 ≥ ‖ω(I0)‖ ≥ mr0/8 and n ≥ 2, we thus have

sr

5ε≥ An K ns

Ω0≥ A2 s

Ω0,

so at least the same stability time is obtained.

Further Results. The estimates of Theorem 1 apply uniformly to all orbits inphase space. Thus they reflect the worst possible cases either for the stability radius orthe stability time. In certain regions of phase space, however, much better bounds areavailable: the stability radius is much smaller in the completely nonresonant domain,while the stability times are much larger in the close vicinity of resonances.

The nonresonant domain is the home of maximal invariant tori of KAM-type,on which orbits enjoy perpetual stability, with a bound on the variation of their actionsof the order of

√ε [1,22]. But although of large measure, the union of all these tori

is a nowhere dense Cantor set, hence has no interior points. Restricting oneself to afinite, but exponentially large time interval, the much simpler Nekhoroshev estimateprovides the same stability radius on a relatively open subset of phase space of thesame large measure. Moreover, since resonances need not be considered, neither thedetailed geometric analysis nor the resonant stability estimates are required for thisresult, and the convexity assumption may also be relaxed if necessary.

From now on we again write s0 rather than s for the width of the analyticitydomain in θ .

Theorem 2. Suppose h is l, m -quasi-convex and ε ≤ ε0 . Then

‖I (t) − I0‖ ≤ R

√ε

ε0for |t | ≤ T

√ε0

εexp

( s0

6

(ε0

ε

)a)

with

R = r0

An, T = 27 Ans0

mr0

for every orbit with initial position in a relatively open subset of P × Tn of relativemeasure 1 − O(r0) .

Proof. Consider the covering of P used in the previous proof. Within thecompletely nonresonant block D the stability radius is r = r0/An K n , and the

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24 Section 5: Proof of Theorem 1 and Further Results

stability time is bounded from below by eKs0/6 times

s0r

5ε≥ 27 Ans0

mr0· K n.

With K n = √ε0/ε the stability estimates follow.

For the measure estimate we may assume that P is bounded. Let G be itsbounded image under the frequency map. The complement of the nonresonant blockB is covered by Z = Z∗

1 , the union of all first order resonance zones, for which wefound the measure estimate

µ(Z ∩ G) = O(mr0)

at the end of the previous section. The pull back of Z via the frequency map coversthe complement of D in P . Hence, if h is m -convex, then the complement’smeasure is bounded by

µ(ω−1(Z) ∩ P) ≤ m−1µ(Z ∩ G) = O(r0) ,

as we wanted to show.This argument needs some modification, if h is only l, m -quasi-convex. First

of all, by definition h is m -convex on that subdomain of P which is mapped into theball ω : ‖ω‖ ≤ l . So here the preceding argument applies. In the complementarydomain we consider the restriction ωE of ω to a nonempty energy surface h =E , and in frequency space the retraction π : ω → ω/‖ω‖ of the domain X = ω : ‖ω‖ > l to the unit sphere. The l, m -quasi-convexity implies that the Jacobianof π ωE is m -convex, as one verifies in suitable coordinates. Moreover, in frequencyspace,

µ(π(Z ∩ X)) = O(mr0) .

It follows that the (n − 1) -dimensional measure of the complement of D on eachenergy surface is bounded by

µ(ω−1E (Z) ∩ P) ≤ m−1µ(π(Z ∩ X)) = O(r0) .

Taking the union over a finite range of energy values yields an upper bound of thesame order for the n -dimensional measure of the complement of D in P .

Next we look closer at orbits in the vicinity of resonances. Let be a non-trivial integer lattice of dimension d , let R be the linear space of exact -resonant

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Section 5: Proof of Theorem 1 and Further Results 25

frequencies of dimension ν = n − d , and

S = I ∈ P : ω(I ) ∈ R

the associated resonance surface in action space P . At or very near S the resonancerelations play the role of d additional integrals of motion reducing the effectivenumber of degrees of freedom to ν = n − d . In addition, convexity keeps orbits nearS for a long time. The upshot is that the stability estimates depend on the smallernumber ν of free frequencies rather than the number n of all frequencies.

This stabilizing effect of resonances in convex systems has been observed byseveral authors. In the controversial papers by Molchanov [16,17] the stability ofthe solar system — if it is stable — is attributed to the presence of many almostresonances among the planets rather than the presence of many invariant KAM-tori.Later, this idea was successfully pursued by a group of Italian mathematicians aroundGalgani. In studying systems with many degrees of freedom they emphasized the roleof resonances and the use of Nekhoroshev estimates in order to control the dynamicsin their vicinity by providing bounds which depend essentially only on the smallernumber of free frequencies.

Recall that is a K -lattice if it is generated by vectors in |k| ≤ K .

Theorem 3. Suppose h is l, m -quasi-convex, and S is a nonempty reso-nance surface in P associated with a nontrivial K -lattice of dimension d andcodimension ν = n − d . If

ε ≤ ε

K 2ν

, ε = mr20

210 A2ν ||2 ,

with r0 ≤ 4l/m , then for every orbit with initial position in Uρ S × Tn , ρ =4M−1√εm , one has

‖I (t) − I0‖ ≤ R0

ε

)a

for |t | ≤ T0 exp( s0

6

ε

)a)

,

except when ‖ω(I0)‖ ≤ mr0/8 in which case ‖I (t) − I0‖ ≤ r0 for all t . Theparameters are

a = 1

2ν, R0 = r0

A, T0 = s0

Ω0

with Ω0 as in Theorem 1. Moreover, with probability 1 − O(r0) within the domainUρ S × Tn the stability radius is even 32

√ε/m .

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26 Section 5: Proof of Theorem 1 and Further Results

This result, as well as Theorem 5 below, may be generalized in the same wayas Theorem 1 in Section 1.

Proof. We use the same kind of covering as in the proof of Theorem 1 but thistime choose K larger. Namely,

K =(ε

ε

)a ≥ K ⇐⇒ ε = mr2

210≤ ε

K 2ν

in view of the definition of ε and r . In frequency space the blocks BL with ⊆ L ∈ MK cover the space of exact -resonances R , and r ≤ rL ≤ r0 for allthose . Hence their pull backs DL cover a neighbourhood of S in P of radius

ρ = δ

M= mr

8M= 4

M

√εm.

The resonant stability estimate applies to all these blocks simultaneously. The upperbound for the stability radius is the same as before, and the stability time is boundedfrom below by eKs0/6 times

s0

288Ω0

mr2

ε≥ s0

Ω0

as claimed. Finally, within the block D the stability radius is r = 32√

ε/m , andthe relative measure of all sets DL ∩ Uρ S with L is again 1 − O(r0) as oneeasily verifies.

Incidentally, the proof shows that the same estimates hold in fact on the largerneighbourhood of S , on which the unperturbed frequencies are δ = 4

√εm -close

to exact -resonances.The estimates of Theorem 3 are the better the smaller the number ν of free

frequencies. They are best in small neighbourhoods of unperturbed periodic orbits.This special case deserves a special statement, since periodic orbits are better charac-terized by their minimal period rather than by an integer lattice of resonance relations.More importantly, the pertaining proof does not depend on the Covering Lemma, butreduces directly to the Resonant Stability Estimate. In addition, the construction ofthe Normal Form only requires an averaging with respect to a single fast variable— the longitude of the periodic orbit — and thus may dispense with Fourier seriesexpansions, small divisors and exponentially weighted norms as was pointed out byLochak [13]. Here, however, we rely on the Normal Form Lemma as stated.

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Section 5: Proof of Theorem 1 and Further Results 27

Theorem 4. Suppose h is l, m -quasi-convex, and I∗ is the action of anunperturbed periodic orbit of minimal period T . If

ε ≤ mr2

28, r ≤ min

MT,

4l

m, r0

),

then for every orbit with ‖I0 − I∗‖ ≤ mr

8Mone has

‖I (t) − I0‖ ≤ r for |t | ≤ T∗ exp( s0

2MT r

),

where T∗ = s0

4‖ω(I∗)‖ .

In particular, choosing r minimal we obtain

‖I (t) − I0‖ ≤ 16

√ε

mfor |t | ≤ T∗ exp

(s0

32MT

√m

ε

)

for every orbit with ‖I0 − I∗‖ ≤ 2

M

√εm .

Proof. By assumption there exists a nonzero integer vector k∗ in lowest termssuch that

ω(I∗) = α∗k∗, α∗ = 2π

T.

It follows that the resonant point I∗ itself is α∗, K -nonresonant modulo ∗ for allK , where ∗ is the (n − 1) -dimensional integer lattice perpendicular to k∗ .

The ball B of radius ρ = mr/8M around I∗ is then α, K -nonresonant mod-ulo ∗ and δ -close to the exact ∗ -resonance ω(I∗) with

α = α∗ − M Kρ, δ = Mρ = mr

8.

Choosing

K = π

MT r≥ 1,

one easily checks that α ≥ pM K r with p = 3/2. Thus, all the hypotheses of theResonant Stability Lemma are satisfied with p = 3/2, q = 3. As a result, for every

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28 Section 5: Proof of Theorem 1 and Further Results

orbit starting in B × Tn we have the stability radius r and the stability time

s0

288Ω0

mr2

εeKs0/6 ≥ s0

2Ω0exp

( s0

2MT r

),

where Ω0 ≤ sup‖I−I∗‖≤r+ρ ‖ω(I )‖ . Since, for ‖I − I∗‖ ≤ r + ρ ,

‖ω(I ) − ω(I∗)‖ ≤ M(r + ρ) ≤ 2Mr ≤ 2π

T≤ ‖ω(I∗)‖,

we have Ω0 ≤ 2‖ω(I∗)‖ .

In the work of Lochak [12,13] the exponential stability of periodic orbits is thestarting point for the more general Nekhoroshev estimates. The idea is to approximatean arbitrary initial position by periodic ones. The one additional ingredient required isDirichlet’s theorem on simultaneous diophantine approximations: for every ω ∈ Rn

and natural number Q ≥ 1,

minq=1,2,...,Q

〈qω〉 ≤ 1

Q1/n,

where 〈x〉 = mink∈Zn |x − k| denotes the distance of x to the integer lattice withrespect to the sup-norm | · | .

This implies that after fixing any positive δ ≤ 1 and any integer Q ≥ 1, thereexists a real T and a nonzero integer vector k for every ω in |ω| ≥ δ such that

∣∣ω − T −1k∣∣ ≤ 1

T Q1/(n−1), 1 ≤ T ≤ Q/δ.

To cover all phase space we thus want to apply Theorem 4 simultaneously to allperiodic frequencies T −1k with

ρ ∼ r ∼ 1

T Q1/(n−1).

The worst smallness condition arises for T ∼ Q , hence we need

ε r2 ∼ Q2n

n−1 .

Conversely, we may choose Q ∼ ε− n−12n . As a result, the common stability time is

again of the order of the exponential of

1

T r∼ Q

1n−1 ∼ ε− 1

2n .

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Section 5: Proof of Theorem 1 and Further Results 29

We do not make explicit the constants involved here.

Linear Integrable Systems. We conclude this last section by discussing twosimple special cases. First we consider a linear integrable hamiltonian h ,

h(I ) = ω · I,

where ω is a fixed frequency vector satisfying the diophantine conditions

|k · ω| ≥ γ0

|k|τ−1 , 0 = k ∈ Zn

with some γ0 > 0 and τ ≥ n . In this case, all of P is in the nonresonant regime,and the Nonresonant Stability Estimate applies to the hamiltonian H = h + fε . Noconvexity is required, and no geometry is involved, either. Such hamiltonians werestudied, for example, in [4,5,8].

Theorem 5. Suppose h is linear, ω satisfies the preceding diophantine con-ditions, and

fε P,r0,s0≤ ε ≤ ε0 = γ0r0

27.

Then for every orbit with initial position in P × Tn one has

‖I (t) − I0‖ ≤ r0 for |t | ≤ T0

εexp

(s0

6

(ε0

ε

)1/τ)

,

where T0 = s0r0/5 .

Proof. The domain P is completely αK , K -nonresonant for every K withαK = γ0/K τ−1 . Letting M → 0 and q → 1, the Nonresonant Stability Estimateapplies with r = r0 , if

ε ≤ 1

27

γ0r0

K τ= ε0

K τ.

That is, for ε ≤ ε0 we may choose K = (ε0/ε)1/τ ≥ 1. Then, for every orbit

with initial position in P × Tn we have the bound ‖I (t) − I0‖ ≤ r0 for |t | ≤(s0r0/5ε) · eKs0/6 as claimed.

Kinetic Energy plus Potential. Finally, consider the classical hamiltonian

H = 12 〈I, I 〉 + U (θ)

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30 Section 5: Proof of Theorem 1 and Further Results

on the phase space Rn × Tn . Thus, h = 12 〈I, I 〉 , and fε = U (θ) is independent of

I and ε . We assume that

U s0≤ 1

by a proper scaling of time and space. Of course, in this autonomous system, all orbitsare bounded on account of energy conservation and the convexity of h . It is the non-autonomous case, where U depends analytically on t in a uniform strip, which is ofreal interest, and which is analyzed in [10]. But it turns out that the same estimateshold in both cases. So we include the autonomous case here as an illustration, whilethe non-autonomous case will appear elsewhere.

Theorem 6. Let H be as above. Then for every orbit with initial action I0

in the annulus R ≤ ‖I‖ ≤ 2R , with R sufficiently large, one has

‖I (t) − I0‖ ≤ c0 R1−1/n for |t | ≤ c1s0

Rexp

(s0

c0c1R1/n

),

where c0 = 25/n and c1 = 66 .

Proof. We are going to apply Theorem 1 to sufficiently large annuli

AR = I : R ≤ ‖I‖ ≤ 2R ,

choosing r0 = R . The integrable hamiltonian h = 12 〈I, I 〉 is uniformly convex with

ω(I ) = I and Q(I ) = In for all I , whence m = 1 and M = 1, and l may bechosen arbitrarily large. So we only need to ensure that

U s0≤ 1 ≤ ε0 = R2

210 A2n, A = 11.

That is, we need R ≥ 25 An . We obtain the stability radius

R0ε−a0 = R

A·(

25 An

R

)1/n

= 25/n R1−1/n,

since on AR we have ‖ω(I )‖ ≥ R . The stability time is bounded from below by

A2 s0

Ω0exp

(s0ε

a0

6

)≥ 40s0

Rexp

(s0

6

(R

25 An

)1/n)

,

since Ω0 ≤ supI∈Vr0 AR‖ω(I )‖ = 3R .

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Appendix A: Proof of the Iterative Lemma 31

A Proof of the Iterative Lemma

The Iterative Lemma is proven in the usual fashion using the Lie-transformmethod in its simplest form. The canonical transformation to be constructed iswritten as the time-1-map of the flow Xt

φ of a hamiltonian vectorfield Xφ :

= Xtφ

∣∣t=1

.

Given a hamiltonian H0 = h + f0 we may then use Taylor’s formula to write

H0 = h Xtφ

∣∣t=1

+ f0 Xtφ

∣∣t=1

= h + h, φ +∫ 1

0(1 − t) h, φ , φ Xt

φ dt

+ f0 +∫ 1

0 f0, φ Xt

φ dt

= h + h, φ + f0 +∫ 1

0(1 − t) h, φ + f0, φ Xt

φ dt.

We apply this expansion with f0 = g + TK f and choose φ in such a way thath, φ + f0 = g+ is again in resonant normal form with respect to . Equivalently,setting χ = g+ − g we solve

φ, h + χ = TK f

for φ and χ . For the given hamiltonian H = H0 + f − TK f we then find H =h + g+ + f+ with

f+ =∫ 1

0g + ft , φ Xt

φ dt + ( f − TK f ) X1φ,

where ft = (1 − t)χ + t TK f .Formally, a solution of φ, h + χ = TK f is given by

χ = PTK f, φ = L−1 (TK f − PTK f ) ,

where L−1 denotes the unique inverse of the linear operator · , h restricted to thespace of functions u : TK u = u, Pu = 0 . In terms of Fourier coefficients,

χk = fk for k ∈ ZnK ∩

φk = fk

ik · ω(I )for k ∈ Zn

K \,

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32 Appendix A: Proof of the Iterative Lemma

with all other coefficients being set to zero.For the estimates let δ = f r,s < αρσ/2q . On the domain D we have

|k · ω(I )| ≥ α for k ∈ ZnK \ , hence

|k · ω(I )| ≥ α − MKr ≥(

1 − 1

p

)α = α

q

on Vr D by the mean value theorem and our assumptions on Q and r . In view of thedefinition of the norm · we then have

χ r,s ≤ δ, φ r,s ≤ q

αδ.

The Cauchy estimates of Appendix B then yield

|φI |∞ ≤ qδ

αρ≤ σ

2, ‖φθ‖ ≤ |φθ |1 ≤ qδ

ασ≤ ρ

2

uniformly on Vr−ρ,s−σ . Equivalently, the hamiltonian vectorfield Xφ satisfies

∣∣WXφ

∣∣P

≤ qδ

αρσ≤ 1

2, W = diag

(ρ−1 In, σ

−1 In),

uniformly on Vr−ρ,s−σ . Consequently, its flow Xtφ maps Vr− 3

2 ρ,s− 32 σ into Vr−ρ,s−σ

for 0 ≤ t ≤ 1 and satisfies

∣∣W (Xtφ − id)

∣∣P

≤ qδ

αρσ, 0 ≤ t ≤ 1,

on that smaller domain. The |W · |P -distance of its boundary to its subdomainVr−2ρ,s−2σ is 1/2, so that by the generalized Cauchy inequality of Appendix C wealso have

W (DXtφ − I2n)W −1

P≤ 2 · qδ

αρσ, 0 ≤ t ≤ 1,

uniformly on Vr−2ρ,s−2σ . This proves our claims for the map = Xtφ

∣∣t=1

.It remains to estimate f+ . Obviously, ft r,s ≤ f r,s and thus

ft , φ r−ρ,s−σ ≤ 1

ρσft r,s φ r,s ≤ q

αρσf 2

r,s

by Lemma B.4. Similarly, for each term gj in g ,

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Appendix B: Weighted Norms 33

gj , φ

r−ρ,s−σ

≤ 1

e

(1

(rj − r + ρ)σ+ 1

(sj − s + σ)ρ

)gj rj ,sj

φ r,s

≤ 1

2

rj − r + ρ+ σ

sj − s + σ

)gj rj ,sj

· q

αρσf r,s

and hence

g, φ r−ρ,s−σ ≤ q

αρσg ∗ f r,s

in the notation of the Iterative Lemma. Finally, f − TK f r,s−σ ≤ e−Kσ f r,s , andwe have the general estimate

u Xtφ r−2ρ,s−2σ

≤(

1 − 2

ρσφ r,s

)−1

u r−ρ,s−σ

uniformly for 0 ≤ t ≤ 1 by Lemma B.5. Putting these pieces together we obtain

f+ r−2ρ,s−2σ ≤(

1 − 2qδ

αρσ

)−1 (g ∗ + f r,s

αρσ/q+ e−Kσ

)f r,s

as claimed.

B Weighted Norms

We consider functions that are analytic on complex neighbourhoods of a fixeddomain D × Tn . For fixed I we set

|u|I,s =∑k∈Zn

|uk(I )| e|k|s,

so that u r,s = u D,r,s = supI∈Vr D |u|I,s . We also write Vr for Vr D .

Lemma B.1. For the sup-norm | · |r,s on Vr,s one has, for all σ > 0 ,

|u|r,s ≤ u r,s ≤ cothnσ |u|r,s+2σ .

Proof. The first inequality is obvious. The second one follows from the famil-iar estimate |uk(I )| ≤ e−|k|s |u|I,s for the Fourier coefficients of analytic functionsand the identity

∑k e−2|k|σ = cothnσ .

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34 Appendix B: Weighted Norms

Lemma B.2. For all I ∈ Vr , |uv|I,s ≤ |u|I,s |v|I,s .

Proof. The k -th Fourier coefficient of uv is (uv)k = ∑l uk−lvl . Hence,

|uv|I,s =∑

k

|(uv)k(I )| e|k|s

≤∑

k

∑l

|uk−l(I )| e|k−l|s · |vl(I )| e|l|s

=∑

k

|uk(I )| e|k|s ·∑

l

|vl(I )| e|l|s

= |u|I,s |v|I,s

as claimed.

Lemma B.3. For 0 < σ < s , 0 < ρ < r and I ∈ Vr−ρ ,

∑1≤i≤n

∣∣uθi

∣∣I,s−σ

≤ 1

eσu r,s, max

1≤i≤n

∣∣uIi

∣∣I,s

≤ 1

ρu r,s .

Proof. For the θ -derivatives we have∑1≤i≤n

∣∣uθi

∣∣I,s−σ

=∑

1≤i≤n

∑k

|ki | |uk(I )| e|k|(s−σ)

=∑

k

|k| |uk(I )| e|k|(s−σ)

≤ supt≥0

te−tσ · |u|I,s

≤ 1

eσ|u|I,s

uniformly for all I ∈ Vr . For the partial derivative with respect to Ii at a point I inVr−ρ we write

uIi = 1

2π i

∫γ

u(I + ζ, θ)

ζdζ

with a circle γ in the Ii -plane around the origin of radius ρ . We obtain

∣∣uIi

∣∣I,s

≤ 1

∫γ

|u|I+ζ,s

|ζ | |dζ | ≤ 1

ρu r,s

uniformly in i and I ∈ Vr−ρ .

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Appendix B: Weighted Norms 35

Lemma B.4.

u, v r−ρ,s−σ ≤ 1

e

(1

(r0 − r + ρ)σ+ 1

(s0 − s + σ)ρ

)u r0,s0

v r,s

for 0 < r − ρ < r0 , 0 < s − σ < s0 and ρ, σ > 0 .

Proof. Fix I ∈ Vr−ρ . Then, by the preceding lemmata,

|〈uI , vθ 〉|I,s−σ ≤∑

1≤i≤n

∣∣uIi

∣∣I,s−σ

∣∣vθi

∣∣I,s−σ

≤ max1≤i≤n

∣∣uIi

∣∣I,s−σ

∑1≤i≤n

∣∣vθi

∣∣I,s−σ

≤ 1

r0 − (r − ρ)u r0,s0

· 1

eσv r,s .

Likewise for |〈uθ , vI 〉|I,s−σ .

Lemma B.5. If φ r0,s0<

ρσ

2, then

u X1φ r−ρ,s−σ

≤(

1 − 2

ρσφ r0,s0

)−1

u r,s

for 0 < ρ < r ≤ r0 − ρ and 0 < σ < s ≤ s0 − σ .

The smallness condition implies Xtφ : Vr−ρ,s−σ → Vr,s for 0 ≤ t ≤ 1. This

fact, however, is not used explicitly in the following proof.

Proof. Consider the Lie series expansion

u X1φ =

∑h≥0

1

h!adh

φ u,

where

ad0φ u = u, adh

φ u = adh−1

φ u, φ, h ≥ 1.

For h ≥ 1, let ρ = ρ/h , σ = σ/h . Let · i = · r−i ρ,s−i σ for 1 ≤ i ≤ h . We thenhave

adiφ u

i≤

(1

eρ(s0 − s + i σ )+ 1

eσ (r0 − r + i ρ)

)φ r0,s0

adi−1φ u

i−1

≤ 2

eρσ

1

h + iφ r0,s0

adi−1φ u

i−1.

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36 Appendix C: The General Cauchy Inequality

Hence,

adhφ u

r−ρ,s−σ≤

(2

eρσ

)h h!

(2h)!φ h

r0,s0u r,s .

Observing that

(2

eρσ

)h 1

(2h)!≤

(2h2

eρσ

)he2h

(2h)2h≤

(2

ρσ

)h

we arrive at

u X1φ r−ρ,s−σ

≤∑h≥0

1

h!adh

φ ur−ρ,s−σ

≤∑h≥0

(2

ρσ

)h

φ hr0,s0

u r,s

=(

1 − 2

ρσφ r0,s0

)−1

u r,s .

C The General Cauchy Inequality

Let A and B be two complex Banach spaces with norms | · |A and | · |B , andlet F be an analytic map from an open subset of A into B . The first derivative dv Fof F at v is a linear map from A into B , whose induced operator norm is

dv F B,A = maxu =0

|dv F(u)|B

|u|A.

The Cauchy inequality can be stated as follows.

Lemma C.1. Let F be an analytic map from the open ball of radius r aroundv in A into B such that |F |B ≤ M on this ball. Then the inequality

dv F B,A ≤ M

r

holds.

Proof. Let u = 0 in A . Then f (z) = F(v + zu) is an analytic map from thecomplex disc |z| < r/ |u|A in C into B that is uniformly bounded by M . Hence,

|d0 f |B = |dv F(u)|B ≤ M

r· |u|A

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References 37

by the usual Cauchy inequality. The above statement follows, since u = 0 wasarbitrary.

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