Instability for a priori unstable

Hamiltonian systems: a dynamical

approach

Abed Bounemoura ∗ and Edouard Pennamen †

March 10, 2011

Abstract

In this article, we consider an a priori unstable Hamiltonian systemwith three degrees of freedom, for which we construct a drifting solutionwith an optimal time of instability. Such a result has been alreadyproved by Berti, Bolle and Biasco using variational arguments, and byTreschev using his separatrix map theory. Our approach is new: itis based on a special type of symbolic dynamics corresponding to therandom iteration of a family of twist maps of the annulus, and it givesthe first concrete application of this idea introduced by Moeckel in anabstract setting and further studied by Marco. Our method shouldalso be useful in obtaining the optimal time of instability in the moredifficult context of a priori stable Hamiltonian systems.

1 Introduction

The theory of perturbations of Hamiltonian systems is essentially the studyof near-integrable Hamiltonian systems, generated by functions of the form

H(θ, I) = h(I) + f(θ, I), (θ, I) ∈ An = Tn × Rn,

where H is sufficiently smooth and f sufficiently small.

1. For f = 0, H = h is an integrable system and the situation is well-understood. In this case, the variables (θ, I) are called angle-action coordi-nates for h and the Hamiltonian depends only on the action variables. Thephase space An is trivially foliated by invariant Lagrangian tori parametrizedby the action variables: indeed the equations of motion of H read

{θ = ∇h(I)I = 0,

∗abed@impa.br, IMPA†edouard@math.jussieu.fr, Institut Mathematiques de Jussieu

1

so for each I0 ∈ Rn, the torus T0 = Tn × {I0} is Lagrangian and invariantunder the Hamiltonian flow. The latter is complete and restricts to a linearflow on T0 with frequency ω0 = ∇h(I0), that is

ΦHt (θ0, I0) = (θ0 + tω0 [Zn], I0) ∈ T0, t ∈ R.

The action variables remain fixed for all times, and all solutions are quasi-periodic.

2. Now if we let f to be non zero but small, that is

|f | = ε << 1,

with respect to some norm | . |, the situation is dramatically different. Forsuch a near-integrable system, the action variables are no longer first in-tegrals and we may find “unstable” solutions (θ(t), I(t)) that experience asubstantial drift in the action direction. Such an instability phenomenonis usually referred to Arnold diffusion, after the original work of Arnold([Arn64]). In this paper, the author devised a beautiful mechanism to con-struct an example of an analytic near-integrable system with three degreesof freedom (n = 3), possessing an orbit (θ(t), I(t)) drifting in the space ofactions, that is

|I(τ) − I(0)| ≥ 1

for some time τ = τ(ε) > 0, which is called the time of instability (or timeof diffusion).

3. However, showing that such drifting solutions exist in some large classof near-integrable Hamiltonian systems is a very difficult task. Indeed, onthe one hand KAM theory ([Kol54], [Mos62] and [Arn63]) gives, under somenon-degeneracy condition on h and provided ε is small enough, the existenceof a set of positive measure of quasi-periodic solutions. These solutions areperpetually stable, in the sense that the variation of their action componentsis of order at most

√ε for all time. Moreover, for n = 2 and if the integrable

Hamiltonian is isoenergetically non degenerate ([Arn63]), one can even showthat all solutions are stable for all time. On the other hand, for n ≥ 3we know from Nekhoroshev theory ([Nek77], [Nek79]) that all solutions arestable, not for all time, but for an interval of time which is exponentially longwith respect to ε−1, provided that h meets some quantitative transversalitycondition and the perturbation is sufficiently small.

But even though the existence of drifting orbits for near-integrable Hamil-tonian systems seems to be quite exceptional, Arnold conjectured that thistopological instability is in fact a “typical” phenomenon (see [Arn63] or[AKN06]). We refer to [Loc99] for a very lucid and enlightening discussionon Arnold mechanism (and on Arnold diffusion in general). Under some“generic” conditions and for n = 3, Mather ([Mat04]) announced that this

2

conjecture holds if the unperturbed Hamiltonian is convex (the convexity isrequired by the use of his variational methods). This is so far the best resultin this direction, however, his proof is highly technical and still incomplete.Another connected question, which is even harder, is to find orbits thatdensely fill the energy level (this is the so-called quasi-ergodic hypothesis): for instance, in [Her98] it is asked whether there exists a Hamiltoniansystem, C∞-smooth and for r ≥ 2, Cr-close to the integrable Hamiltonianh(I) = 1

2 |I|2, which has a dense orbit on an energy level (progress towardsthis question have been made recently in [KLS10] and [KZZ09]).

4. Yet there is another case, derived from Arnold’s original example, whichis much simpler and hence more studied and understood, in which the un-perturbed Hamiltonian is completely integrable (in the sense of symplecticgeometry) but possesses some “hyperbolicity”. The prototype for such asystem is given by an uncoupled product of rotators with a pendulum. Thiscase is usually referred to as a priori unstable (following the terminologyintroduced in [CG94]), in contrast with the a priori stable case where theunperturbed system is integrable in angle-action coordinates and thereforehas no hyperbolic feature (one can also say that a priori stable systems are“fully elliptic”). The interest in studying such a priori unstable systems isin fact double. First, one can use the original strategy of Arnold to findunstable orbits, by constructing and following a “transition chain” made ofsets with suitable hyperbolic properties (or minimizing properties when us-ing variational arguments). The second interest lies in the fact that, due tonormal form theory, the study of a priori stable systems in a neighbourhoodof a simple resonance reduces to the a priori unstable case (see [LMS03] forexample). Results on the topological instability of a priori unstable sys-tems under fairly general conditions can be found in [DdlLS06] and [Tre04]by means of geometrical methods and in [CY04], [Ber08] and [CY09] by vari-ational methods. However there are many difficulties in using these resultsto tackle Arnold’s conjecture which concerns a priori stable systems.

5. Another feature of a priori unstable systems is that one avoids all ex-ponentially small phenomena which are typical for analytic (or Gevrey) apriori stable systems, as the lower bound on the time of instability imposedby Nekhoroshev’s theorem. Indeed, if µ denotes the small parameter inthe a priori unstable case, then it was realized by Lochak that the timeof instability should be polynomial, and then Bernard ([Ber96]), adaptingBessi’s work ([Bes96]), showed that one can obtain an upper bound on thetime of instability which is of order µ−2. In [Loc99], it was conjectured thatthe optimal time of instability should be µ−1 lnµ−1, and this was provedby Berti, Bolle and Biasco [BBB03], using Bessi’s ideas and suitable a pri-ori estimates, obtained by convexity arguments, to localize the minima ofthe action functional. This time of instability was also found by Treschev([Tre04]) using fairly different geometric methods.

3

6. The goal of this article is to give yet another method to construct anexample with this optimal time of instability, which should be useful toobtain the optimal time of instability for a priori stable systems. Our ap-proach is dynamical: it uses the notion of polysystem introduced by Moeckel([Moe02]) in the context of Arnold diffusion, and which corresponds to therandom iteration of a family of maps (the term “polysystem” is due toMarco, such systems are just locally constant skew-products over a Bernoullishift and they are also known as iterated function systems). More precisely,our example uses an explicit construction of a polysystem, which can be seenas a concrete realisation of an abstract mechanism introduced by Marco in[Mar08]. Finally, let us also note that polysystems are a crucial ingredientin the recent work of Marco towards the instability for general a priori sta-ble systems with 3 degrees of freedom ([Mar10b], [Mar10c], [Mar10d] and[Mar10a]).

2 Main results

7. For n ≥ 1, let us recall that a function f ∈ C∞(An) is α-Gevrey, forα ≥ 1, if for any compact subset K ⊆ An there exist two positive constantsAK , BK such that

|∂kf |C0(K) ≤ AKB|k|K (k!)α, k ∈ N2n,

with the standard multi-index notation. We shall denote by Gα(An) thespace of such functions. For α = 1, these are exactly the analytic functions,but for α > 1, the space Gα(An) contains non zero compactly-supportedfunctions (we refer to [MS02], Appendix A, for more details). Our pertur-bation will be α-Gevrey for α > 1.

In all this paper, we will consider a Hamiltonian system h with twodegrees of freedom defined by

h(θ1, θ2, I1, I2) =1

2(I2

1 + I22 ) + cos 2πθ1, (θ1, θ2, I1, I2) ∈ A2,

which is the direct product of a pendulum P (θ1, I1) = 12I

21 + cos 2πθ1 on

the first factor and a standard rotator S(θ2, I2) = 12I

22 on the second factor.

It is the simplest example of an a priori unstable integrable Hamiltoniansystem.

Below we shall state two versions of our theorem, one for Hamiltonian dif-feomorphisms (Theorem 2.2) and one for Hamiltonian flows (Theorem 2.1).For the discrete case, our unperturbed diffeomorphism is the time-one mapof the Hamiltonian flow generated by h, Φh : A2 → A2. In the continu-ous case, our unperturbed system is the a priori unstable Hamiltonian withthree degrees of freedom defined by

h(θ, I) = h(θ1, θ2, I1, I2) + I3 =1

2(I2

1 + I22 ) + I3 + cos 2πθ1.

4

8. Let us state our main result.

Theorem 2.1. For α > 1, there exist positive constants C, µ0 and a func-tion f ∈ Gα(A3) such that if 0 < µ ≤ µ0, the Hamiltonian system

H(θ, I) = h(θ, I) + µf(θ, I), (θ, I) ∈ A3,

has an orbit (θ(t), I(t))t∈R such that

limt→±∞

I2(t) = ±∞,

with the estimates

|I2(τ) − I2(0)| ≥ 1, τ ≤ Cµ−1 lnµ−1.

We note that even though our perturbation is only Gevrey regular, us-ing the techniques developed in [LM05] one can obtain Theorem 2.1 in theanalytic case, but with considerably more work. However, as opposed to[BBB03] where their method works for an open set of perturbations, ourconstruction really requires a specific perturbation (this will be explainedbelow in section 3.2, point 3, when discussing the choice of our perturbation).

The unstable orbit constructed in the theorem above is bi-asymptoticto infinity, and the estimates show that the time of instability is of orderµ−1 lnµ−1. As we have already said, similar examples have been constructedby means of very different methods, in [BBB03], [Tre04] (see also [CG03]).Moreover, in [BBB03] the authors proved that for such a system the followingstability estimates hold true in the analytic case: given any ρ > 0, there existpositive constants µ′0 and c such that for µ ≤ µ′0,

|I(t) − I(0)| ≤ ρ, |t| ≤ cµ−1 lnµ−1.

Their proof relies on Nekhoroshev’s estimates, for both convex and steep in-tegrable systems, that lead to exponential stability in the part of the phasespace (away from the separatrices) where one can introduce angle-actioncoordinates for the integrable part (thus reducing the system to an a prioristable one), as well as some direct arguments leading to the time of stabilityµ−1 lnµ−1 close to the separatrices. Therefore this time of instability is op-timal within the analytic category. But as it is only polynomial (and in factalmost linear) and since the arguments close to the separatrices are essen-tially independent of the regularity, it should be also optimal in the Gevreycase and in the Ck case for k large enough, provided that the correspondingNekhoroshev’s estimates hold true. In [MS02] (resp. in [Bou10b]), expo-nential (resp. polynomial) stability estimates have been proved for Gevrey(resp. Ck) perturbations of quasi-convex systems. Therefore to obtain astability result similar to [BBB03] in lower regularity, one only needs themore general steep case, and this has been done in ([Bou10a]).

5

9. Let us recall that one of our main motivation in using a new mechanismto construct an unstable orbit with an optimal time of instability in this apriori unstable case, is to tackle the corresponding problem for the a prioristable case, which is a much more challenging problem. In fact, in this casethe optimal time of instability τ(ε) is of order exp (ε−a), for some positiveexponent a, and it was shown recently in [BM11] that one has the lowerbound

a ≥ (2α(n − 1))−1,

for α-Gevrey perturbations, with α ≥ 1 (this include the analytic case α =1). In the same paper, the authors conjectured that this value is indeedoptimal, in the sense that the upper bound a ≤ (2α(n − 1))−1 should alsohold (for the moment, one only knows that a ≤ (2α(n − 2))−1, this is dueto Herman, Marco and Sauzin ([MS02]) for the case α > 1, and to KeZhang ([Zha09]) for the more difficult analytic case α = 1). We believe thatusing some technical tools introduced in the present work, in particular theconstruction of our polysystem, one should finally be able to reach the valuea = (2α(n − 1))−1 (first in the easiest case α > 1).

Indeed, one can show that polysystems do appear in the a priori stablecase, as in the examples of [Mar05] and [LM05], our example being just ana priori unstable version of these. But of course, whereas in the a prioriunstable case, the splitting (which is the “distance” between stable andunstable manifolds of partially hyperbolic tori, or equivalently, the size ofthe drift in the action after one homoclinic excursion in the annulus) issimply of order µ, in the a priori stable case it is exponentially small (withrespect to ε) and therefore much more difficult to deal with. Nevertheless,we believe that one can “embed” our polysystem into an a priori stablesystem, either by refining the splitting estimates of [Mar05] and [LM05],or by using an embedding mechanism analogous to [MS02], in which theauthors use a very clever lemma to estimate the time of instability withoutany splitting estimates (they easily recover the latter as a consequence oftheir construction).

10. We shall not prove Theorem 2.1 directly, but the following equivalentversion in terms of diffeomorphisms. Given a function H on An, we shalldenote by ΦH

t : An → An the time-t map of its Hamiltonian flow and byΦH = ΦH

1 the time-one map.

Theorem 2.2. For α > 1, there exist positive constants C, µ0 and a func-tion f ∈ Gα(A2) such that if 0 < µ ≤ µ0, the diffeomorphism

Φh ◦ Φµf : A2 −→ A2

has an orbit (θk, Ik)k∈Z such that

limk→±∞

Ik2 = ±∞,

6

with the estimates

|IN2 − I0

2 | ≥ 1, N ≤ Cµ−1 lnµ−1.

Theorem 2.1 follows easily from Theorem 2.2 by a classical suspension ar-gument (see [MS02] for a simple method in the Gevrey case using compactly-supported functions), so we shall not repeat the details. Of course, the con-stants µ0, C and the function f are not the same in both theorems, butwe have kept the same notation for simplicity. Moreover, in the sequel wewill not give explicit values for these constants, in fact sometimes it will bemore convenient to use asymptotic notations: given u(µ) and v(µ) definedfor µ ≥ 0, we shall write u(µ) = O(v(µ)) if there exist positive constants µ′

and c′ independent of µ such that the inequality u(µ) ≤ c′v(µ) holds truefor 0 ≤ µ ≤ µ′.

11. The plan of this paper is the following. In section 3, we shall describethe perturbation. We will show that the perturbed system has an invariantnormally hyperbolic manifold, the stable and unstable manifolds of whichintersect transversely along a homoclinic annulus.

Then, this will be used in section 4 to show the following generalisationof the Birkhoff-Smale theorem to the normally hyperbolic case: near thehomoclinic manifold, there exist an invariant set on which a suitable iterateof the system is conjugated to a symbolic dynamic, more precisely to askew-product on the annulus A over a Bernoulli shift. Let us give a precisedefinition.

Given an alphabetA ⊆ N, we let ΣA = AZ be the Cantor set of bi-infinitesequence of elements in A. We will denote by σ = σA the left Bernoulli shifton ΣA, that is if n = (nk)k∈Z belongs to ΣA, then σ(n) = (n′k)k∈Z is definedby

n′k = nk+1, k ∈ Z.

Definition 2.3. Let M be a manifold. A skew-product on M over σ is amap G : ΣA ×M −→ ΣA ×M of the form

G(n, x) = (σ(n), Fn(x)), x ∈M,

where Fn : M →M is an arbitrary map, for n ∈ ΣA.

However we are not able to prove the existence of our orbit using di-rectly this symbolic dynamic. Hence in section 5, we will first constructthis orbit for a simplified model called polysystem, which appears as a ran-dom iteration of “standard” maps of the annulus, and which is close to ourskew-product. Here is the definition.

Definition 2.4. A polysystem on M is a skew-product over a Bernoulli shiftof the form

Fn(x) = fn0(x), n ∈ ΣA, x ∈M,

with fn : M →M for n ∈ A.

7

I1 I2

θ1 θ2

ΦP ΦS

•O

S

×

Figure 1: Integrable diffeomorphism F0

In other terms, a polysystem is just a locally constant skew-product overthe shift, and this special property will greatly help us in the constructionof our orbit. Moreover, polysystems are equivalent to iterated functionssystems on the annulus, and these dynamical systems are known to havedrifting orbits under generic assumptions (see [Moe02]). Then, in section 6,the orbit for the polysystem will be considered as a pseudo-orbit for theskew-product and, using some hyperbolicity of our orbit, we will concludethe existence of an orbit for the general skew-product using shadowing ar-guments.

Finally we have gathered in an appendix some estimates on the so-calledtime-energy coordinates for the simple pendulum that are used in section 4.

3 Construction of the perturbation

This section is devoted to a description of our system, which is similar tothe one introduced in [Mar05]. We will first consider the integrable case,and then explains the construction of the perturbation.

3.1 Integrable system

Our integrable diffeomorphism F0 : A2 → A2 is the time-one map of theHamiltonian flow generated by h, that is

F0 = Φh = Φ1

2(I2

1+I2

2)+cos 2πθ1 ,

which is the product of the pendulum map ΦP = Φ1

2I21+cos 2πθ1 and the

integrable twist map ΦS = Φ1

2I22 (see figure 1).

It is an “a priori unstable” map in the sense that it possesses an invariantnormally hyperbolic annulus. Indeed, let O = (0, 0) be the hyperbolic fixed

8

point of the pendulum map ΦP , its stable and unstable manifolds obviouslycoincide, and if S is the upper part of the separatrix, then

W±(O,ΦP ) = S = {(θ1, I1) ∈ A | I1 = 2 sinπθ1}.

Due to the product structure of the map F0, it is easy to see that the annulusA = O × A is invariant by F0, and one can check that it is symplectic forthe canonical structure of A2. But the most important feature is that thisannulus is normally hyperbolic, more precisely it is r-normally hyperbolicfor any r ∈ N, in the sense of [HPS97]. To see this, just decompose thetangent bundle of A2 along A as

TAA2 = TA⊕ Es ⊕ Eu,

where Es (resp. Eu) is the one-dimensional contracting (reps. expanding)direction associated to the hyperbolic fixed point. Then note that the re-striction of F0 to A coincides with the integrable twist map ΦS, hence thereis zero contraction and expansion in the tangent direction TA.

The 3-dimensional stable and unstable manifolds of A also coincide, andare given by the product

W±(A,F0) = S × A.

3.2 Perturbed system

Now we will describe the geometric properties of our perturbed system. Itwill be of the form

Fµ = Φµf ◦ F0,

where µ > 0 is the small parameter, and f : A2 → R a function of the form

f(θ1, θ2, I1, I2) = χ(θ1)f1(θ1, I1)f2(θ2)

to be defined precisely below.

1. First χ : T → R will be a bump function. Of course such a function canbe chosen to be α-Gevrey for α > 1 but not analytic. More precisely, let uschoose θ ∈ T such that

(θ, I) ∈ S =⇒ ΦP (θ, I) = (1 − θ, I).

Since the separatrix S is symmetric about the section {θ1 = 1/2}, θ is well-defined. Then choose any θ ∈ ]θ, 1/2[, and a function χ ∈ Gα(T), α > 1,such that

χ(θ) =

{1 if θ ∈ [θ, 1 − θ],

0 if θ /∈ [θ, 1 − θ].

9

Since χ is identically zero at 0, the perturbation Φµf is the identity on thenormally hyperbolic annulus A, and therefore the latter remains invariantand normally hyperbolic for the map Fµ. Moreover, as χ vanishes in aneighbourhood of 0, some pieces of the stable and unstable manifolds for Fµ

will coincide with the ones of F0. The use of a bump function here is onlyto simplify the construction, it can be relaxed but at the expense of a moreinvolved analysis (as in [LM05]).

2. Now to define the function f1 : A → R, we will use time-energy coordi-nates on the first factor, so we introduce the symplectic diffeomorphism

Ψ : E × A −→ E∗ × A

(θ1, I1, θ2, I2) 7−→ (τ, e, θ2, I2)

whereE = {(θ1, I1) ∈ A | 0 < θ1 < 1, I1 > 0}

and

E∗ = {(τ, e) ∈ R2 | e > −2, |τ | < 1

2T (e)}. (1)

We refer to Appendix A for more details on those coordinates. The functionf1 is simply defined by

f1(τ, e) = 1 − 1

2τ2, (τ, e) ∈ E∗,

and it will create a transverse intersection between the stable and unstablemanifolds W+(A,Fµ) and W−(A,Fµ).

3. Finally the function f2 will be defined by

f2(θ2) = −π−1(2 + sin 2πθ2), θ2 ∈ T.

The fact the f2 is nowhere zero on T (for any θ2 ∈ T, |f2(θ2)| ≥ π−1)will imply that W+(A,Fµ) and W−(A,Fµ) intersect transversely along anannulus, and the explicit form of f2 will be responsible for the non trivialdynamics along this homoclinic annulus. The existence of this homoclinicannulus is a crucial ingredient in our construction but it is also non-generic,and this explains why our method cannot handle more general perturbations.

4. Let us define the domains

D = ([θ, 1 − θ] × R+∗ ) × A ⊆ A2, D∗ = Ψ(D) ⊆ E∗ × A,

on which the perturbation (either in the original coordinates or in time-energy coordinates) is non zero, and

D = ([θ, 1 − θ] × R+∗ ) × A ⊆ A2, D∗ = Ψ(D) ⊆ E∗ × A.

10

For (τ, e, θ2, I2) ∈ D∗, our perturbation can be written explicitly as

Φµf (τ, e, θ2, I2) = (τ, e − µf ′1(τ)f2(θ2), θ2, I2 − µf1(τ)f′2(θ2))

= (τ, e + µτf2(θ2), θ2, I2 + 2µf1(τ) cos(2πθ2)), (2)

and otherwise the diffeomorphism Φµf is the identity.

5. In the sequel we shall need the following property.

Proposition 3.1. The immersed manifolds W+(A,Fµ) and W−(A,Fµ)intersect transversely along the annulus

Iµ = {(τ, e, θ2, I2) ∈ E∗ × A | τ = e = 0}.

Let us remark that this annulus is also given by

Iµ = {(θ1, I1, θ2, I2) ∈ A2 | θ1 = 1/2, I1 = 2}

in the original coordinates.

Proof. By definition of D, one easily sees that the sets

W+(A,F0) ∩ F0(D), W−(A,F0) ∩ F−10 (D),

are disjoint from D. Since Fµ and F0 coincide outside D, we can definepieces of stable and unstable manifolds by

W+θ

(A,Fµ) = W+(A,F0) ∩ F0(D) ⊆W+(A,Fµ)

andW−

θ(A,Fµ) = W−(A,F0) ∩ F−1

0 (D) ⊆W−(A,Fµ).

Therefore

F−1µ (W+

θ(A,Fµ)) ⊆W+(A,Fµ), Fµ(W−

θ(A,Fµ)) ⊆W−(A,Fµ). (3)

Then on the one hand,

F−1µ (W+

θ(A,Fµ)) = F−1

0 ◦ Φ−µf (W+θ

(A,Fµ))

= F−10 (W+

θ(A,Fµ))

= W+(A,F0) ∩ D,

and on the other hand,

Fµ(W−θ

(A,Fµ)) = Φµf ◦ F−10 (W−

θ(A,Fµ))

= Φµf (W−(A,F0) ∩ D).

11

Now in time-energy coordinates, one simply has

W±(A,F0) = {(τ, e, θ2, I2) ∈ E∗ × A | e = 0},

and therefore

W+(A,F0) ∩ D = {(τ, e, θ2, I2) ∈ D∗ | e = 0},

while, using the expression of the perturbation (2),

Φµf (W−(A,F0) ∩ D) = {(τ, e, θ2, I2) ∈ D∗ | e = µτf2(θ2)}.

Since f2 is nowhere zero, one easily sees that the manifolds F−1µ (W+

θ(A,Fµ))

and Fµ(W−θ

(A,Fµ)) intersect transversely along the annulus

Iµ = {(τ, e, θ2, I2) ∈ D∗ | τ = e = 0},

and the conclusion follows by (3).

4 Construction of a symbolic dynamic

6. In this section, we will take advantage of the fact that our diffeomorphism

Fµ : A2 → A2

possesses an invariant normally hyperbolic annulus A, whose stable andunstable manifolds intersect transversely along a homoclinic annulus Iµ.

In the case where the normally hyperbolic manifold is a point, under sucha transverse homoclinic intersection it is well-known that chaotic dynamicsarise: the system has an invariant Cantor set on which a suitable iterate isconjugated to a shift map (this is the Horseshoe theorem, due to Birkhoff,Smale and Alexeiev). In our more general situation, the symbolic dynamicis more complicated.

7. Recall from definition 2.3 that a skew-product (over σ) is completelydefined by the family of maps (Fn)n∈ΣA

: M → M , and the skew-productwill be denoted by [[Fn]]n∈ΣA

. Then one can easily see that a sequence(nk, xk)k∈Z ∈ A×M gives rise to an orbit (σk(n), xk)k∈Z ∈ ΣA ×M , wheren = (nk)k∈Z, for the skew-product [[Fn]]n∈ΣA

if and only if

Fσk(n)(xk) = xk+1, k ∈ Z.

In this section, we will need our alphabet A to contain integers of orderlnµ−1, but for subsequent arguments, we will also need it to contain integersn ∈ N as large as µ−

1

2 lnµ−1, so we may already fix

A = Aµ = {[lnµ−1], . . . , 2[µ−1

2 lnµ−1]}.

12

For simplicity, we shall get rid of the subscript µ.

8. Let Oµ ⊆ D∗ be a neighbourhood (in time-energy coordinates) of thehomoclinic annulus Iµ. For x ∈ Oµ, let us set

nx0 = inf{n ∈ N∗ | Fn

µ (x) ∈ Oµ} ∈ N∗ ∪ {+∞}and assuming nx

0 < +∞, we can define inductively

nxk = inf

{n ∈ N∗ | Fn

µ

(Fnx

k−1µ (x)

)∈ Oµ

}∈ N∗ ∪ {+∞},

for k ≥ 1 provided nxk−1 < +∞. Similarly we can define

nx−1 = inf{n ∈ N∗ | F−n

µ (x) ∈ Oµ} ∈ N∗ ∪ {+∞},and by induction

nx−k = inf

{n ∈ N∗ | Fn

µ

(Fnx

−k+1µ (x)

)∈ Oµ

}∈ N∗ ∪ {+∞}

if nx−k < +∞ for k ≥ 1.We will show below that there exists Λµ ⊆ Oµ such that for any x ∈ Λµ,

the doubly infinite sequence nx = (nxk)k∈Z is a well-defined element of ΣA. In

fact, Λµ is homeomorphic to a Cantor set of annuli, and we will be interestedin the dynamics restricted to this set. For that, following Moser ([Mos73])we define the transversal map

Fµ(x) = Fnx0

µ (x), x ∈ Oµ, nx0 < +∞,

which is the first return map to the neighbourhood Oµ, and if the sequencenx is well-defined, then so is Fn

µ (x) for n ∈ Z.

9. In the sequel, we will consider the discrete topology on the alphabet A,and the sets AN and ΣA = AZ will be endowed with the product topology,for which they are compact and metrizable. The goal of this section is toprove the following proposition.

Proposition 4.1. For µ small enough, there exist a neighbourhood Oµ, aset Λµ ⊆ Oµ invariant by the transversal map Fµ, a homeomorphism

Υµ : ΣA × A −→ Λµ

(n, (θ, I)) 7−→ (ψn(θ, I), (θ, I))

with ψn : A → E∗, where E∗ is defined in (1), and a family of maps

Fn(θ, I) = (θ + n0I, I + 2µf1(φn(θ, I)) cos 2π(θ + n0I)) , n ∈ ΣA, (θ, I) ∈ A,

such that for x = (τx, ex, θx, Ix) ∈ Λµ, then nx ∈ ΣA and

Fµ ◦ Υµ(nx, θx, Ix) = Υµ(σ(nx), Fnx(θx, Ix)).

Moreover the map φn : A → R is Lipschitz and satisfies

|φn|C0(A) = O(µ2π−1

), Lip(φn) = O

(µ2π− 1

2 lnµ−1). (4)

13

The statement of the above proposition seems complicated, but in fact itsimply means that the restriction of Fµ to the invariant set Λµ is conjugatedto the skew-product on A given by the maps

Fn(θ, I) = (θ+n0I, I+2µf1(φn(θ, I)) cos 2π(θ+n0I)), n ∈ ΣA, (θ, I) ∈ A.

The proof of this result is long and technical. We will first prove an abstractresult in section 4.1, which is contained in Proposition 4.4, and then we willapply this proposition to our example in section 4.2.

4.1 Symbolic dynamic

1. Here we will use the framework developed by Chaperon ([Cha04],[Cha08]), which was designed to obtain rather general invariant manifoldtheorems, in particular in the normally hyperbolic case.

Consider a complete metric space X and a complete subspace Y of ametric space F . We endow the product space X × F with the productmetric, that is for (x, y), (x′, y′) ∈ X × F we define

d((x, y), (x′, y′)) = sup{d(x, x′), d(y, y′)}.

Let us set Z = X × Y . Then we consider a Lipschitz map

h = (f, g) : Z −→ X × F,

and we make the following two assumptions.

(H1) There exists a constant ρ−10 > 0 such that for all x ∈ X, y, y′ ∈ Y

d(g(x, y), g(x, y′)) ≥ ρ−10 d(y, y′).

Hence, for all x ∈ X, the map

gx : Y −→ gx(Y )

is a bijection.

(H2) For all x ∈ X, we have Y ⊆ gx(Y ). Hence the map

g−1x : Y −→ Y

is well-defined, and ifG : Z −→ Y

is defined by G(x, y) = g−1x (y), then G is Lipschitz.

We will say that the map h satisfies hypothesis (H) if it satisfies bothhypotheses (H1) and (H2). Under these assumptions, we will consider three

14

positive constants ν0, σ0 and κ0 where ν0 is the Lipschitz constant of G withrespect to Y , that is

d(G(x, y), G(x, y′)) ≤ ν0d(y, y′), ∀x ∈ X, ∀y, y′ ∈ Y,

which from (H1) is smaller than ρ0, σ0 is the Lipschitz constant of G withrespect to X, that is

d(G(x, y), G(x′ , y)) ≤ σ0d(x, x′), ∀x ∈ X, ∀y, y′ ∈ Y,

and κ0 the Lipschitz constant of f . Let us now formulate another hypothesis.

(L) Assume that σ0 + ν0 max{κ0, 1} < 1. Hence for any x ∈ X, y ∈ Ythe maps

G(., y) : X −→ Y, G(x, .) : Y −→ Y,

are contractions.

The following result is due to Chaperon ([Cha04]).

Theorem 4.2 (Chaperon). With the previous notations, assume that Y isbounded and the map

h : Z −→ X × F

satisfies hypotheses (H) and (L). Then the set

Λ =⋂

n∈N

h−n(Z)

is the graph of a contracting map Φ : X → Y , with

Lip(Φ) ≤ σ0(1 − ν0κ0)−1 < 1.

2. The above theorem is concerned with the iteration of a single map h, butwith this formalism we obviously have an analogous result for the iterationof a family of maps. More precisely, consider a family of maps

hn = (fn, gn) : Z −→ X × F, n ∈ A,

where A is the given alphabet. We will assume that each map satisfieshypothesis (H) and a uniform version of hypothesis (L), that is:

(L’) supn∈A {σn + νn max{κn, 1}} < 1.

Then one can state the following proposition.

Proposition 4.3. With the previous notations, assume that Y is boundedand that for any n ∈ A the maps

hn : Z −→ X × F

15

satisfy hypotheses (H) and (L’). Then for any sequence n+ = (nk)k∈N ∈ AN,the set

Λ+(n+) =⋂

k∈N

(hnk◦ · · · ◦ hn0

)−1(Z)

is the graph of a contracting map Φn+: X → Y , with

Lip(Φn+) ≤ sup

n∈A{σn(1 − νnκn)−1} < 1.

Moreover, if we endow the space of continuous function C(X,Y ) with thetopology of pointwise convergence, then the map

n+ ∈ AN 7−→ Φn+∈ C(X,Y )

is continuous.

The proof is the same as in Theorem 4.2, with some obvious modifica-tions.

3. Now we consider two metric spaces F+ and F−, and two completesubspaces X ⊆ F− and Y ⊆ F+. Let V another complete metric space, and

Z = X × Y × V.

One has to think of V as a central direction and F− (resp. F+) as acontracting (resp. expanding) direction. We consider two families of maps(h+

n )n∈A and (h−n )n∈A of the form

h+n : Z −→ X × F+ × V, h−n : Z −→ F− × Y × V,

and we decompose them as h±n = (f±n , g±n ), where

f+n : Z −→ X × V, f−n : Z −→ Y × V

andg+n : Z −→ F+, g−n : Z −→ F−.

The hypotheses (H) and (L’) for h±n refer to these decompositions. Let usalso denote by

F+n : Z −→ V

the second component of the map f+n , and consider sets Z+

n ⊆ Z and Z−n ⊆ Z

such thatZ ∩ (h+

n )−1(Z) ⊆ Z+n , Z ∩ (h−n )−1(Z) ⊆ Z−

n ,

for any n ∈ A.

4. The aim of this section is to prove the following result.

16

Proposition 4.4. With the previous notations, assume that X,Y are bounded,V is locally compact and that for any n ∈ A the maps

h+n : Z −→ X × F+ × V, h−n : Z −→ F− × Y × V

satisfy hypotheses (H) and (L’). Let us also assume that for any n ∈ A,

(i) the maps (h±n )|Z±n

are invertible and

((h+

n )|Z+n

)−1= (h−n )|Z+

n,

((h−n )|Z−

n

)−1= (h+

n )|Z−n

;

(ii) the sets Z+n (resp. Z−

n ) are pairwise disjoint.

Then for any sequence n ∈ ΣA, the map h+n0

has an invariant set Λ andthere exist a homeomorphism

Υ : ΣA × V −→ Λ(n, v) 7−→ (Φn(v), v)

which conjugates h+n0|Λ

to the skew-product on V given by

Fn(v) = F+n0

(Φn(v), v), n ∈ ΣA, v ∈ V,

where Φn : V → X × Y is a contracting map, with

Lip(Φn) ≤ supn∈A

{(σ+n (1 − ν+

n κ+n )−1), (σ−n (1 − ν−n κ

−n )−1)} < 1.

As a consequence, the functions Fn : V → V , for n ∈ ΣA, are also definedby the following equation

h+n0

(Φn(v), v) = (Φn(Fn(v)), Fn(v)), v ∈ V. (5)

This remark will be useful later on to obtain the estimates (4) in Proposi-tion 4.4.

Proof. For n ∈ ΣA, let us define

n+ = (nk)k∈N ∈ AN, n− = (n−k)k∈N∗ ∈ AN∗

.

Since each family of maps (h+n )n∈A and (h−n )n∈A satisfies hypotheses (H)

and (L’), we can apply Proposition 4.3 so both sets

Λ+(n+) =⋂

k∈N

(h+nk

◦ · · · ◦ h+n0

)−1(Z)

andΛ−(n−) =

⋂

k∈N∗

(h−n−k◦ · · · ◦ h−n−1

)−1(Z)

17

are graphs of contracting maps

Φ+n+

: X × V −→ Y, Φ−n−

: Y × V −→ X,

that isΛ+(n+) = {(x,Φ+

n+(x, v), v) | x ∈ X, v ∈ V }

andΛ−(n−) = {(Φ−

n−(y, v), y, v) | y ∈ Y, v ∈ V }.

Moreover, one has

Lip(Φ+n+

) ≤ supn∈A

{σ+n (1 − ν+

n κ+n )−1} < 1, (6)

Lip(Φ−n−

) ≤ supn∈A

{σ−n (1 − ν−n κ−n )−1} < 1. (7)

Therefore for each v ∈ V , the maps

Φ+n+,v = Φ+

n+(., v) : X −→ Y, Φ−

n−,v = Φ−n−

(., v) : Y −→ X,

are also contracting, and so are the maps

Φ−n−,v ◦ Φ+

n+,v : X −→ X, Φ+n+,v ◦ Φ−

n−,v : Y −→ Y.

Since X and Y are complete, these maps have fixed points x(n, v) ∈ X andy(n, v) ∈ Y from the contraction principle, and by uniqueness they satisfy

Φ+n+

(x(n, v), v) = y(n, v), Φ−n−

(y(n, v), v) = x(n, v).

Now let us defineΛ(n) = Λ+(n+) ∩ Λ−(n−).

Then by the previous relation this set is non-empty since it is the graph ofthe contraction

Φn : V −→ X × Yv 7−→ (x(n, v), y(n, v)),

and, from (6) and (7),

Lip(Φn) ≤ supn∈A

{(σ+n (1 − ν+

n κ+n )−1), (σ−n (1 − ν−n κ

−n )−1)} < 1.

Moreover, as the maps

n+ ∈ AN 7→ Φ+n+

∈ C(X × V, Y ), n− ∈ AN∗ 7→ Φ−n−

∈ C(Y × V,X),

are continuous, from the contraction principle one also has the continuity ofthe map

n ∈ ΣA 7−→ Φn(v) ∈ X × Y.

18

Now setΛ =

⋃

n∈ΣA

Λ(n).

The fact that this set is invariant under h+n0

will follow from our condition(i). Indeed, if z ∈ Λ, then z ∈ Λ(n) for some n ∈ ΣA and so by definition

z ∈⋂

k∈N

(h+nk

◦ · · · ◦ h+n0

)−1(Z), z ∈⋂

k∈N∗

(h−n−k◦ · · · ◦ h−n−1

)−1(Z).

From the first relation we get

h+n0

(z) ∈⋂

k∈N∗

(h+nk

◦ · · · ◦ h+n1

)−1(Z) = Λ+(σ(n)+), (8)

and since by hypothesis

((h+

n0)|Z+

n0

)−1= (h−n0

)|Z+n0

and Λ(n) ⊆ Z+n0

, we get from the second relation

h+n0

(z) ∈⋂

k∈N

(h−n−k◦ · · · ◦ h−n0

)−1(Z) = Λ−(σ(n)−). (9)

Now (8) and (9) means exactly that h+n0

(z) ∈ Λ(σ(n)), so Λ is positivelyinvariant under h+

n0. In fact, a completely similar argument using condition

(i) shows thath+

n0(Λ(n)) = Λ(σ(n)),

and hence Λ is totally invariant under h+n0

. More generally, one obtains

h±nk−1◦ · · · ◦ h±n0

(Λ(n)) = Λ(σ±k(n)), k ∈ Z.

Next let us prove that as a consequence of (ii) the union

Λ =⋃

n∈ΣA

Λ(n)

is disjoint. Let n 6= n′, so there exists l ∈ Z such that nl 6= n′l. First supposethat l = 0, then on the one hand

Λ(n) ⊆ Λ(n+) ⊆ Z+n0

and on the other hand

Λ(n′) ⊆ Λ(n′+) ⊆ Z+n′

0

.

19

As Z+n0

and Z+n′

0

are disjoint by hypothesis, then so are Λ(n) and Λ(n′).

Now if l ≥ 1, we can assume without loss of generality that nk = n′k for0 ≤ k ≤ l − 1, and as before

h+nl−1

◦ · · · ◦ h+n0

(Λ(n)) = Λ(σl(n)) ⊆ Λ(σl(n)+) ⊆ Z+nl

andh+

nl−1◦ · · · ◦ h+

n0(Λ(n′)) = Λ(σl(n′)) ⊆ Λ(σl(n′)+) ⊆ Z+

n′l

,

so Λ(n) and Λ(n′) have to be disjoint. Finally, the case l ≤ −1 is completelysimilar using the hypothesis that Z−

n are pairwise disjoint for n ∈ A.To conclude, as Λ is a disjoint union, every point z ∈ Λ can be uniquely

written as z = (Φn(v), v) for v ∈ V and n ∈ ΣA, so the map

Υ : ΣA × V −→ Λ(n, v) 7−→ (Φn(v), v)

is a well-defined continuous bijection, and as V is locally compact, one cancheck that it is a homeomorphism with respect to the product topology onΣA × V . Then for z = (Φn(v), v) ∈ Λ(n), as h+

n0(z) ∈ Λ(σ(n)) we have from

the definition of F+n ,

h+n0

(z) =(Φσ(n)(F

+n0

(z)), F+n0

(z))

=(Φσ(n)(F

+n0

(Φn(v), v)), F+n0

(Φn(v), v)).

The last equality exactly means that the map

h+n0

: Λ −→ Λ

is conjugated by Υ to the skew-product

G : ΣA × V −→ ΣA × V(n, v) 7−→ (σ(n), F (n, v)).

where F (n, v) = F+n0

(Φn(v), v). This ends the proof.

4.2 Proof of Proposition 4.1

This section is entirely devoted to the proof of Proposition 4.1, this willbe done in several steps. We will have to use notations and estimates ontime-energy coordinates, contained in Appendix A, and this will requireto choose µ sufficiently small. Moreover, we shall use various coordinatesbut we shall keep the same notation for the diffeomorphisms expressed indifferent coordinates.

Step 1. Straightening of the invariant manifolds.

20

Using time-energy coordinates on the first factor, our homoclinic annulusis given by

Iµ = {(τ, e, θ2, I2) ∈ D∗ | τ = e = 0}and in a neighbourhood of it, the stable manifold of Fµ is given by

{(τ, e, θ2, I2) ∈ D∗ | e = 0}

while the unstable manifold is

{(τ, e, θ2, I2) ∈ D∗ | e = µf2(θ2)τ}.

Now we introduce the change of coordinates

Θ : E∗ × A −→ E ′∗ × A

(τ, e, θ2, I2) 7−→ (τ ′, e′, θ2, I2)

whereτ ′ = τ − (µf2(θ2))

−1e, e′ = (µf2(θ2))−1e.

Its inverse is given by

Θ−1 : E ′∗ × A −→ E∗ × A

(τ ′, e′, θ2, I2) 7−→ (τ ′ + e′, µf2(θ2)e′, θ2, I2).

It follows that in these new coordinates, denoting by D′∗ = Θ(D∗), the stable

manifold{(τ ′, e′, θ2, I2) ∈ D′

∗ | e′ = 0}and the unstable manifold

{(τ ′, e′, θ2, I2) ∈ D′∗ | τ ′ = 0}

are straightened out.

Step 2. Choice of the box Z.

Our goal is to use Proposition 4.4, so we will explain how to choosethe domain Z. Our central direction V = A will be the annulus given bythe coordinates (θ2, I2), our contracting and expanding directions will beone-dimensional, so X ⊆ F− = R and Y ⊆ F+ = R. We will choose

X = [−τ ′0, τ ′0], Y = [0, e′0],

with τ ′0 = c1µ2π−1, e′0 = c2µ

2π−1, and c1 > 2π, c2 > 2π. Eventually

Z = [−τ ′0, τ ′0] × [0, e′0] × A.

For (θ2, I2) ∈ A, let us also define the section

Z(θ2, I2) = [−τ ′0, τ ′0] × [0, e′0] × {θ2, I2},

21

e = µf2(θ2)τ

τ

e

−τ ′0 τ ′0

µf2(θ2)e′0

Z(θ2, I2)

Figure 2: Section Z(θ2, I2) of the domain Z

where all the constructions will take place (see figure 2).This domain Z is located above the homoclinic annulus Iµ, and for µ

small enough it is contained in the domain D′∗ where the manifolds are

straightened.For n ∈ A = {[lnµ−1], . . . , 2[µ−

1

2 lnµ−1]}, the point an = (0, en) ∈ D∗,or a′n = (−e′n, e′n) ∈ D′

∗, is by definition n-periodic for the pendulum mapΦP . We want the annulus {a′n}×A, which is n-periodic for the unperturbedmap F0, to be included in our domain Z, and this requires that e′n < e′0 and−τ ′0 < −e′n for any n ∈ A. First note that

e0 = µf2(θ2)e′0 = c2f2(θ2)µ

2π,

and as f2(θ2) ≥ π−1, c2f2(θ2) > 2 so e0 > 2µ2π. Then using (21) and the

fact that A = {[lnµ−1], . . . , 2[µ−1

2 lnµ−1]}, for µ is small enough (and hencen is large enough) one obtains

supn∈A

{en} < 2µ2π,

and therefore en < e0 is satisfied for any n ∈ A, which gives e′n < e′0.Similarly, we obtain

−τ ′0 < −2πµ2π−1 < −e′n.

Step 3. Construction of the domains Z+n .

For n ∈ A, the domains Z+n ∩ Z(θ2, I2) will be rectangles (see figure 3,

they are parallelograms in the coordinates (τ, e)), with vertices

A1n =

(−τ ′0,

en − δen

µf2(θ2), θ2, I2

), A2

n =

(−τ ′0,

en + δen

µf2(θ2), θ2, I2

),

22

A3n =

(τ ′0,

en + δen

µf2(θ2), θ2, I2

), A4

n =

(τ ′0,

en − δen

µf2(θ2), θ2, I2

),

where the parameter δen is defined by

δen =

c3µ2π−1

|T ′n|

, n ∈ A

for some constant c3 > max{c1 + c2, c1 + 2π}, and T ′n = T ′(en) (see Ap-

pendix A and (22)). Hence the domains Z+n are small neighbourhoods of

the annuli {a′n} × A, for an = (−e′n, e′n) ∈ D′∗.

For µ small enough, one can easily check from the definition of δen and

the mean value theorem that

n+ c3µ2π−1 ≤ T (en − δe

n) ≤ n+ 2c3µ2π−1 (10)

andn− 2c3µ

2π−1 ≤ T (en + δen) ≤ n− c3µ

2π−1. (11)

Let us show that these domains Z+n are pairwise disjoint. By construction

it is enough to prove that e′(A1n) > e′(A2

n+1). Choosing µ small so

n+ 2c3µ2π−1 < n+ 1 − 2c3µ

2π−1,

from (10) and (11) we get

T (en − δen) < T (en+1 + δe

n+1).

But as the period function T is decreasing, this gives

en − δen > en+1 + δe

n+1

which implies that e′(A1n) > e′(A2

n+1).

Let us also remark that by definition of our domain D′∗, one can ensure

that for µ small enough (and so n− T (e) is small enough)

Fk0 (Z+

n ) ∩ D′∗ = ∅, 0 < k < n. (12)

Step 4. Construction of the domains Z−n .

The construction is similar (see figure 4), namely Z−n ∩ Z(θ2, I2) is a

rectangle with vertices

B1n =

(− enµf2(θ2)

− δτn, e

′0, θ2, I2

), B2

n =

(− enµf2(θ2)

+ δτn, e

′0, θ2, I2

),

B3n =

(− enµf2(θ2)

+ δτn, 0, θ2, I2

), B2

n =

(− enµf2(θ2)

− δτn, 0, θ2, I2

),

23

τ

e

A1n

A2n

A3n

A4n

Z+n

Figure 3: Section Z+n (θ2, I2) of the domain Z+

n

where δτn is defined by

δτn =

c4µ2π−2

|T ′n|

,

for a constant c4 > πmax{c1+4π, c1+c2}, and T ′n = T ′(en) (see Appendix A

and (22)). As in the previous step, one can check that τ ′(B1n+1) > τ ′(B2

n)so the domains Z−

n are pairwise disjoint.

Step 5. Expressions of the maps F±nµ restricted to Z±

n .

For any (τ, e, θ2, I2) ∈ Θ−1(Z+n ), one has the following explicit expression

for the unperturbed map

Fn0 (τ, e, θ2, I2) = (τ + n− T (e), e, θ2 + nI2, I2).

In those coordinates, our perturbation is given by

Φµf (τ, e, θ2, I2) = (τ, e+ µτf2(θ2), θ2, I2 − µf1(τ)f′2(θ2)).

Then, from (12) we know that Fnµ = Φµf ◦Fn

0 when restricted to Θ−1(Z+n ),

so

Fnµ (τ, e, θ2, I2) = (τ + n− T (e), e + µ(τ + n− T (e))f2(θ2 + nI2),

θ2 + nI2, I2 − µf1(τ + n− T (e))f ′2(θ2 + nI2)).

Using the expression for Θ−1, for (τ ′, e′, θ2, I2) ∈ Z+n we compute

Fnµ (τ ′, e′, θ2, I2) = (−(f2(θ2 + nI2))

−1f2(θ2)e′,

(1 + f2(θ2 + nI2))−1f2(θ2))e

′ + τ ′ + n− T (µf2(θ2)e′),

θ2 + nI2,

I2 − µf1(τ′ + e′ + n− T (µf2(θ2)e

′))f ′2(θ2 + nI2)).

24

τ

e

B3n

B4n

B1n B2

n

Z−n

Figure 4: Section Z−n (θ2, I2) of the domain Z−

n

If we fix (θ2, I2) ∈ A, then the first two components of this map are linearwith respect to τ ′ and e′, therefore the image of the parallelogram Z+

n bythe map Fn

µ is still a parallelogram (in a different section), with vertices

Ain = Fn

µ (Ain) for i = 1, 2, 3, 4 which can be explicitly computed.

Similarly, for (τ ′, e′, θ2, I2) ∈ Z−n we compute

F−nµ (τ ′, e′, θ2, I2) = ((1 + (f2(θ2))

−1f2(θ2))τ′ + e′ − k + T (−µf2(θ2)τ

′),

(f2(θ2))−1f2(θ2)τ

′,

θ2 + nI2 − nµf1(τ′ + e′)f ′2(θ2),

I2 + µf1(τ′ + e′)f ′2(θ2)),

so the image of Z−n by the map F−n

µ is a parallelogram with vertices Bin =

F−nµ (Bi

n) for i = 1, 2, 3, 4.

Step 6. Relative position of Z±n and F±n

µ (Z±n ).

Here we will prove that the figures (5) and (6) make sense, that is we willshow that the horizontal (resp. vertical) edges of Fn

µ (Z+n ) (resp. F−n

µ (Z−n ))

are not contained in Z. The upper horizontal edge of Z+n is the segment

joining A2n to A3

n, therefore the upper horizontal edge of Fnµ (Z+

n ) is the

segment joining A2n to A3

n. We can compute

e′(A2n) =

(1

f2(θ2 + nI2)+

1

f2(θ2)

)µ−1en − τ ′0 + n− T (en + δe

n)

and

e′(A3n) =

(1

f2(θ2 + nI2)+

1

f2(θ2)

)µ−1en + τ ′0 + n− T (en + δe

n).

25

τ

e

A1n

A2n

A3n

A4n

A1n A4

n

A3nA2

n

Z+n

Fnµ (Z+

n )

Figure 5: Position of Z+n and Fn

µ (Z+n )

From (11) we have T (en + δen) ≤ n − c3µ

2π−1, and as τ ′0 = c1µ2π−1 and

c3 > c1 + c2 this gives

e′(A2n) > −τ ′0 + n− T (en + δe

n)

> (c3 − c1)µ2π−1 > c2µ

2π−1 > e′0.

We also havee′(A3

n) = e′(A2n) + 2τ ′0 > e′0,

and so the upper horizontal edge of Z+n is not contained in Z.

For the lower horizontal edge of Fnµ (Z+

n ), which is the segment joining

A4n to A1

n, one has to prove that e′(A4n) < 0 and e′(A1

n) < 0. We compute

e′(A4n) =

(1

f2(θ2 + nI2)+

1

f2(θ2)

)µ−1en + τ ′0 + n− T (en − δe

n)

≤ 2πµ−1en + (c1 − 2c3)µ2π−1

≤ (c1 − 2c3 + 2π)µ2π−1,

and as 2c3 > c3 > c1 + 2π, this gives e′(A4n) < 0. We also have

e′(A1n) = e′(A4

n) − 2τ ′0 < 0,

and so the lower horizontal edge of Z+n is not contained in Z.

Similarly, one can check that the vertical edges of F−nµ (Z−

n ) are notcontained in Z, and this follows from the choice of the constant c4.

Step 7. Definition of the maps h±n .

Our maps h+n and h−n will be suitable extensions of the maps Fn

µ andF−n

µ . First we set

(h+n )|Z+

n= (Fn

µ )|Z+n, (h−n )|Z−

n= (F−n

µ )|Z−n,

26

τ

e

B3n

B4n

B1n B2

n F−nµ (Z−

n )Z−n

B1n

B4n

B2n

B3n

Figure 6: Position of Z−n and F−n

µ (Z−n )

and we want to define Lipschitz extensions of h±n to Z in order to have

Z ∩ (h±n )−1(Z) ⊆ Z±n .

Let us begin with the maps h+n . Take z = (τ ′, e′, θ2, I2) ∈ Z \ Z+

n , thenwe can find a unique z = (τ ′, e′(z), θ2, I2) ∈ Z+

n : indeed, either

en + δen

µf2(θ2)< e′ < e′0

in which case we choose

e′(z) =en + δe

n

µf2(θ2),

or

0 < e′ <en − δe

n

µf2(θ2)

and then we choose

e′(z) =en − δe

n

µf2(θ2).

Then we set

h+n (z) = Fn

µ (z) + (0, α+n (e′(z) − e′(z)), 0, 0),

for some positive constant α+n yet to be chosen. Hence the map

h+n : Z −→ [−τ ′0, τ ′0] × F+ × A

is a well-defined Lipschitz extension of Fnµ and, using the form of the exten-

sion, one can check that Z ∩ (h+n )−1(Z) ⊆ Z+

n .

27

For the maps h−n , this is completely analogous. For z = (τ ′, e′, θ2, I2) ∈Z \ Z−

n , then we can find a unique z = (τ ′(z), e′, θ2, I2) ∈ Z−n where

τ ′(z) = − enµf2(θ2)

+ δτn

if− enµf2(θ2)

+ δτn < τ ′ < τ ′0,

orτ ′(z) = − en

µf2(θ2)− δτ

n

if−τ ′0 < τ ′ < − en

µf2(θ2)+ δτ

n.

Then we define

h−n (z) = F−nµ (z) + (α−

n (τ ′(z) − τ ′(z)), 0, 0, 0)

for some constant positive α−n . The map

h+n : Z −→ F− × [0, e′0] × A

is a well-defined Lipschitz extension of F−nµ , with Z ∩ (h−n )−1(Z) ⊆ Z−

n .

Step 8. Verification of Hypotheses (H) and (L’).

Now let us show that the maps h+n and h−n satisfy hypotheses (H) and

(L’). Recall that

X = [−τ ′0, τ ′0] ⊆ F− = R, Y = [0, e′0] ⊆ F+ = R,

and let us write h+n = (f+

n , g+n ) where

f+n : X × Y × A −→ X × A

andg+n : X × Y × A −→ F+.

For x ∈ X × A, consider the partial map

g+n,x : Y −→ F+.

By step 6 (see the figure 5), the image under (h+n )|Z+

n= (Fn

µ )|Z+n

of the

horizontal edges of Z+n do not belong to Z, and this implies that

Y = [0, e′0] ⊆ g+n,x(Y ∩ Z+

n ) ⊆ g+n,x(Y ),

so the hypothesis (H2) is satisfied.

28

Now it remains to show hypothesis (H1) and (L’). This follows from thechoice of α+

n and lengthy calculations of the various partial derivatives ofFn

µ , using both the explicit expression obtained in step 5 and the estimatesof Appendix A. We find the following: if we define

α+n = π−1(µ|T ′

n|),

then (H1) is satisfied with

ρ+n = O

(µ|T ′

n|),

and we estimate the Lipschitz constants

ν+n = O

(µ−1|T ′

n|−1), σ+

n = O(µ2π−1

), κ+

n = O(µ2π−1|T ′

n|),

so (L’) is satisfied for µ small enough as

σ+n + ν+

n max{1, κ+n } = σ+

n + ν+n κ

+n = O

(µ2π−1

).

The situation for h−n is of course similar.

Step 9. Conclusions.

All the hypotheses of Proposition 4.4 are satisfied, so we can apply itto any sequence n ∈ A: there exist a set Λ ⊆ Z, invariant by h+

n0, and a

homeomorphism

Υ : ΣA × A −→ Λ(n, (θ, I)) 7−→ (Φn(θ, I), (θ, I))

which conjugates h+n0|Λ

to the skew-product on A given by

Fn(θ, I) = F+n0

(Φn(θ, I), θ, I), (θ, I) ∈ A,

where Φn : A → X × Y is a contracting map, with

Lip(Φn) ≤ supn∈A

{(σ+n (1 − ν+

n κ+n )−1), (σ−n (1 − ν−n κ

−n )−1)} = O

(µ2π−1

). (13)

Recall also from (5) that

h+n0

(Φn(θ, I), θ, I) = (Φn(Fn(θ, I)), Fn(θ, I)), (θ, I) ∈ A. (14)

So we let

Oµ = Θ−1(Z), Λµ = Θ−1(Λ) ⊆ Oµ, Υµ = Θ−1 ◦ Υ.

For x ∈ Λµ,

h+nx

0(x) = Fnx

0µ (x) = Fµ(x),

29

where Fµ is the transversal map of Fµ associated to Oµ. Then by def-inition of Λµ, the sequence nx is a well-defined element of ΣA. Settingx = (τx, ex, θx, Ix) ∈ Λµ, the above conjugacy gives

Fµ ◦ Υµ(nx, θx, Ix) = Υµ(σ(nx), Fnx(θx, Ix)).

Now it remains to explain the form of Fn and the estimates (4). We canwrite

Υµ : ΣA × A −→ Λµ

asΥµ(n, (θ, I)) = (ψn(θ, I), (θ, I)) = (τn(θ, I), en(θ, I), θ, I),

where τn : A → R and en : A → R are easily deduced from

Φn = (Φ1n,Φ

2n) : A −→ X × Y

and from Θ−1. Now by definition of X and Y ,

|Φ1n|C0(A) = O

(µ2π−1

), |Φ2

n|C0(A) = O(µ2π−1

),

and this gives

|τn|C0(A) = O(µ2π−1

), |en|C0(A) = O

(µ2π

).

Moreover using (13) these maps are Lipschitz and

Lip(τn) = O(µ2π−1

), Lip(en) = O

(µ2π−1

).

Now from (14) we can write

Fn(θ, I) = (θ+n0I, I+2µf1(φn(θ, I)) cos 2π(θ+n0I)), n ∈ ΣA, (θ, I) ∈ A,

where the Lipschitz function φn : A → R satisfies the equation

φn(θ, I) = τσ(n)(θ + n0I, I + 2µf1(φn(θ, I)) cos 2π(θ + n0I)). (15)

The estimate|φn|C0(A) = O

(µ2π−1

)

is obvious, but for the Lipschitz constant this is more subtle. Let us define

gn(θ, I) = (θ + n0I, I + 2µf1(φn(θ, I)) cos 2π(θ + n0I)).

Then on the one hand, using the definition of f1 we can estimate

Lip(gn) ≤ sup{1 + µ−1

2 lnµ−1,Lip(φn)},but on the other hand, from (15)

Lip(φn) ≤ Lip(τn)Lip(gn) < Lip(gn),

soLip(gn) = O

(µ−

1

2 lnµ−1)

and thereforeLip(φn) = O

(µ2π− 1

2 lnµ−1).

This concludes the proof.

30

5 Construction of a pseudo-orbit

1. In this section, we will restrict our study to a special type of skew-product over a Bernoulli shift, which is called “polysystem” (see [Mar08], itis also known as Iterated Function System).

Recall from definition 2.4 that a polysystem is a skew-product (over σ)such that for any n = (nk)k∈Z ∈ ΣA, one has Fn = fn0

. So a polysystem doesnot depend on the whole sequence n ∈ ΣA but only on its first componentn0 ∈ A, hence instead of being defined by a family of maps indexed by ΣA, itis defined by a family of maps indexed by A. Then one can easily see that asequence (nk, xk)k∈Z ∈ A×M gives rise to an orbit (σk(n), xk)k∈Z ∈ ΣA×M ,where n = (nk)k∈Z, for the polysystem [[fn]]n∈A if and only if

fnk(xk) = xk+1, k ∈ Z,

and its projection onto M corresponds to the iteration of the maps (fn)n∈A

in the order prescribed by the sequence n.

2. In the previous section, we showed that the dynamics of the first returnmap of our diffeomorphism Fµ in a neighbourhood of the homoclinic annulusis conjugated to the skew-product map

Fn(θ, I) = (θ+n0I, I+2µf1(φn(θ, I)) cos 2π(θ+n0I)), n ∈ ΣA, (θ, I) ∈ A.

This family of maps depend on the whole sequence n ∈ ΣA, but as φn issmall, f1(φn(θ, I)) is close to one, and so Fn is close to

fn0(θ, I) = (θ + n0I, I + 2µ cos 2π(θ + n0I)), n0 ∈ A, (θ, I) ∈ A.

These “standard” maps can be seen as perturbations of iterates of the inte-grable twist map T (θ, I) = (θ+I, I): more precisely we can write fn = V ◦T n

whereV (θ, I) = (θ, I + 2µ cos 2πθ)

is a “vertical” map which is close to identity if µ is small.We will first construct an orbit for the polysystem [[fn]]n∈A defined by

the maps fn, and in the next section, we will use it as a pseudo-orbit for theskew-product [[Fn]]n∈ΣA

defined by the maps Fn.

3. The goal of this section is to prove the following proposition.

Proposition 5.1. There exists a positive constant C such that for µ smallenough, there exists an orbit (σk(n), xk)k∈Z ∈ ΣA × A for the polysystem[[fn]]n∈A defined by

fn(θ, I) = (θ + nI, I + 2µ cos 2π(θ + nI)), n ∈ A, (θ, I) ∈ A,

31

such thatlim

k→±∞Ik = ±∞,

and the estimates

|IN − I0| ≥ 2,N∑

k=0

nk ≤ Cµ−1 lnµ−1,

hold true.

We will see that this proposition holds only if we can choose the integersnk, k ∈ Z, as large as µ−

1

2 lnµ−1, so this explains the choice of

A = {[lnµ−1], . . . , 2[µ−1

2 lnµ−1]}.

The upper bound on the sum of the integers nk needed to produce a driftof order one is basically the “time of instability” in this context.

In the sequel, we shall only explain the construction of the positive se-quence (nk, θk, Ik)k≥0, since, of course, the construction of the negative se-quence (nk, θk, Ik)k≤−1 is completely similar.

4. Let us now describe the construction of our orbit. In all this section wewill need to introduce two real numbers 0 < K < K ′ <

√2/2. First, we fix

K such that 0 < K <√

2/2 and we define the non-empty domain

BK = {(θ, I) ∈ A | cos 2πθ ≥ K, sin 2πθ ≤ −K}.

We shall only need the first condition cos 2πθ ≥ K in this section, the othercondition sin 2πθ ≤ −K will be used in the next section.

One can also write

BK = {(θ, I) ∈ A | − arccosK

2π≤ θ ≤ −arcsinK

2π[Z]},

that is BK = Jδ × R (see figure 7), where Jδ is an interval of T of length

δ =arccosK

2π− arcsinK

2π.

Now K being fixed, we shall also consider K ′ ∈ ]K,√

2/2[ such that thedomain BK ′ = Jδ/2 × R, where Jδ/2 is an interval of length δ/2. Now recallthat we have fn = V ◦ T n with

V (θ, I) = (θ, I + 2µ cos 2πθ), T (θ, I) = (θ + I, I), (θ, I) ∈ A

and by definition, the map V leaves the set BK invariant and produce inBK a drift in the I-direction which is at least equal to 2µK by the conditioncos 2πθ ≥ K. Therefore, to prove the first part of our proposition, we

32

0 K

−K

T

Jδ

Figure 7: Interval Jδ (projection of BK onto T)

will construct a sequence of points (θk, Ik)k∈Z ∈ A for which we can find asequence of integers (nk)k∈Z ∈ A such that

T nk(θk, Ik) ∈ BK . (16)

Indeed, in this case (θk+1, Ik+1) = fnk(θk, Ik) = V ◦ T nk(θk, Ik) satisfies

Ik+1 − Ik ≥ 2µK.

Then to prove our second part, we shall need estimates on these integers nk,k ∈ Z. In fact the relation (16) can be written as

Rnk

Ik(θk) = θk + nkIk ∈ Jδ ,

where RI is the rotation on the circle T of angle I. Most of the time, thatis if either Ik is irrational or if Ik = p/q with |q| ≥ δ−1, this will be realized,and the integers nk, k ∈ Z, can be estimated. This is an easy consequenceof the “ergodization” theorem recalled below, that we shall use crucially inour construction.

5. For I ∈ R, consider the rotation RI(θ) = θ + I defined on the circle T.Given 0 < δ < 1, let Jδ be the set of intervals of T of length δ. We definethe δ-ergodization time N(I, δ) ≤ +∞ by

N(I, δ) = inf{n ∈ N | {θ, . . . , RnI (θ)} ∩ J 6= ∅, ∀θ ∈ T, ∀J ∈ Jδ},

or equivalently

N(I, δ) = inf{n ∈ N |⋃

0≤k≤n

R−kI (J) = T, ∀J ∈ Jδ}.

33

One can easily see that N(I, δ) < +∞ except if I is a rational number witha denominator smaller than δ−1, but it is more difficult to prove that whenit is defined, this number is essentially given by the inverse of the distanceto these “bad” rationals. This is the content of the theorem below, which isdue to Berti, Biasco and Bolle ([BBB03]).

Theorem 5.2 (Berti-Bolle-Biasco). There exist a positive constant M suchthat if

Rδ = {p/q ∈ Q | |q| ≤Mδ−1},then for I ∈ R \ Rδ,

N(I, δ/2) ≤ d(I,Rδ)−1.

This is a consequence of Theorem 4.2 in [BBB03] (see also the estimate(5.3) in [BBB03]), where the above proposition is proved both for the con-tinuous and multi-dimensional case. Of course one can give an explicit valuefor the numerical constant M but this will not be useful.

In fact, most of the time we shall use this result in the following form.

Lemma 5.3. Let I ∈ R \ Rδ, θ ∈ T and J ⊆ T any interval of length δ/2.Then for any m ∈ N, one can find an integer

n ∈ [m,m+ d(I,Rδ)−1]

such that θ + nI ∈ J .

6. In order to construct our orbit, now we know that we have to considerthe distance to this set of “bad rationals” Rδ.

So given µ > 0, we first define the domain of “fast drift”

DF (µ) =

{(θ, I) ∈ A | d(I,Rδ) ≥

1

lnµ−1

},

the domain of “slow drift”

DS(µ) =

{(θ, I) ∈ A | µ

1

2

lnµ−1< d(I,Rδ) <

1

lnµ−1

},

and finally the domain of “resonances”

DR(µ) =

{(θ, I) ∈ A | d(I,Rδ) ≤

µ1

2

lnµ−1

}.

Obviously we have

A = DF (µ) ∪DS(µ) ∪DR(µ).

34

But in the sequel, it will be more convenient to further decompose thesesets. Indeed, for p/q ∈ Rδ, if we define

DS(µ, p/q) =

{(θ, I) ∈ A | µ

1

2

lnµ−1< |I − p/q| < 1

lnµ−1

}

and

DR(µ, p/q) =

{(θ, I) ∈ A | |I − p/q| ≤ µ

1

2

lnµ−1

},

then, for µ small enough, one has

DS(µ) =⋃

p/q∈Rδ

DS(µ, p/q), DR(µ) =⋃

p/q∈Rδ

DR(µ, p/q)

and

A = DF (µ)⋃

⋃

p/q∈Rδ

DS(µ, p/q)

⋃

⋃

p/q∈Rδ

DR(µ, p/q)

. (17)

From now on we will assume that µ is sufficiently small, with respect to δ andM which are fixed, so the above decomposition (17) is in fact a partitionof A (since the set Rδ is discrete) and moreover the following propertieshold true: for any (θ, I) ∈ DS(µ), there exist a unique p/q ∈ Rδ such that(θ, I) ∈ DS(µ, p/q), and if

m = inf{n ∈ N∗ | fn(θ, I) /∈ DS(µ, p/q)},

then m is well defined and necessarily the point (θm, Im) = fm(θ, I) doesnot belong to DS(µ) (either it is in DR(µ) or DF (µ), since the size of thejump, which is of order µ, is much smaller than the width of the connectedcomponents of DR(µ) or DF (µ)). We also ask the same for a point (θ, I) ∈DR(µ): under the iteration of a map fn the first time it escapes the domainDR(µ, p/q), it also escapes the domain DR(µ) (in fact it enters into thedomain DS(µ)). This situation is depicted in figure 8.

7. The construction of our orbit will be inductive, and we will start with apoint (θ0, I0) ∈ DF (µ). Then we have the following easy application of theergodization theorem.

Lemma 5.4. Let (θ, I) ∈ DF (µ). There exists an integer

n ∈ {[lnµ−1], . . . , 2[lnµ−1]}

such that T n(θ, I) ∈ BK ′.

35

p1

q1

p2

q2

DR(µ)

DF (µ)

DS(µ)

Figure 8: Domains DF (µ), DS(µ) and DR(µ)

Proof. Recall that T n(θ, I) ∈ BK ′ if and only if θ + nI ∈ Jδ/2. Now bydefinition of DF (µ),

d(I,Rδ) ≥1

lnµ−1

so the conclusion follows from Lemma 5.3 by choosing m = [lnµ−1].

8. As long as the orbit stays inDF (µ), we can use the previous lemma. Thenit will enter into the domain of slow drift, and this is where the ergodizationtheorem gives integers as large as µ−

1

2 lnµ−1. However in the lemma belowwe will see how, after iterating a finite number of maps fnk

, nk ∈ A, with∑nk of order µ−1 lnµ−1, one can actually cross through this domain of slow

drift.

Lemma 5.5. Let (θ, I) ∈ DS(µ). There exist an index j ∈ N and integersn0, . . . , nj satisfying:

(i) nk ∈ A, for 0 ≤ k ≤ j;

(ii) fnk◦ · · · ◦ fn0

(θ, I) ∈ BK ′, for 0 ≤ k ≤ j;

(iii) fnj◦ · · · ◦ fn0

(θ, I) /∈ DS(µ).

Moreover, if j ∈ N∗, then

(iv)∑j

k=0 nk ≤ (j + 1) lnµ−1 + (2K ′)−1µ−1 lnµ−1.

Let us point out that in the proof of Proposition 5.1 below, the abovelemma will always be used with j ≥ 1 so item (iv) will be available.

36

Proof. There exists a unique p/q ∈ Rδ such that (θ, I) ∈ DS(µ, p/q). ByLemma 5.3 (with m = [lnµ−1]) and since

d(I,Rδ) = |I − p/q|−1 ≤ µ−1

2 lnµ−1,

we can find an integer n0 ∈ A such that fn0(θ, I) ∈ BK ′ . Now if fn0

(θ, I) /∈DS(µ), then we can take j = 0 in the statement and assertions (i), (ii) and(iii) are proven.

Otherwise, we construct inductively (nk)1≤k≤j where j ∈ N is defined by

j = inf{k ∈ N∗ | fnk◦ · · · ◦ fn0

(θ, I) /∈ DS(µ, p/q)}.

Since our orbit always stays in B′K , then j is obviously well-defined, and at

each step we have used Lemma 5.3 so conditions (i) and (ii) are satisfied.Moreover by a previous remark we can also write

j = inf{k ∈ N∗ | fnk◦ · · · ◦ fn0

(θ, I) /∈ DS(µ)},

sofnj

◦ · · · ◦ fn0(θ, I) /∈ DS(µ),

and condition (iii) is also satisfied.Finally, let us write (θ0, I0) = (θ, I) and (θk+1, Ik+1) = fnk

◦ · · · ◦fn0

(θ0, I0), for 0 ≤ k ≤ j. Since these points belong to BK ′ we haveIk+1 − Ik ≥ 2µK ′ for 0 ≤ k ≤ j and hence

j∑

k=0

nk ≤j∑

k=0

lnµ−1 +

j∑

k=0

1

|Ik − p/q|

≤ (j + 1) lnµ−1 + (2K ′)−1µ−1j∑

k=0

Ik+1 − Ik|Ik − p/q| .

The second term on the right-hand side is a Riemann sum, hence for µ smallenough it can be estimated by an integral, namely

j∑

k=0

Ik+1 − Ik|Ik − p/q| ≤ 2

∫

µ12

ln µ−1≤|I−p/q|≤ 1

ln µ−1

dI

|I − p/q| = 2

∫ 1

lnµ−1

µ12

ln µ−1

dI

I,

and as

2

∫ 1

lnµ−1

µ12

ln µ−1

dI

I= 2 lnµ−

1

2 ≤ lnµ−1,

this eventually gives

j∑

k=0

nk ≤ (j + 1) lnµ−1 + (2K ′)−1µ−1 lnµ−1.

This is exactly (iv), and so this ends the proof.

37

9. Now that we have escaped the domain of slow drift, we are in theresonant domain. Here we cannot use any ergodization result. However ourpoint belongs B′

K , and in the lemma below we will show that by iteratingj times a map fn, with n of order lnµ−1, it can cross the resonant domainwhile staying in the larger domain BK .

Lemma 5.6. Let (θ, I) ∈ DR(µ) ∩ BK ′. If µ is small enough, there existintegers

n ∈ {[lnµ−1], . . . , 2[lnµ−1]}and j ∈ N∗ such that:

(i) fkn(θ, I) ∈ DR(µ) ∩BK , for 0 ≤ k ≤ j − 1;

(ii) f jn(θ, I) /∈ DR(µ).

Note that it is possible that the point f jn(θ, I) does not belong to BK

either, but then we are back in the domain of slow drift and we can find aninteger n′ ∈ A such that T n′

(f jn(θ, I)) ∈ BK ′ ⊆ BK .

Proof. There exists a unique p/q ∈ Rδ such that (θ, I) ∈ DR(µ, p/q) ∩BK ′.We choose µ small enough so

Mδ−1 ≤ lnµ−1,

and as q ≤Mδ−1, we can find a integer d such that n = dq satisfies

n ∈ {[lnµ−1], . . . , 2[lnµ−1]}.

We will also add a further smallness condition on µ by requiring that

2

K lnµ−1<

1

2πmax{arccosK − arccosK ′, arcsinK ′ − arcsinK}.

Let us define inductively

fkn(θ, I) = (θk, Ik), 0 ≤ k ≤ j,

where j is defined as follows: j = inf{j1, j2} with

j1 = inf{k ∈ N∗ | (θk, Ik) /∈ DR(µ, p/q)}

and

j2 =

[(Kµ

1

2 lnµ−1)−1

]+ 1.

It follows from the definition of j that (θk, Ik) ∈ DR(µ), for 0 ≤ k ≤ j − 1,so now let us show that (θk, Ik) ∈ BK .

38

In fact we will prove below by induction on k that

−arccosK ′

2π− knµ

1

2

lnµ−1≤ θk ≤ −arcsinK ′

2π+knµ

1

2

lnµ−1, 0 ≤ k ≤ j − 1. (18)

Since n ≤ 2 lnµ−1 this implies

−arccosK ′

2π− 2kµ

1

2 ≤ θk ≤ −arcsinK ′

2π+ 2kµ

1

2 ,

and as k ≤ j2, then

−arccosK ′

2π− 2

K lnµ−1≤ θk ≤ −arcsinK ′

2π+

2

K lnµ−1,

−arccosK

2π≤ θk ≤ −arcsinK

2π.

This means that cos 2πθk ≥ K and sin 2πθk ≤ −K, and therefore this gives(θk, Ik) ∈ BK for 0 ≤ k ≤ j − 1.

So now let us go through the induction, that is let us prove (18). This isobviously true for k = 0, so let us assume it is satisfied for some 0 ≤ k ≤ j−2.We have

θk+1 = θk + nrk,

but since Ik ≥ p/q − µ1

2 / lnµ−1 and n = dq then

θk+1 ≥ θk + dp− nµ1

2 / lnµ−1 = θk − nµ1

2 / lnµ−1 [Z],

as dp is an integer. Then, using the hypothesis of induction this gives

θk+1 ≥ −arccosK ′

2π− knµ

1

2

lnµ−1− nµ

1

2

lnµ−1

≥ −arccosK ′

2π− (k + 1)nµ

1

2

lnµ−1.

Similarly using the fact that Ik ≤ p/q + µ1

2 / lnµ−1 one obtains

θk+1 ≤ −arcsinK ′

2π+

(k + 1)nµ1

2

lnµ−1,

therefore (18) is proven and so is (i).Now assume that j = j2, then since (θk, Ik) ∈ BK for 0 ≤ k ≤ j − 1,

Ij2 ≥ I0 + 2j2Kµ > I0 + 2µ1

2 / lnµ−1 > p/q + µ1

2/ lnµ−1,

which means that j2 ≥ j1. This is absurd, therefore j = j1, so (θj , Ij) /∈DR(µ, p/q) and this means that (θj , Ij) /∈ DR(µ). This proves (ii).

39

10. Now we can finally conclude the proof of Proposition 5.1.

Proof of Proposition 5.1. We choose any point (θ0, I0) ∈ DF (µ). Applyingsuccessively Lemma 5.4, Lemma 5.5, Lemma 5.6 and Lemma 5.5 once again,in this precise order, one obtains a positive sequence (nk, θk, Ik)k∈N ∈ A×A

which gives a positive orbit for our polysystem, that is

fnk(θk, Ik) = (θk+1, Ik+1), k ∈ N.

Note that since we have started in DF (µ), at each time Lemma 5.5 is appliedwith an integer j ≥ 1.

Now by construction, T nk(θk, Ik) ∈ BK for any k ∈ N, and since

(θk+1, Ik+1) = V ◦ T nk(θk, Ik),

by definition of BK one obtains

Ik+1 ≥ Ik + 2µK.

This clearly shows thatlim

k→+∞Ik = +∞,

which proves the first part of the statement.Then as

Ik − I0 ≥ 2kµK,

if we set N = [(µK)−1] + 1,

IN − I0 ≥ 2.

It remains to estimate the sum of integers

S =

N∑

k=0

nk.

To do that, we will write

σ1 = {k ∈ [0, N ] | (θk, Ik) ∈ DF (µ) ∪DR(µ)}

andσ2 = {k ∈ [0, N ] | (θk, Ik) ∈ DS(µ)},

soS =

∑

k∈σ1

nk +∑

k∈σ2

nk = S1 + S2.

For each k ∈ σ1, we know that nk ≤ 2 lnµ−1 and hence

S1 ≤ 2∑

k∈σ1

lnµ−1 ≤ 2N lnµ−1 ≤ 4K−1µ−1 lnµ−1.

40

To estimate S2, first observe that the set Rδ is discrete, so its intersectionwith the compact set {I0 ≤ I ≤ IN} is finite. But the latter set is includedin {I0 ≤ I ≤ I0 + 3}, so the constant

M = card{p/q ∈ Rδ | I0 ≤ p/q ≤ I0 + 3},

is independent of µ. Then setting

σ2(p/q) = {k ∈ [0, N ] | (θk, Ik) ∈ DS(µ, p/q)},

we obtainS2 =

∑

k∈σ2

nk ≤M∑

k∈σ2(p/q)

nk.

Now each σ2(p/q) is not reduced to a point, so by Lemma 5.5∑

k∈σ2(p/q)

nk ≤ N lnµ−1 + (2K ′)−1µ−1 lnµ−1

≤ 2K−1µ−1 lnµ−1 + (2K ′)−1µ−1 lnµ−1

≤(2K−1 + (2K ′)−1

)µ−1 lnµ−1.

This finally gives

S ≤ (4K−1)µ−1 lnµ−1 +M(2K−1 + (2K ′)−1

)µ−1 lnµ−1

≤ Cµ−1 lnµ−1,

with C = 2 sup{4K−1,M

(2K−1 + (2K ′)−1

)}, and this proves the proposi-

tion.

6 Proof of Theorem 2.2

11. In this section we will prove Theorem 2.2, and in view of Proposi-tion 4.1, this will follow easily from the next result.

Proposition 6.1. There exists a positive constant C such that for µ smallenough, there exists an orbit (σk(n), xk)k∈Z ∈ ΣA × A for the skew-product[[Fn]]n∈A defined by

Fn(θ, I) = (θ+n0I, I+2µf1(φn(θ, I)) cos 2π(θ+n0I)), n ∈ ΣA, (θ, I) ∈ A,

such thatlim

k→±∞I ′k = ±∞

and the estimates

|I ′N − I ′0| ≥ 1,

N∑

k=0

nk ≤ Cµ−1 lnµ−1,

hold true.

41

The strategy will be to consider the orbit constructed in Proposition 5.1as a pseudo-orbit for the skew-product. So in order to find a true orbitnearby, we shall need some hyperbolicity and this will be described below.

12. Recall that

BK = {(θ, I) ∈ A | cos 2πθ ≥ K, sin 2πθ ≤ −K}.

The condition sin 2πθ ≤ −K will be used here to prove the following lemma.

Lemma 6.2. Let x ∈ A such that T n(x) ∈ BK , for n ∈ A, and x ∈ R2 alift of x. Then the eigenvalues λ± of dfn(x) are real and for µ small enough,they satisfy

λ+ > 1 + 2√nπµK > 1, λ− < 1 − 2

√nπµK + 4nπµK < 1.

Moreover, if e± ∈ R2 are eigenvectors associated to λ±, and for v ∈ R2,v = v+e+ + v−e−, then

1

2|v| ≤ sup{|v+|, |v−|} ≤ a|v|,

with a = O(µ−

3

4 (lnµ−1)1

2

), and where | . | is the supremum norm on R2.

Proof. Let us write x = (θ, I) ∈ A and x = (θ, I) ∈ R2. If we define

s = −2πµ sin 2π(θ + nI),

then s > 2πµK since T n(θ, I) = (θ + nI, I) ∈ BK . As

fn(θ, I) = (θ + nI, I + 2µ cos 2π(θ + nI)), n ∈ A, (θ, I) ∈ A,

we have

dfn(θ, I) =

(1 n2s 1 + 2ns

)∈M2(R).

Then the eigenvalues are real and given by

λ± = 1 + ns±√ns(2 + ns).

Therefore we easily obtain

λ+ > 1 +√

2ns > 1 + 2√nπµK,

and then using the equality λ+λ− = 1, for µ small enough one finds

λ− < 1 − 2√nπµK + 4nπµK.

This proves the first part of the statement.

42

Then if we define

α± = n−1(λ± − 1) = s±√n−1s(2 + ns),

one can easily check that the vectors

e± =

(1α±

)∈ R2,

are eigenvectors associated to λ±. Since |e+| = |e−| = 1, for v ∈ R2 if wewrite

||v|| = sup{|v+|, |v−|}, v = v+e+ + v−e−,

it is trivial that|v| ≤ |v+| + |v−| ≤ 2||v||.

As for the other inequality, note that

|v| = sup{|v+ + v−|, |α+v+ + α−v−|}.

If we define

r± =α+α− − 1 ± (α+ − α−)

1 − α2+

,

then one can check that

|α+v+ + α−v−| ≤ |v+ + v−|

if and only if v− = 0, or (v−)−1v+ ≤ r−, or (v−)−1v+ ≥ r+. Hence

|v| =

{|v+ + v−| if v− = 0, or (v−)−1v+ ≤ r−, or (v−)−1v+ ≥ r+, ,

|α+v+ + α−v−| if (v−)−1v+ ≥ r−, or (v−)−1v+ ≤ r+.

If we study all the cases, one finds

||v|| ≤ sup{1, |1 + r+|−1, |1 + r−1

+ |−1, |1 + r−|−1, |1 + r−1− |−1

}|v|.

Then we can estimate

r± = −1 ±√

2n−1s+ o(√

n−1µ),

and using the fact that n ∈ A, one finds

||v|| ≤ a|v|,

with a = O(µ−

3

4 (lnµ−1)1

2

).

43

13. Let us now describe an abstract fixed point theorem that will be usedto find an orbit close to our pseudo-orbit.

Consider a Banach space (E, | . |) and T : E → E a continuous linearmap. Recall that the spectrum Sp(T ) of T is the set of complex numbers λsuch that TC − λIdC is not an automorphism of EC, where EC and TC arethe complexifications of E and T .

Given two real numbers κs, κu satisfying 0 < κs < 1 < κu, we say thatT is (κs, κu)-hyperbolic if

Sp(T ) ∩ {κs ≤ |z| ≤ κu} = ∅.

In such a case, there exists a T -invariant decomposition

E = Es ⊕ Eu, T (Es) ⊆ Es, T (Eu) ⊆ Eu,

and a constant c > 0 such that

|(T|Es)n| ≤ cηn

s , |(T|Eu)−n| ≤ cηn

u ,

for any n ∈ N, ηs < κs, ηu > κu and where | . | is the induced norm on linearoperators.

In fact, one can always find a norm ‖ . ‖ on E which is adapted to T inthe following sense: ‖ . ‖ is equivalent to | . | and satisfies

(i) ‖xs + xu‖ = sup{‖xs‖, ‖xu‖}, xs ∈ Es, xu ∈ Eu;

(ii) ‖T|Es‖ ≤ κs, ‖(T|Eu

)−1‖ ≤ κ−1u .

The following theorem is proved in [Yoc95], section 2.1.

Theorem 6.3. Let T : E → E be (κs, κu)-hyperbolic and let U : E → E bea Lipschitz map such that

ε = Lip(U − T ) < ε0 = inf{1 − κs, 1 − κ−1u }.

Then U has a unique fixed point p ∈ E, and if ‖ . ‖ is a norm adapted to T ,

‖p‖ < (ε0 − ε)−1‖U(0)‖.

14. Now we can prove Proposition 6.1.

Proof of Proposition 6.1. Consider the orbit (σk(n), θk, Ik)k∈Z ∈ ΣA × A

given by Proposition 5.1. Then the proof is an immediate consequenceof the following claim: there exist a sequence (θ′k, I

′k)k∈Z ∈ A such that

(σk(n), θ′k, I′k)k∈Z ∈ ΣA × A is an orbit for the skew product and

|Ik − I ′k| ≤ µ2, k ∈ Z.

44

So let us construct this orbit.Let xk = (θk, Ik)k∈Z ∈ A, and xk = (θk, Ik)k∈Z one of its lift in R2. First

we define a linear map

T : (R2)Z −→ (R2)Z

v 7−→ T (v)

by(T (v))k = dfnk−1

(xk−1).vk−1, k ∈ Z.

Using the supremum norm on R2 let us define

E = {v = (vk)k∈Z ∈ (R2)Z | supk∈Z

|vk| <∞}.

Then E is obviously a Banach space with the norm

|v| = supk∈Z

|vk|, v ∈ E.

Now recall that for any x = (θ, I) ∈ R2, if s = −2πµ sin 2π(θ + nI), then

dfn(x) =

(1 n2s 1 + 2ns

)∈M2(R).

Since the norm on M2(R) induced by the supremum norm on R2 is givenby the maximum of the sums of the absolute values of the elements in eachrow, we obtain

supk∈Z

{|dfnk

|C0(R2)

}= sup

k∈Z

{1 + nk} = 1 + µ−1

2 lnµ−1 <∞,

then T (F ) ⊆ F and therefore T is continuous. Moreover, by constructionthe sequence (xk)k∈Z satisfies T nk(xk) ∈ BK , so we can apply Lemma 6.2and each map

Tk−1 : vk−1 7−→ dfnk−1(xk−1).vk−1

is a (κk−1s , κk−1

u )-hyperbolic linear map of R2, with

κks = 1 − 2

√nkπµK + 4nkπµK < 1, k ∈ Z,

andκk

u = 1 + 2√nkπµK > 1, k ∈ Z.

This implies that T is (κs, κu)-hyperbolic, with

κs = supk∈Z

{κks}, κu = inf

k∈Z

{κku}.

Since nk ∈ A, one finds

κs = 1 − 2√πK(lnµ−1)

1

2µ1

4 + 4πK lnµ−1µ1

2 < 1,

45

andκu = 1 + 2

√πK(lnµ−1)

1

2µ1

2 > 1.

Let us setε0 = inf{1 − κs, 1 − κ−1

u } = O(µ

1

2 (lnµ−1)1

2

).

Now if‖v‖ = sup

k∈Z

‖vk‖k,

where ‖ . ‖k is a norm in R2 adapted to Tk, k ∈ Z, then ‖ . ‖ is adapted toT , and from Lemma 6.2

1

2|v| ≤ ‖v‖ ≤ a|v|, v ∈ E,

with a = O(µ−

3

4 (lnµ−1)1

2

).

Next we defineU : (R2)Z −→ (R2)Z

v 7−→ U(v)

by(U(v))k = Fσk−1(n)(xk−1 + vk−1) − xk, k ∈ Z.

Note that if v is a fixed point of U , then

Fσk−1(n)(xk−1 + vk−1) = xk + vk,

so (σk(n), x′k)k∈Z ∈ ΣA × A, where x′k ∈ A is the projection onto A ofx′k = xk + vk ∈ R2, is an orbit for the skew-product. To prove that U has afixed point, we will use Theorem 6.3 and for that we need to estimate theLipschitz constant of ∆ = U − T with respect to the adapted norm ‖ . ‖.

First if v, v′ ∈ E satisfy |v| ≤ µ2, |v| ≤ µ2, then from Taylor formula onecan compute

|∆(v) − ∆(v′)| ≤ (L+ 2Mµ2)|v − v′|,with

L = supk∈Z

{Lip

(fnk

− Fσk(n)

)}, M = sup

k∈Z

{|d2fnk

|C0(R2)

}.

Using the estimates (4) obtained in Proposition 4.1, one finds

L = O(µ4π−3/2 lnµ−1

),

whereasM = O

((lnµ−1)2

)

is obvious from the definition of the maps (fn)n∈A.

46

However, far away from 0, the maps U and T are not necessarily close,so for v ∈ (R2)Z we define U(v) by

U(v) =

{U(v) if |v| ≤ µ2,

U(µ2|v|−1v

)+

(1 − µ2|v|−1

)T (v) if |v| ≥ µ2.

Then setting ∆ = U − T , for any v, v′ ∈ E one easily obtains

|∆(v) − ∆(v′)| ≤ 2(L+ 2Mµ2)|v − v′|,

and this gives

‖∆(v) − ∆(v′)‖ ≤ 4a(L+ 2Mµ2)‖v − v′‖.

In particular, this shows U(F ) ⊆ F , and the Lipschitz constant of U − Twith respect to the adapted norm ‖ . ‖ is

ε = Lip(U − T ) ≤ 4a(L+ 2Mµ2) = O(µ

5

4 (lnµ−1)5

2

).

Asε = O

(µ

5

4 (lnµ−1)5

2

)< ε0 = O

(µ

1

2 (lnµ−1)1

2

)

we can finally apply Theorem 6.3: U has a unique fixed point v′ ∈ E suchthat

‖v′‖ ≤ (ε0 − ε)−1‖U(0)‖,and hence

|v′| ≤ 2a(ε0 − ε)−1|U (0)|.Note that

|U (0)| = supk∈Z

{|Fσk(n)(xk) − xk+1|} = supk∈Z

{|Fσk(n)(xk) − fnk(xk)|}.

Using the estimates (4) in Proposition 4.1,

supk∈Z

|Fσk(n)(xk) − fnk(xk)| = O

(µ4π−1

)

and as 2a(ε0 − ε)−1 = O(µ−5/4

), we have in particular

|v′| ≤ µ2. (19)

By definition of U , this shows that v′ is in fact a fixed point of U , andtherefore the sequence (σk(n), x′k)k∈Z ∈ ΣA × A, where x′k is the projectiononto A of

x′k = xk + v′k = (θ′k, I′k), k ∈ Z,

is an orbit for the skew-product. If we set x′k = (θ′k, I′k) ∈ A, then from (19)

we obtain|Ik − I ′k| ≤ µ2, k ∈ Z.

This concludes the proof.

47

15. Now we can eventually prove Theorem 2.2.

Proof of Theorem 2.2. Let (σk(n), θ′k, I′k)k∈Z ∈ ΣA × A be the orbit for the

skew-product obtained in Proposition 6.1. Then, by Proposition 4.1,

Υµ(nk, θ′k, I

′k) = (τk, ek, θ

′k, I

′k) ∈ E × A, k ∈ Z,

and in the original coordinates

Ψ−1(τk, ek, θ′k, I

′k) = (θk

1 , Ik1 , θ

k2 , I

k2 ) ∈ A2, k ∈ Z,

is an orbit for the transversal map Fµ. By definition of the latter one has

Fnkµ (θk

1 , Ik1 , θ

k2 , I

k2 ) = (θk+1

1 , Ik+11 , θk+1

2 , Ik+12 ), k ∈ Z,

and since I ′k = I2k for k ∈ Z, the theorem follows.

A Time-energy coordinates for the pendulum

In this appendix, we recall some elementary facts about the time-energycoordinates for the simple pendulum. We refer to [MS04], [Mar05] and[LM05] for more details.

16. Consider the simple pendulum defined by the Hamiltonian

P (θ, I) =1

2I2 + cos 2πθ,

and the open domain

E = {(θ, I) ∈ A | 0 < θ < 1, I > 0}.

We define the energy function

e(θ, I) = P (θ, I) − 1 =1

2I2 + cos 2πθ − 1

and, using {θ = 1/2} as a reference section, the time function

τ(θ, I) =

∫ θ

1

2

dθ√2(e(θ, I) − V (θ))

where V (θ) = cos 2πθ − 1. For positive energy e > 0, the period of motionis given by

T (e) =

∫ 1

2

− 1

2

dθ√2(e− V (θ))

and it is a decreasing function. Moreover, we have the equivalent

T (e) ∼0 −(2π)−1 ln e. (20)

48

For negative energy e < 0, if θ(e) is such that cos 2πθ(e) = 1 + e, then

T (e) =

∫ θ(e)

−θ(e)

dθ√2(e − V (θ))

.

Therefore if we define

E∗ = {(τ, e) ∈ R2 | e > −2, |τ | < 1

2T (e)}

we have a diffeomorphism

Ψ : E −→ E∗(θ, I) 7−→ (τ, e)

and one can check that it is symplectic. Moreover, in those coordinates (τ, e)the flow of the pendulum is straightened out, that is

ΦtP (τ, e) = (τ + t, e).

for t small enough.

17. To conclude, for the proof of Proposition 4.1 in section 4.2 we shallneed some estimates on the energy and the period of the periodic orbits (ofpositive energy) for the pendulum. For n ∈ N∗, we let en be the energy ofthe n-periodic orbit for ΦP , that is T (en) = n, we have

en ∼+∞ exp(−2πn). (21)

Then if we define T ′n = T ′(en), T ′′

n = T ′′(en) where T ′ and T ′′ are the firstand second derivatives of the period function, we can deduce from (20) and(21) that

T ′n ∼+∞ −(2π)−1 exp(2πn) (22)

andT ′′

n ∼+∞ 2π(T ′n)2 = (2π)−1 exp(4πn). (23)

Acknowledgments.The authors are thankful to their advisor Jean-Pierre Marco for his guid-

ance, for many helpful discussions, comments and corrections. The firstauthor thanks the University of Warwick, the Marie Curie training net-work “Conformal Structures and Dynamics” and the Instituto Nacional deMatematica Pura e Aplicada (IMPA) for their support.

References

[AKN06] V.I. Arnold, V.V. Kozlov, and A.I. Neishtadt, Mathematical as-pects of classical and celestial mechanics, [Dynamical SystemsIII], Transl. from the Russian original by E. Khukhro, Thirdedition, Encyclopedia of Mathematical Sciences, 3 ed., Springer-Verlag, Berlin, 2006.

49

[Arn63] V.I. Arnol’d, Proof of a theorem of A.N. Kolmogorov on theinvariance of quasi-periodic motions under small perturbations,Russ. Math. Surv. 18 (1963), no. 5, 9–36.

[Arn64] V.I. Arnold, Instability of dynamical systems with several degreesof freedom, Sov. Math. Doklady 5 (1964), 581–585.

[BBB03] M. Berti, L. Biasco, and P. Bolle, Drift in phase space: a newvariational mechanism with optimal diffusion time, J. Math.Pures Appl. (9) 82 (2003), no. 6, 613–664.

[Ber96] P. Bernard, Perturbation of a partially hyperbolic hamilto-nian system (Perturbation d’un hamiltonien partiellement hyper-bolique), C. R. Math. Acad. Sci. Paris 323 (1996), no. 2, 189–194.

[Ber08] , The dynamics of pseudographs in convex Hamiltoniansystems, no. 3, 615–669.

[Bes96] U. Bessi, An approach to Arnold’s diffusion through the calculusof variations., Nonlinear Anal., Theory Methods Appl. 26 (1996),no. 6, 1115–1135.

[BM11] A. Bounemoura and J.-P. Marco, Improved exponential stabilityfor quasi-convex Hamiltonians, Nonlinearity 24 (2011), no. 1,97–112.

[Bou10a] A. Bounemoura, Effective stability for Gevrey and finitely differ-entiable prevalent Hamiltonians, Submitted (2010).

[Bou10b] , Nekhoroshev theory for finitely differentiable quasi-convex Hamiltonians, Journal of Differential Equations 249

(2010), no. 11, 2905–2920.

[CG94] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space,Ann. Inst. Henri Poincare, Phys. Theor. 60 (1994), no. 1, 1–144.

[CG03] J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion,Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 451–470 (English).

[Cha04] M. Chaperon, Stable manifolds and the Perron-Irwin method,Erg. Th. Dyn. Sys. 24 (2004), no. 5, 1359–1394.

[Cha08] , The lipschitzian core of some invariant manifold theo-rems, Erg. Th. Dyn. Sys. 28 (2008), no. 5, 1419–1441.

[CY04] C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a prioriunstable Hamiltonian systems, J. Differ. Geom. 67 (2004), no. 3,457–517.

50

[CY09] , Arnold diffusion in Hamiltonian systems: a priori un-stable case, J. Differ. Geom. 82 (2009), no. 2, 229–277.

[DdlLS06] A. Delshams, R. de la Llave, and T.M. Seara, A geometricmechanism for diffusion in Hamiltonian systems overcoming inthe large gap problem: Heuristics and rigorous verification on amodel., Mem. Am. Math. Soc. 844 (2006), 141 p. (English).

[Her98] M. Herman, Some open problems in dynamical systems, Doc.Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 1998,pp. 797–808.

[HPS97] M.W. Hirsch, C.C. Pugh, and M. Shub, Invariant Manifolds,Springer, Berlin, 1997.

[KLS10] V. Kaloshin, M. Levi, and M. Saprykina, An example of nearlyintegrable Hamiltonian system with a trajectory dense in a set ofmaximal Hausdorff dimension, Preprint (2010).

[Kol54] A.N. Kolmogorov, On the preservation of conditionally periodicmotions for a small change in Hamilton’s function, Dokl. Akad.Nauk. SSSR 98 (1954), 527–530.

[KZZ09] V. Kaloshin, K. Zhang, and Y. Zheng, Almost dense orbit onenergy surface, Proceedings of the XVI-th ICMP, Prague, 314-322, 2009.

[LM05] P Lochak and J.P. Marco, Diffusion times and stability exponentsfor nearly integrable analytic systems, Central European Journalof Mathematics 3 (2005), no. 3, 342–397.

[LMS03] P. Lochak, J.-P. Marco, and D. Sauzin, On the splitting of invari-ant manifolds in multidimensional near-integrable Hamiltoniansystems, Mem. Am. Math. Soc. 775 (2003), 145 p.

[Loc99] Pierre Lochak, Arnold diffusion ; a compendium of remarks andquestions., Simo, Carles (ed.), Hamiltonian systems with threeor more degrees of freedom. Proceedings of the NATO AdvancedStudy Institute, 1995. Dordrecht: Kluwer Academic Publishers.,1999.

[Mar05] J.-P. Marco, Uniform lower bounds of the splitting for analyticsymplectic systems, preprint (2005).

[Mar08] , Models for skew-products and polysystems, C. R. Math.Acad. Sci. Paris (2008), no. 3-4, 203–208.

[Mar10a] , Arnold diffusion in a priori stable systems on A3, Inpreparation (2010).

51

[Mar10b] , Generic properties of classical systems on the torus T2,Preprint (2010).

[Mar10c] , Nets of hyperbolic annuli in generic nearly integrablesystems on A3, Preprint (2010).

[Mar10d] , Skew-products and polysystems in the neighborhood ofhyperbolic annuli, In preparation (2010).

[Mat04] J. Mather, Arnold diffusion I : Announcement of results, J. ofMath. Sciences 124 (2004), no. 5, 5275–5289.

[Moe02] R. Moeckel, Generic drift on Cantor sets of annuli, Chenciner,Alain (ed.) et al., Celestial mechanics. Dedicated to Donald Saarifor his 60th birthday. Proceedings of an international confer-ence, Northwestern Univ., Evanston, IL, USA, December 15–19, 1999. Providence, RI: AMS, American Mathematical Society.Contemp. Math. 292, 163-171, 2002.

[Mos62] J. Moser, On Invariant curves of Area-Preserving Mappings ofan Annulus, Nachr. Akad. Wiss. Gottingen II (1962), 1–20.

[Mos73] , Stable and random motions in dynamical systems. Withspecial emphasis on celestial mechanics. Hermann Weyl Lectures.The Institute for Advanced Study, Annals of Mathematics Stud-ies. No.77. Princeton, N. J.: Princeton University Press and Uni-versity of Tokyo Press. VIII, 199 pp., 1973.

[MS02] J.-P. Marco and D. Sauzin, Stability and instability for Gevreyquasi-convex near-integrable Hamiltonian systems, Publ. Math.Inst. Hautes Etudes Sci. 96 (2002), 199–275.

[MS04] , Wandering domains and random walks in Gevrey near-integrable systems, Erg. Th. Dyn. Sys. 5 (2004), 1619–1666.

[Nek77] N.N. Nekhoroshev, An exponential estimate of the time of sta-bility of nearly integrable Hamiltonian systems, Russian Math.Surveys 32 (1977), 1–65.

[Nek79] , An exponential estimate of the time of stability of nearlyintegrable Hamiltonian systems II, Trudy Sem. Petrovs 5 (1979),5–50.

[Tre04] D. Treschev, Evolution of slow variables in a priori unstableHamiltonian systems, Nonlinearity 17 (2004), no. 5, 1803–1841.

52

[Yoc95] J.-C. Yoccoz, Introduction to hyperbolic dynamics, Branner,Bodil (ed.) et al., Real and complex dynamical systems. Proceed-ings of the NATO Advanced Study Institute held in Hillerød,Denmark, June 20-July 2, 1993. Dordrecht: Kluwer AcademicPublishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 464, 265-291, 1995.

[Zha09] Ke Zhang, Speed of Arnold diffusion for analytic Hamiltoniansystems, Preprint (2009).

53