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Arnold's Diffusion: from the a priori unstable to the a priori stable case

We expose some selected topics concerning the instability of the action variables in a priori unstable Hamiltonian systems, and outline a new strategy that may allow to apply these methods to a priori stable systems.


Introduction
A very classical problem in dynamics consists in studying the Hamiltonian system on the symplectic manifolds T * T n = T n × R n generated by the Hamiltonian H : T × T * T n = T × T n × R n −→ R (t, x, y) −→ 1 2 y 2 + G(t, x, y) where is a small perturbation parameter. More general unperturbed systems h(y) can be considered instead of y 2 /2, but we restrict to that particular case in the present paper in order to simplify some notations. For = 0, the system is integrable, and the momenta y are integrals of motion. For > 0, these variables undergo small oscillations. KAM theory implies that these oscillations remain permanently bounded for many initial conditions. For other initial conditions, a large evolution might be possible. By Nekhoroshev theory, it must be extremely slow. The questions we discuss in the present text is whether this large evolution is actually possible, and to what geometric structures it is associated. Let us consider a resonant momentum y 0 = (I 0 , 0) ∈ R m ×R r = R n , and assume that I 0 is not resonant, which means that k·I 0 never belongs to Z for k ∈ Z m , k = 0. In order to study the dynamics near the torus {y = y 0 }, it is useful to introduce the notations x = (θ, q) ∈ T m × T r , and y = (I, p) ∈ R m × R r , m + r = n. In the neighborhood of the torus {y = y 0 }, the dynamics is approximated by the averaged system 1 2 where V (q) = T×T m G(t, θ, q, y 0 )dθdt.
Following a classical idea of Poincaré and Arnold, we can try to exploit this observation by considering the system H(t, θ, q, I, p) = 1 2 p 2 + 1 2 I 2 − V (q) − µR(t, θ, q, I, p) with a second perturbation parameter µ independent from . We assume that V has a unique non-degenerate minimum, say at q = 0. Fixing > 0, we can study this system for µ > 0 small enough, which is a simpler problem which may give some hints about the dynamics of (1). The reason why instability is more easily proved in (2) than in (1) is the presence of the hyperbolic fixed point at (0, 0) of the (q, p) component of the averaged system. Studying (2) for µ > 0 small enough is thus called the a priori unstable problem, or the a priori hyperbolic problem. In contrast, the Hamiltonian (1) is called a priori stable. The a priori unstable case is by now quite well understood for m = 1, see [46,17,18,26,5,48] for example. The a priori unstable case for m > 1 and the a priori stable case can be considered as widely open, in spite of the important announcements of John Mather in [42]. The starting point in the study of (2) is the famous paper of Arnold, [1]. In this paper, Arnold introduced a particular a priori unstable system where some geometric structures associated to diffusion, partially hyperbolic tori (that he called whiskered), their stable and unstable manifolds, and heteroclinic connections, can be almost explicitly described. This geometric structure have been called a transition chain. Most of the subsequent works on the a priori unstable problem have consisted in trying to find transition chains in more general cases, but understanding the general a priori unstable Hamiltonian have required a change of paradigm: from partially hyperbolic tori to normally hyperbolic cylinders. The variational methods introduced by John Mather in [41] and Ugo Bessi in [9] have also been very influential. Transforming the understanding gained on the dynamics of (2) to informations on the a priori stable case is not an easy task. Since we understand the system (2) when m = 1 the first attempt should be to study (1) in the neighborhood of an (n − 1)-resonant line, for example the line consisting of momenta of the form y = (I, 0), I ∈ R. We could hope to prove the existence of drift along such a line by using the a priori unstable approximations near each value of y. However, we face the problem that an approximation like (2) holds only in the neighborhood of the torus {y = (I, 0)} when the frequency I ∈ R is irrational. Near the torus {y = (I, 0)} with I rational, one should use an approximation of the form and different methods must be used. This is often called the problem of double resonances when n = 2. We will call it the problem of maximal resonances.
Our general goal in this paper is to study a priori unstable systems with a sufficient generality to be able to gain informations on the a priori stable case. We start with a relatively detailed description of the Arnold's example in Section 2, which is also an occasion to settle some notations and introduce some important objects, like the partially hyperbolic tori, their stable and unstable manifolds, and the associated generating functions. Working with these generating functions allows to highlight the connections between the various classical approaches, geometric methods, variational methods, and weak KAM theory. Then, from the end of Section 2 to Section 3, we progressively generalize the setting and indicate how the methods introduced on the example of Arnold can be improved to face the new occurring difficulties. We present the Large Gap Problem, which prevents Arnold's mechanism from being directly applied to general a priori unstable systems, and explain how the presence of a normally hyperbolic cylinder can be used to solve this Problem and prove instability in general a priori unstable systems. In section 4 we give a new result from [6], on the existence of normally hyperbolic cylinders in the a priori stable situation, which should allow to apply the tools exposed in the previous sections to a priori stable systems. This suggests a possible strategy to prove the following conjecture: Conjecture 1. For a typical perturbation G, there exists two positive numbers 0 and δ, such that, for each ∈]0, 0 [, The system (1) has an orbit (θ(t), q(t),θ(t),q(t)) : We are currently working on this program in collaboration with Vadim Kaloshin and Ke Zhang. The same conjecture can be stated with a more general unperturbed system h(y), and the same proof should work provided h is convex and smooth. Our strategy of proof does not consist in solving the maximal resonance problem, but rather in observing that the conjectured statement can be reached without solving that difficulty. In that respect, what we expose is much easier than the project of Mather as announced in [42]. The result is weaker since only limited diffusion is obtained. The maximal resonance problem has to be solved in order to prove the existence of global diffusion on a whole resonant line, or from one resonant line to another. Our strategy, on the other hand, has the advantage of working with all n 2, while Mather is limited to n = 2 at the moment.

The example of Arnold and some extensions
Following Arnold [1], we consider the Hamiltonian H(t, θ, q, I, p) = 1 2 which merely says that the invariant manifolds are contained in the energy level of the torus. The functions S ± a have an expression in terms of the action: where τ is any real number such that τ mod 1 = t, and (θ ± (s), q ± (s)) is the solution of the Euler-Lagrange equations such that Note that the result does not depend on the choice of τ .

Homoclinic orbits.
obviously belongs both to W + (a) and W − (a), hence it is a homoclinic point. It is a transversal homoclinic point if in addition the Hessian of ∆ a has rank two (it can not have rank 3 because the intersection is necessarily one-dimensional). It is not obvious at this point that the function ∆ a necessarily has critical points on the domain T × T×]1/4, 3/4[. When µ is small enough, this follows from: is a critical point of the function∆ a : (t, q) −→ ∆ a (t, q, 1/2), then (T, Q, 1/2) is a critical point of ∆ a , hence the manifolds W − (a) and W + (a) intersect above (T, Θ, 1/2) ∈ T 3 . This homoclinic point is transversal if and only if the Hessian of∆ a at (T, Q) is a non-degenerate 2 × 2 matrix.
Note that the function∆ a is defined on T 2 , and therefore it has critical points. Proof. We have ∂ t S + (T, Q, 1/2) = ∂ t S − (T, Q, −1/2), let us denote by e this value. We also have ∂ θ S + (T, Q, 1/2) = ∂ θ S − (T, Q, −1/2), we denote by I this value. It is enough to prove that ∂ q S + (T, Q, 1/2) = ∂ q S − (T, Q, −1/2). In order to do so, it is enough to observe that ∂ q S + is the only non-negative solution of the equation e + H(T, Θ, 1/2, a + I, .) = a 2 /2, and that precisely the same characterization is true for ∂ q S − (T, Q, −1/2). Note that the equation above has two solutions, and that we can discriminate between them because we work in a perturbative setting which gives us rough informations on the signs. In more general situation, this is a source of difficulty.

Heteroclinic orbits.
We have proved the existence of homoclinic or- bits. But what is interesting for Arnold diffusion are heteroclinic orbits between different tori. We can deduce the existence of a heteroclinic orbit between T (a) and T (a ) provided we can find a critical point of the function where we have lifted the functions S without changing their names. As before, we can limit ourselves to finding critical points of the function but the term (a − a )θ prevents us from finding them using a global variational method when a = a. This reflects the fact that we are studying a non exact Lagrangian intersection problem. For µ = 0, heteroclinics do not exist. However, recalling that∆ a (t, θ) = ∆ a (t, θ, 1/2), we have: If the function∆ a (t, q) has a non-degenerate critical point, then the functions Σ a,a and Σ a ,a both have a non-degenerate critical point provided a is sufficiently close to a.
Proof. The theory of partial hyperbolicity implies that the stable and unstable manifolds W ± µ (a) depend regularly on the parameter a. As a consequence, their generating functions S ± a also regularly depend on a, and the functions Σ a,a depend regularly on a and a . The result follows since Σ a,a =∆ a .
We say that a 0 , a 1 , . . . , a k is an elementary transition chain if the functions Σ ai−1,ai have non-degenerate critical points. We will sometimes use the same terminology for the different requirement that these functions have isolated local minima. From Lemma 3, we deduce: Proposition 1. Let µ be given and sufficiently small. Let [a − , a + ] be an interval such that each of the functions∆ a,µ , a ∈ [a − , a + ] have a non-degenerate critical point, which means that each of the tori T µ (a), a ∈ [a − , a + ] has a transversal homoclinic orbit. Then there exists an elementary transition chain a − = a 0 , a 1 , . . . , a k = a + .
Proof. Let us consider the set A ⊂ [a − , a + ] of points that can be reached from a − by a transition chain. The set A is open : If a ∈ A, then there exists a transition chain a − = a 0 , a 1 , . . . a k = a and, by Lemma 3, the sequence a − = a 0 , a 1 , . . . a k , a k+1 is a transition chain when a k+1 is sufficiently close to a. The set A is closed : Let a be in the closure of A. By Lemma 3, the pair a, a is a transition chain when a is close to a. Since a is in the closure of A, the point a can be taken in A. Then, there exists a transition chain a − = a 0 , . . . , a k = a , and then the longer sequence a − = a 0 , . . . , a k , a k+1 = a is a transition chain between a 0 and a, hence a ∈ A. Being open, closed and not empty (it contains a 0 ), the set A is equal to [a − , a + ].
The existence of transition chains implies the existence of diffusion orbits. This is proved by Arnold invoking an "obstruction property". This obstruction property is a characteristic of the local dynamics near the partially hyperbolic tori. It has been proved by Jean-Pierre Marco in [38], see also [21,31]. The most appealing way to understand the geometric shadowing of transition chains is to use the following statement of Jacky Cresson [22], which can be seen as a strong obstruction property: Putting everything together, we obtain: Theorem 1. Let µ be given and sufficiently small. Let [a − , a + ] be an interval such that each of the functions∆ a,µ , a ∈ [a − , a + ] have a non-degenerate critical point. Then there exists a heteroclinic orbit between T (a − ) and T (a + ).

Poincaré-Melnikov approximation.
We have constructed diffusion orbits under the assumption that transversal homoclinics exist. We have proved that homoclinic orbits necessarily exist, and one may argue that transversality should hold for typical systems, we will come back on this later. However, it is useful to be able to check whether transversality holds in a given system. A classical approach consists in proving the existence of non-degenerate critical points of the functions∆ a,µ defined in Lemma 3 by expanding them in power series of µ. As a starting point the generating functions S ± a can be expanded as follows: where S 0 (q) = 2 √ π (1 − cos(πq)) is the generating function of the unperturbed manifolds, and M ± are the so-called Poincaré-Melnikov integrals, To better understand these formula, it is worth recalling that is the homoclinic orbit of the system p 2 /2+ (cos(2πq)−1) which takes the value q at time t. The formula above are similar to (6), but the integration is performed on unperturbed trajectories, which are explicitly known. For q ∈]1/4, 3/4[, we obtain: In the specific case studied by Arnold, where f (t, θ, q) = cos(2πθ) + cos(2πt), the Melnikov integral can be computed explicitly through residues, we obtain: it has a non-degenerate minimum at (t, q) = (0, 0). We can conclude, following Arnold: Theorem 2 (Arnold, [1]). Let us consider the Hamiltonian (3) with f (t, θ, q) = cos(2πθ) + cos(2πt) and µ > 0 small enough. Given two real numbers a − < a + , there exists an orbit (θ(t), q(t), I(t), p(t)) and a time T > 0 such that I(0) a − and I(T ) a + .

Bessi's variational mechanism. Ugo Bessi introduced in [9] a very
interesting approach to study the system (3), see also [10,11]. In order to describe this approach, let us define the function where the minimum is taken on the set of When the time interval t 2 −t 1 is very large, the minimizing trajectory in the definition of A a roughly looks like the concatenation of an orbit positively asymptotic to T (a) followed by an orbit negatively asymptotic to T (a). Using this observation, and recalling the formula (6), it is possible to prove that Fixing the real numbers a 0 , a 1 , . . . , a k and the integers τ 1 , . . . , τ k , we consider the discrete action functional It is not hard to check that local minima of this discrete action functional give heteroclinics between the Torus T (a 0 ) and the torus T (a k ). In order to prove that local minima exist, observe that this functional is approximated by when the integers τ i are large enough, with the functions Σ as defined in (7). This limit functional has the remarkable structure that the variables (t i , θ i ) are separated. This break-down of the action functional into a sum of independent functions is sometimes called an anti-integrable limit, it is related to the obstruction property of the invariant tori, to the λ-Lemma, and to the Shilnokov's Lemma, see [15]. The limit functional has an isolated local minimum provided each of the functions Σ ai−1,ai has one, which is equivalent to say that a 0 , a 1 , . . . , a k is an elementary transition chain. In this case, the integers τ i can be chosen large enough so that the action functional above has a local minimum, which gives a heteroclinic orbit between T (a 0 ) and T (a k ). Technically, this method has several advantages. In our presentation we introduced the generating functions S ± ai of the invariant manifolds of the involved tori in order to stress the relations between the geometric and the variational method, and also because this works in a more general setting, see [15] for example. In our context and when µ > 0 is small enough, it is easier to directly approximate the functions A a in terms of the Melnikov integrals, and to use the following approximation for the action functional with large τ i and small µ without the intermediate step through S ± : The corresponding calculations, performed in [9], are much more elementary than those required to derive the expansions (8).

Remarks on estimates.
We have up to that point carefully avoided to discuss the subtle and important aspect of explicit estimates. In order to complete rigorously the proof of Theorem 2, we should prove the existence of a threshold µ 0 ( ) such that the Melnikov approximation holds, simultaneously for all a, when 0 < µ < µ 0 ( ). This can actually been done, with but it is not simple, since it requires to study carefully the expansions of the functions S ± a and how the coefficients depend on a and . This is related to the so-called splitting problem, see [37]. As we mentioned above the approach of Bessi allows to prove that Theorem 2 holds for 0 < µ < µ 0 ( ) without estimating the splitting.
It is also important to give time estimates, that is to estimate the time needed for the variable I to perform a large evolution. Once again, this is closely related to the splitting estimates, although these can be avoided by using the method of Bessi. One should distinguish two different problems. Either we fix , and try to estimate the time as a function of µ, or we take µ as a function of , say µ = µ 0 ( )/2, and try to estimate the time as a function of .
The second problem is especially important, because it is relevant for the study of the a priori stable problem. Once again, Ugo Bessi obtained the first estimate, Estimating the time on examples allows to test the optimality of Nekhoroshev exponents, see [39,36,49] for works in that direction, see also [16] concerning the question of time estimates.
It is worth mentioning also that in the first problem, estimating the time as a function of µ, the estimate is polynomial, and not exponentially small. This was first understood by Pierre Lochak, and proved by Bessi's method in [2], where the estimate T = C/µ 2 is given, see also [23]. The optimal estimate is T = C| ln µ|/µ, as was conjectured by Lochak in [35] and proved by Berti, Biasco and Bolle in [8], see also [7].
Returning to the question of the threshold of validity, let us discuss what happens when µ is increased above µ 0 ( ). The content of Section 2.3 on finding transversal homoclinics via the Poincaré-Melnikov approximation breaks down, but the geometric constructions of the earlier sections is still valid. Theorem 1 holds as long as the invariant tori T (a) remain partially hyperbolic, and that their stable and unstable manifold can be represented by generating functions like (5). Actually, the methods we are now going to expose allow even to relax this last assumption. Being able to treat larger values of µ is especially important in view of the possible applicability to the a priori stable problem.

Higher dimensions.
Let us now discuss the following immediate generalization in higher dimensions of Arnold's example: with (t, θ, q, I, p) ∈ T × T m × T r × R m × R r , where V (q) is a non-negative function having a unique non-degenerate minimum at q = 0, with V (0) = 0. The main difference with the example of Arnold appears for r > 1. In this case, the system is not integrable even for µ = 0. There still exists a family of partially hyperbolic tori of dimension m, parametrized by a ∈ R m , but the system p 2 /2 − V (q) is not necessarily integrable any more. As a consequence we do not know explicitly the stable and unstable manifolds of the hyperbolic fixed point (0, 0), and so we do not have a perturbative setting to describe the stable and unstable manifolds of the hyperbolic tori T (a). This is also what happens for r = 1 if µ is not small enough. There is no obvious generalization of the generating functions S ± a in that setting, because the stable and unstable manifolds are not necessarily graphs over a prescribed domain. The proof of the existence of homoclinic orbits as given in 2.1 thus breaks down. The existence of homoclinic orbits in that setting can still be proved by global variational methods, as is now quite well understood, see [14,28,20,3,27] for example.
The proof is quite easy in our context, let us give a rapid sketch. We first define a function A a similar to the one appearing in Section 2.4, but slightly different: Let M be a submanifold of T × T m × T r which separates T × T m × {q 0 } from T × T m × {q 1 }, and let T i ∈ [0, k i ] be a time such that (T i mod 1, θ i (T i ), q i (T i )) ∈ M , and let τ i be the integer part of T i . It is not hard to check that the curves (θ i (t − τ i ), q i (t − τ i )) converge (up to a subsequence) uniformly on compact sets to a limit (θ ∞ (t), q ∞ (t)) : R −→ T m × T r . This limit curve satisfies and the corresponding orbit is a heteroclinic from T 0 (a) to T 1 (a). We call minimizing heteroclinics (for the lifted system) those which have minimal action, or in other words those which satisfy (9). In the original system (before taking the covering), we call minimizing homoclinic orbit a homoclinic which lifts to a minimizing heteroclinic.
Let us now try to establish some connections between the present discussion and the proof of the existence of homoclinic orbits given in 2.1. We define two functions on T × T m × T r : Note that S + a (0, 0, q 1 ) = −S − a (0, 0, q 0 ) = ξ(a). The functions S ± a , whose definition is basic both in Mather's ( [41]) and in Fathi's ( [29]) theory, share many features with those introduced in (5), that's why we use the same name. Let us state some of their properties: The function S − a is non-positive and it vanishes on T × T m × {q 1 } (and only there). Moreover, it is smooth around this manifold, which is a transversally nondegenerate critical manifold. Let us chose a small δ > 0. The set Similarly, S + a is non-negative, it is null on T × T m × {q 0 }, and smooth around it, and this critical manifold is transversally non-degenerate. The set W + loc (a) := t, θ, q, a + ∂ θ S + a , ∂ q S + a (t, θ, q) , S + a (t, θ, q) < δ is a negatively invariant local unstable manifold of T 0 (a).
The functions S ± a also have a global meaning. Let us give the details for S + . For each point (T, Θ, Q), there exists a real number τ ∈ R and at least one solution (θ(s), q(s)) : (−∞, τ ] −→ T m × T r of the Euler-Lagrange equations such that (τ mod 1, θ(τ ), q(τ )) = (T, Θ, Q), and which is calibrated by S + a in the following sense: The relation S + a (t mod 1, θ(t), q(t)) − S + a (s mod 1, θ(s), q(s)) = t s L(σ, θ(σ), q(σ),θ(σ),q(σ)) − aθ(σ) + a 2 /2 dσ holds for all s < t τ . The corresponding orbit is asymptotic either to T 0 (a) or to T 1 (a) when s −→ −∞. It is not easy in general to determine whether the asymptotic torus is T 0 (a) or T 1 (a) but the following Lemma is not hard to prove: If the function S + a is differentiable at (T, Θ, Q) then there is one and only one calibrated curve as above, it is characterized by the equationṡ Formally, the critical points of the difference S + a − S − a correspond to heteroclinic orbits (in the lifted system). By studying a bit more carefully the relations between the calibrated curves and the differentiability properties of the functions S ± a (which is one of the central aspects of Fathi's Weak KAM theory, see [29]), this idea can be made rigorous as follows: and the orbit of this point is either a heteroclinic between T 0 (a) and T 1 (a) or a homoclinic to T 0 (a) or to T 1 (a) in the system lifted to the covering, and thus it projects to an orbit homoclinic to T (a) in the original system.
Although it is not obvious a priori that a local minimum of the function S + a −S − a exists away from q = q 0 and q = q 1 , this follows from the existence of minimizing heteroclinics, that we already proved. More precisely, we have: • The minimal value of S + a − S − a is ξ(a).
• The point (T, Θ, Q) is a global minimum of S + a − S − a if and only if either Q ∈ {q 0 , q 1 } or the orbit of the point T, Θ, Q, a + ∂ θ S − , ∂ q S − = T, Θ, Q, a + ∂ θ S + , ∂ q S + , is a minimizing heteroclinic between T 0 (a) and T 1 (a).
• The set of minima of the function S + a − S − a properly contains As a consequence, the trajectory (θ(t), q(t),θ(t),q(t)) is a minimizing heteroclinic if and only if (S + a − S − a )(t mod 1, θ(t), q(t)) = ξ(a) for each t ∈ R (and if q(t) is not identically q 0 or q 1 ). This minimizing heteroclinic is called isolated if, for some t ∈ R, the point (θ(t), q(t)) is an isolated minimum of the function (θ, q) −→ (S + a − S − a )(t mod 1, θ, q).
Now we have proved that the stable and unstable manifolds of the torus T (a) necessarily intersect, let us suppose that there exists a compact and connected set A ⊂ R m such that the intersection is transversal for a ∈ A. By a continuity argument as in Proposition 1, we conclude that any two points a − and a + in A can be connected by a transition chain, that is a sequence a 0 = a − , a 1 , . . . , a n = a + such that the unstable manifold of T (a i−1 ) transversally intersects the stable manifold of T (a i ). We would like to deduce the existence of a transversal heteroclinic orbit between T (a − ) and T (a + ), but I do not know whether the higher codimensional analog of Cresson's transitivity Lemma 4 holds. However, the weaker obstruction property proved in [21,31] is enough to imply the existence of orbits connecting any neighborhood of T (a − ) to any neighborhood of T (a + ). It is also possible to build shadowing orbits using a variational approach. We need the slightly different assumption that A ⊂ R m is a compact connected set such that, for all a ∈ A, all the minimizing homoclinics of T (a) are isolated. For each a − and a + in A, it is then possible to construct by a variational method similar to Section 2.4 a heteroclinic orbit between T (a − ) and T (a + ).

The general a priori unstable case
A very specific feature of all the examples studied so far is that the perturbation preserves the partially hyperbolic invariant tori T (a), a ∈ R m . We now discuss the general a priori unstable system (2).

The Large Gap
Problem. Let us assume that r = 1 and try to apply the method of Section 2. There is no explicit invariant torus any more, but KAM methods can be applied to prove the existence of many partially hyperbolic tori. More precisely, there exists a diffeomorphism ω µ (a) : R m −→ R m , close to the identity, such that an invariant quasiperiodic Torus T µ (a) of frequency ω µ (a) exists, and is close to T (a), provided the frequency ω µ (a) satisfies some Diophantine condition. Moreover, for such values of a, the local stable and unstable manifolds W ± µ (a) can be generated by functions as earlier. So we have exactly the same picture as in Section 2, except that the objects are defined only on a subset A µ ⊂ R m of parameters. In order to reproduce the mechanism of Section 2, we must find elementary transitions chains a 0 , . . . , a k in R m , with the additional requirement that a i ∈ A µ . It is necessary at this point to describe a bit more the set A µ . Roughly, the KAM methods allow to prove the existence of the Torus T µ (a) provided a belongs to for some constant τ m + 1. This set A µ is totally disconnected, hence it is not possible to apply a continuity method like in Proposition 1 in order to prove the existence of a transition chain. We must be more quantitative, which is possible when µ is so small that the Poincaré-Melnikov approximation is valid. In that regime, we have∆ a,µ ≈ µM a , where M a has a non-degenerate critical point. The conclusion of Lemma 3 can then be proved to hold under the more explicit condition that a − a Cµ. In other words, the sequence a 0 , a 1 , . . . , a k is an elementary transition chain if a i − a i−1 Cµ. However, the gaps in A µ have a width of size √ µ > Cµ. As a consequence, for small µ, it seems impossible to build long transition chains, and the method fails. This is the Large Gap Problem, see [35]. Even if there are classes of examples where the method can be applied because more tori exist in some regions of phase space, see [13,19,8] for example, the generic case seems out of range.

Normally hyperbolic invariant cylinder. The Large Gap prob-
lem has now been solved, at least in the case where m = 1, see [46,17,18,26,5,44]. We will not discuss and compare all these solutions here, but just expose some general ideas which arise from them. An important new point of view is to focus on the whole cylinder C = ∪ a T (a) rather than on each of the tori T (a) individually. This cylinder is Normally hyperbolic in the sense of [34,30], and thus it is preserved in the perturbed system. This new point of view is very natural, it appears in [43,24], and then in many other papers. The deformed cylinder C µ contains all the preserved tori T µ (a) obtained by KAM theory. The restricted dynamics is described by an a priori stable system on T × T m × R m . If m > 1, we are confronted to our lack of understanding of the a priori stable situation. If m = 1, however, the restricted system is the suspension of an area preserving twist map, and we can exploit the good understanding of these systems given by Birkhoff theory which has also been interpreted (and extended) variationally in the works of Mather [40,41]. We consider this case (m = 1) from now on. The invariant 2-tori which are graphs are of particular importance (they correspond to rotational invariant circles of the time-one map). To each of these invariant graphs, we can associate two real numbers, the rotation number ω (defined from Poincaré theory of circle homeomorphisms), and the area a, which is the symplectic area of the domain of the cylinder C µ ∩ {t = 0} delimited by the zero section and by the invariant graph under consideration. If a given invariant graph T of the restricted dynamics has irrational rotation number (or is completely periodic), then there is no other invariant graph with the same area a. We can take a two-covering and associate to this graph two functions S ± a by formula similar to (10). They generate the local stable and unstable manifold of the Torus T , the correspond to the global minima of the difference of the so-called barrier function S + a − S − a . Minimal homoclinics and isolated minimal homoclinics to T can be defined as in Section 2.6. The existence of minimal homoclinics can be proved basically in the same way as it was there. Transition tori can be used to build transition chains in the same way as partially hyperbolic quasiperiodic tori with transversal homoclinics. Let A ⊂ R be the set of areas of transition tori. To each a ∈ A is attached a unique transition torus T µ (a) (note that this torus may be only Lipschitz, and is not necessarily quasiperiodic). If A contains an interval [a − , a + ], then the existence of a heteroclinic orbit between T µ (a − ) and T µ (a + ) can be proved by already exposed methods (considering the way we have chosen our definitions, a variational method should be used, but a parallel geometric theory could certainly be given).
In general, the set A is totally disconnected, and transition chains can't be obtained by a simple continuity method. If we make the additional hypothesis that all invariant graphs of the restricted dynamics are transition tori, then the set A is closed and a connected component ]a − , a + [ of its complement corresponds to a "region of instability" of the restricted system in the terminology of Birkhoff. More precisely, the tori T µ (a − ) and T µ (a + ) enclose a cylinder which does not contain any invariant graph. The theory of Birkhoff then implies that there exist orbits of the restricted dynamics connecting an arbitrarily small neighborhood of T µ (a − ) to an arbitrarily small neighborhood of T µ (a + ). This gives an indication about how to solve the large gap problem: use the Birkhoff orbits to cross regions of instability, and the Arnold homoclinic mechanism to cross transition circles. It is by no means obvious to prove the existence of actual orbits shadowing that kind of structure. In order to do so, one should first put these mechanisms into a common framework. The variational framework seems appropriate, although a geometric approach is also possible. The Birkhoff theory was described and extended using variational methods by Mather in [40], and he proposed a new variational formalism adapted to higher dimensional situations in [41]. On the other hand, Bessi's method indicates how to put Arnold's mechanism into a variational framework. These heuristics lead to: Theorem 3. Let [a − , a + ] be a given interval. If all the invariant graphs of area a ∈ [a − , a + ] of the restricted dynamics are transition tori, then there exists an orbit (θ(t), q(t),θ(t),q(t)) and a time T > 0 such thatθ(0) a − andθ(T ) a + .
This theorem is proved using variational methods and weak KAM theory in [5], Section 11, where it is deduced from more general abstract results. It also almost follows from [18], Theorem 5.1, which is another general abstract result proved by elaborations on Mather's variational methods [41], see also [4]. Applying that result of Cheng and Yan, however, would require a minor additional generic hypothesis on the restricted dynamics. In the case where r = 1, a slightly weaker version of Theorem 3 could also be deduced from the earlier paper of Chen and Yan [17]. Under different sets of hypotheses, results in the same spirit have been obtained by geometric methods in [32,33]. At the moment, these methods do not reach statements as general as Theorem 3, but they apply in contexts where the variational methods can't be used.
The following variant of Theorem 3 may deserve attention in connection to the Arnold Mechanism: Assume that a − and a + belong to A, or in other words that there exist transition tori T µ (a ± ). These tori enclose a compact invariant piece C µ [a − , a + ] of the invariant cylinder. If all the invariant graphs contained in C µ [a − , a + ] are transition tori, then we say that C µ [a − , a + ] is a transition channel.
The proof of Theorem 3 also implies that, if C µ [a − , a + ] is a transition channel, then there exists a heteroclinic orbit connecting T µ (a − ) to T µ (a + ).
Theorem 3 proves the existence of diffusion under "explicit" conditions. These conditions are hard to check on a given system, but they seem to hold for typical systems. It is much harder than one may expect to prove a precise statement in that direction, but it was achieved by Cheng and Yan in [17,18]. The main difficulty comes from the condition on the isolated minimal homoclinics. Actually, it is not hard to prove that the homoclinics to a given torus are isolated for a typical perturbation, but we need the condition to hold for all the tori simultaneously. Since there are uncountably many tori, it is necessary to understand the regularity of the map a −→ S ± a . Recall that the functions S ± a are well-defined provided there exists an invariant graph of area a which has irrational rotation number or is foliated by periodic orbits. We callÃ this set of areas, it contains A. Cheng and Yan prove that the map a −→ S ± a is Hölder continuous onÃ, and deduce the genericity result using an unpublished idea of John Mather.

Back to the a priori stable case
The main objects in Arnold's mechanism are partially hyperbolic tori, that he called whiskered tori. It was proved by Treshchev [46], that whiskered tori exist in the a priori stable situation, see also [27,45]. However, because of the Large Gap Problem, it seems difficult to prove directly the existence of transition chains made of whiskered tori. Actually, small transition chains do exist, because the density of KAM tori increases near a given one, but the length of these chains gets small when gets small, hence these chains do not produce instability of the action variables in general.
The modern paradigm on the a priori unstable case that we exposed in Section 3.2 elects 3-dimensional normally hyperbolic invariant cylinders as the important structure. It is well-known that normally hyperbolic invariant cylinders exist in the a priori stable case. For example, each 2-dimensional whiskered torus has a center manifold, which is a 3-dimensional normally hyperbolic invariant cylinder, see e. g. [12]. Actually, it is simpler to prove directly the existence of normally hyperbolic invariant cylinders, this involves no small divisors. However, the most direct proofs seem to produce "small" normally hyperbolic cylinders, which means that their size is getting small with , so that we face the same problem as above when we had small transition chains. The main statement of [6] is that "large" normally hyperbolic cylinders exist, meaning that their size is bounded from below independently of .
In order to be more specific, let us select a resonant momentum of the form y 0 = (I 0 , 0) ∈ R × R n−1 , with I 0 Diophantine. Assuming that the corresponding averaged potential V has a unique minimum and that this minimum is nondegenerate, we have: Theorem 4 ( [6]). There exists two intervals [a − , a + ] ⊂ J, J open, both independent from , and 0 > 0 such that, for ∈]0, 0 [ the following holds: There exists a C 1 map (Q, P ) : T × T × J (t, θ, I) −→ Q(t, θ, I), P (t, θ, I) ∈ T n−1 × R n−1 such that the flow is tangent to the graph Γ of (Q, P ). Moreover, there exist two real numbers a 0 < a − and a 1 > a + in J (which depend on ) such that the Treshchev tori T (a 0 ) and T (a 1 ) exist and are contained in Γ. The part Γ 1 0 of Γ delimited by these two tori is then a compact invariant manifold with boundary of the flow, it is normally hyperbolic. It is equivalent to say that it is partially hyperbolic with a central distribution equal to the tangent space of Γ. The inner dynamics is the suspension of an area-preserving twist map (where the area is the one induced from the ambient symplectic form).
It is then reasonable to expect that, under generic additional hypotheses, Γ 1 0 is a transition channel as defined in Section 3.2, and thus that T (a 0 ) and T (a 1 ) are connected by a heteroclinic orbit. We are currently exploring that program in collaboration with Vadim Kaloshin and Ke Zhang. It is important to observe that the map (Q, P ) is not C 1 -close to (0, 0), and that the inner dynamics is not close to integrable. Fortunately, Theorem 3 allows such a generality.