Geometry of Arnold Diffusion

Transcription

1 SIAM REVIEW Vol. 50, No. 4, pp c 008 Society for Industrial and Applied Mathematics Geometry of Arnold Diffusion Vadim Kaloshin Mark Levi Abstract. The goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of Arnold diffusion. We choose to speak of a specific model that of geometric rays in a periodic optical medium. This model is equivalent to that of a particle in a periodic potential in R n with energy prescribed and to the geodesic flow in a Riemannian metric on R n. Key words. Hamiltonian systems, instability, integrability, resonances AMS subject classifications. 37J50, 37J40, 37J45 DOI / Introduction A Brief Background on Complete Integrability. The KAM (Kolmogorov Arnold Moser) theory deals with perturbations of completely integrable Hamiltonian systems. A Hamiltonian system { ( ) ẋ = Hy (x, y), 0 I (1) or ż = J H, J =, ẏ = H x (x, y), I 0 with a Hamiltonian H : R n R, is said to be completely integrable if it possesses n independent integrals I k : R n R in involution that is, if the Hamiltonian vector fields J I k, k =1,...,n, are linearly independent and commute. For a fixed set c =(c 1,...,c n )ofn constants the level set T c = {z R n : I k = c k,k=1,...,n} is invariant under the Hamiltonian flow. A compact component of the level set T c is an n-dimensional torus for an open set of constants c. In other words, an open set in R n is foliated by an n-parameter set of n-dimensional tori, the parameters being the values c k of the integrals; we refer to [A] and to [MZ] for details. The KAM theory states that, under a small and sufficiently smooth perturbation of the completely integrable Hamiltonian, the majority of the invariant tori survive as parametrized tori invariant under the flow of the perturbed Hamiltonian system, the majority being in the metric sense: the relative measure of surviving tori approaches one as the size of the perturbation approaches zero. (For precise formulations of different versions of KAM theory we refer to [Mo], [H], [D], and [P]; the last reference contains the proof of the positivity of the measure of invariant tori.) Received by the editors September 19, 007; accepted for publication (in revised form) December 7, 007; published electronically November 5, Mathematics Department, Penn State University, University Park, PA The first author was partially supported by the Sloan and NSF grant DMS The second author was partially supported by NSF grant DMS

2 GEOMETRY OF ARNOLD DIFFUSION 703 Stability for Two Degrees of Freedom. In the case of n = degrees of freedom, KAM theory implies that the actions I 1, I stay close to their initial values for all time, provided the perturbation of H is small enough; no resonances will ever accumulate. The reason for this stability is topological: for the unperturbed Hamiltonian, the three-dimensional surface H = const. is foliated by a one-parameter family of invariant -tori. Under a small perturbation, a Cantor set of these tori survives. These surviving tori have codimension one on the energy surface. Consequently, any solution starting between two surviving invariant tori must stay between them for all time. Instability with 3 Degrees of Freedom. With n 3 degrees of freedom there is no a priori reason for stability, which led Arnold [A1] to conjecture that a general small perturbation of a completely integrable system leads to instability, i.e., to the existence of a solution along which the action vector (I 1,...,I n ) consisting of the integrals of the unperturbed system changes by a finite amount. This instability came to be known as the Arnold diffusion. The phenomenon has nothing to do with the standard diffusion, and would be better described as the resonance-based drift of action. There has been much work done trying to prove the ubiquity of Arnold diffusion since Arnold s original paper appeared. Douady [Do] proved that a symplectic map with an elliptic fixed point in R n, n, can be perturbed so as to destroy topological stability of the fixed point without affecting any of the Taylor coefficients of the map at the fixed point. Such a map can be viewed as a Poincaré map of a periodic orbit of a Hamiltonian system with n degrees of freedom. Douady s result thus shows the presence of Arnold diffusion in the neighborhood of an elliptic periodic orbit under some mild perturbations. In Arnold s original example [A1], as well as in ours, the action drifts in a particular direction, namely, in the direction of a particular resonance; in our example, this drift corresponds to the geodesic turning in the direction of the (x, y)-plane. There is a rich literature on Arnold diffusion; we mention [AKN], [BB], [MS], [LM], and references therein, going far beyond the example discussed here. In this paper we discuss the same example as in [KL] but from a different point of view. This example is a variation on Arnold s example and we treat it by the method of broken geodesics. The nature of the proof is similar to those of Bessi [Bs] and Mather [Ma6], but we avoid substantial technical difficulties that these authors had to overcome by choosing a localized perturbation. We sacrifice generality and technical strength for the sake of transparency. Our primary goal is to present the geometrical essence of Arnold diffusion in what seems to be the simplest possible setting. We outline our construction in the introduction, giving details in later sections. 1.. An Illustration on a Mechanical/Optical/Geodesic Example. Without loss of generality, one can illustrate all of the above on the example of a particle in a Newtonian potential εu(x) onr n : () ẍ = ε U(x), x =(x 1,...,x n ), with U periodic of period 1 in each x k. This system can be recast in the Hamiltonian form (1) with H(x, y) = y + εu(x). The free particle case ε = 0 is completely integrable: the n integrals are the n coordinates of the momentum=velocity vector y = ẋ. The invariant n-tori are given by {(x, y) :x R n mod 1, y = v}, where v R n is a constant. Each such torus, viewed in the configuration space, is simply the velocity vector field ẋ = v on the configuration torus R n mod 1.

3 704 VADIM KALOSHINAND MARK LEVI For ε 0 in () the invariant KAM tori have a similar interpretation: they are the vector fields y =ẋ = v(x) on the torus T 3 which are invariant in the sense that if a particle subject to () is given the initial velocity v(x) at some x, then its velocity at any other point x it reaches later is v(x ). The KAM trajectories are the diffeomorphic images of the straight trajectories x =(ωt)mod 1 of the unperturbed case ε = 0. The KAM frequencies ω satisfy certain Diophantine irrationality conditions. These are the conditions that single out a Cantor set of the perturbation-surviving tori. As ε 0, the relative measure of the invariant tori approaches one. In particular, for most initial conditions, the action variables ẋ = y remain close to their initial values. Thus KAM theory answers the question of the long-time behavior for most (in the measure) initial data. But what about the remaining solutions, the ones that do not lie on the invariant tori? 1.3. Arnold Diffusion. The invariant tori of dimension n lie on the energy surface H = const. of dimension n 1. If n =, the tori separate the three-dimensional energy surface into two components, and thus no trajectory can pass from one side of the torus to the other. Since the torus is defined by setting each action to a constant, this amounts to the statement that the variation of each action stays bounded for all time. For n the topological obstruction is no longer valid, and there is no reason to expect the boundedness of the action for all time, even for small perturbations. In fact, Arnold outlined an example of a near-integrable system with.5 degrees of freedom 1 for which one of the two action variables changes by a prescribed amount no matter how small the perturbation. The change of action by a bounded amount, which takes place for arbitrarily small perturbations of a near-integrable system, came to be called the Arnold diffusion An Optical/Geodesic Interpretation of Arnold Diffusion. According to the Maupertuis principle, trajectories of () with fixed energy ẋ /+εu(x) =1are also geodesics in the Jacobi (kinetic energy) metric, (3) dj = (1 εu(x)) ds, where ds is the Euclidean metric. Alternatively, one can think of trajectories as the rays in an optical medium with the index of refraction n(x) = (1 εu(x)). We point out that the geodesics give only the trajectories, i.e., the paths in the configuration space. The position x(t) at time t can be recovered from the speed v(x) = (1 εu(x)). The topological difference between the cases n = and n = 3 has a nice interpretation in the geodesic case. The geodesic flow can be described by a symplectic mapping as follows. Consider first the case of n =. Let S(q, Q; ε) be the distance in the Jacobi metric dj between the points (0,q) and (1,Q)inR. Let us define the momenta p and P by (4) { p = Sq (q, Q; ε), P = S Q (q, Q; ε). 1 That is, of a system in R 4 with additional periodic time dependence in the Hamiltonian. This system can be viewed as a subsystem of an autonomous Hamiltonian system with n = 3 degrees of freedom. It has to be pointed out that this diffusion has nothing in common with the more conventional meaning of diffusion.

4 GEOMETRY OF ARNOLD DIFFUSION p q P Q 0 0 p (q, p) (Q, P ) 1 q Fig. 1 An annulus mapping arising from the variational principle via (4). Geometrically, p and P are approximately the sines of the angles between the trajectory joining the two endpoints and the horizontal coordinate axis; see Figure 1. The system (4) defines the symplectic mapping ψ ε := (q, p) (Q, P ) for all q R, p M, where the constant M depends on ε. The constant M can be chosen arbitrarily large for sufficiently small ε. For ε small, the mapping ψ is close to the integrable twist map ψ 0 := (q, p) (q + p, p). For the latter map, p is the conserved quantity this is the manifestation of the fact that for ε = 0 the trajectories are straight lines. The map ψ ε satisfies all the conditions of Moser s twist theorem [Mo], and thus possesses invariant circles; see Figure 1. In particular, all iterates of any point in the annular region between two invariant circles is bounded therein. The corresponding values of P are therefore bounded as well, and there is no Arnold diffusion in this case. For the case of n = 3 this argument no longer holds. Indeed, ψ ε defined by (4) is a map of R 4, and there is no a priori reason for the momentum p to be confined under the iterations by ψ. Thus, geometrically, to prove Arnold diffusion for () amounts to showing that for arbitrarily small ε there exist geodesics in the Jacobi metric (3) which change direction by a prescribed angle const. > 0. An equivalent optical interpretation is that there exist rays which change direction by a prescribed finite angle in a medium whose index of refraction is arbitrarily close to 1.. The Result: A Simple Metric with Diffusion. We state our main result: a simple metric, arbitrarily close to Euclidean, with some of the geodesics changing directions by a large amount. At the same time, the majority of the geodesics in this metric are of KAM type..1. The Metric. Our metric dρ is defined as a C k -small perturbation of the standard Euclidean metric ds = dx + dy + dz in R 3 : ( (5) dρ = 1+εcos πz ) εk+1 β(θ,ε) ds, where k, and β is a C -smooth bump function, periodic in x, y, z, supported on ε-balls centered at integer points in Z 3 with even coordinates: 3 (6) β(θ,ε)= ( ) θ n η. ε n Z 3 To be specific, we pick η([0, 1 ])=1,η([1, )) = 0 with η being C -smooth monotone decreasing on [ 1, 1], even, and smooth on R. Under these conditions the bump 3 This choice is made for later convenience.

5 706 VADIM KALOSHINAND MARK LEVI y z x z 1 0 (x, y)-plane sv Fig. The mechanism of Arnold diffusion. Centers of the balls form a cubic lattice. Only transitions with σ i =+are shown in this illustration. ε k+1 β is indeed C k -small with ε. We will show that for arbitrarily small ε the metric (5) possesses geodesics which change their direction by any prescribed amount, given enough time. This amounts to the proof of Arnold diffusion. Speaking in the equivalent terms of particles in potentials, this amounts to proving that perturbations can accumulate in a consistent way to cause a large change of direction, no matter how small the perturbations are. Our example (5) is in some sense the simplest possible: the leading term in the perturbation depends only on z (this problem is equivalent to the mathematical pendulum in the z-direction and a free motion in the (x, y)-direction), with the higherorder perturbation due to the lenses. A heuristic explanation of the dynamics of this example is given after the formulation of the main theorems. The Results. Theorem 1 (Arnold diffusion). There exist ε 0 > 0 and c>0 such that for all 0 <ε<ε 0 the metric (5) possesses geodesics which make U-turns. More precisely, there exists a geodesic segment with endpoints A and B such that the tangent vectors are nearly opposite, T A = T B + o(ε 0 ) (Figure ), and with the length of the segment cε (k+ 9 ).

6 GEOMETRY OF ARNOLD DIFFUSION 707 p α p D 1 Σ D p + Σ Σ + Fig. 3 If D 1 and D are large and α is small, then S has a minimum inside Σ. A broken geodesic and a shadowing geodesic are shown. The z-axis is perpendicular to the page. The following theorem states that the transitions from one ball to another can be prescribed arbitrarily within a certain latitude. This theorem is illustrated in Figure. Theorem. There exists ε 0 > 0 such that for any 0 <ε<ε 0 the following holds. Given any infinite sequence of integer vectors n i Z with (7) n i >ε (k+), e i+1 e i <ε k+ 5, where ei = n i n i, and any infinite sequence of σ i = ±, there exists a geodesic θ(s) of (5), where s is the length parameter, with the itinerary {n i,σ i } i=, in the following sense: For a sequence of parameters <s i <s i+1 < we have θ(s i+1 ) θ(s i )=(n i,σ i )+δ, where δ < ε. Moreover, each θ(s i ) lies in an ε-neighborhood of a vertex of the lattice Z 3. Finally, s i+1 s i = n i (1 + o(ε 0 )). Corollary 1. The (fastest achievable) speed of Arnold diffusion in our example is O(ε k+4 1 ). More precisely, there exist ε 0 and a constant c such that for all 0 < ε ε 0 there exists a geodesic θ(s) parametrized by the Euclidean length s such that θ(s ) θ(s) > 1 for some s, s with s s <cε k+4 1. According to this corollary, the smoother the perturbation, the slower the diffusion. This is consistent with Nekhoroshev s estimates which show that the diffusion caused by analytic perturbations is exponentially slow; see [Ni] and references therein. To formulate another consequence of the last theorem we define the rotation set of a geodesic γ, γ(t) R 3, as the set of existing limits of all convergent sequences { } γ(tn ),t n, γ(t n ) where is the Euclidean norm. We denote the rotation set of γ by r(γ); this is a subset of S. If r(γ) is a single point, we call it the rotation vector. Using this notation we have: Theorem 3. There exists ε 0 > 0 such that for any 0 <ε<ε 0 and for any r R 3, r =1, 1. there exists a geodesic of the metric (5) with r as its rotation vector;. there exists a geodesic whose rotation set is equal to S. A Heuristic Outline of the Proof. The support of β is the lattice of balls in R 3. Consider three of these balls, nearly aligned as shown in Figure 3, and let Σ denote the equatorial disks of these balls. We also assume, to be specific, that the centers of the balls have the respective z-coordinates z =0,z =, and z + = 4. Let us fix any

7 708 VADIM KALOSHINAND MARK LEVI pair of points p ± Σ ± and pick any point p Σ. A unique minimal geodesic connects p and p, and similarly for p and p + (Lemma ). The length of these geodesics will be denoted by L(p,p) and L(p, p + ). Consider the length of the broken geodesic: S(p) =L(p,p)+L(p, p + ), p Σ. Now the key observation is that S has a minimum strictly inside Σ, provided the balls are sufficiently far apart (D 1, >ε (k+) ) and provided they are well aligned (α <ε k+ 5 ); see Figure 3. To explain the previous statement heuristically, the geodesic can be thought of as an elastic under tension and in a potential force. The negativity of the β-term in (5) means that the string is pulled into the well (a supporting ball of β); on the other hand, the string wants to straighten out, i.e., to pull out of Σ. In this competition, we want the force of sliding into the well to win over the straightening force. The near-alignment mentioned above is necessary for this to happen. In the proofs below we do not use those mechanical terms, although the gradients of the actions can interpreted precisely as the forces of tension of an imaginary string. Using the above observation we will concatenate geodesic segments so as to accumulate a large turning angle. Concatenation is also the key ingredient in proving the shadowing results of Theorem. Remark 1. The velocity of the deflecting geodesics drifts while staying close to the horizontal plane ż =0. This motion corresponds to the resonance k 1 ẋ+k ẏ+k 3 ż =0, 4 with k 1 = k =0. This drift of action is a manifestation of a general phenomenon: the action of a Hamiltonian system drifts along resonances [A]. Remark. The geodesics of (5) are trajectories of a near-integrable Hamiltonian system of the form (). The relative measure of quasi-periodic geodesics is nearly full. Therefore, all the geodesics we produce in Theorems 1 and coexist with the KAM geodesics which form the set of nearly full relative measure 1 o(ε 1 ). 3. Lemmas. In this section we formulate four lemmas that capture the essential properties of our metric. Lemma 1. Consider the metric dσ = λds, where λ>0 is a smooth function on R n. Fix O R n, and let L(O, p) = λds denote the geodesic distance from O to Op p R n. 5 Assume that there are no points conjugate to O on the geodesic segment Op, so that the gradient L(O, p) is well defined. Consider, on the other hand, the solution θ(t) of ( λ (8) θ ) (θ) = passing through O and p: θ(0) = O, θ(t )=p for some T, satisfying (9) θ = λ(θ). We note that (9) holds for all t if it holds for some t, as follows from the conservation of the energy θ λ (θ). Then (see Figure 4) (10) p L(O, p) = θ(t). 4 We identify velocities with the momenta. 5 We are interested only in the case n =3.

8 GEOMETRY OF ARNOLD DIFFUSION 709 L = θ L = const. p = θ(t ) O Fig. 4 Gradient of the action equals the velocity of the corresponding particle. Proof of Lemma 1. By Maupertuis principle, trajectories of θ = V (θ) with fixed energy θ /+V (θ) =E are geodesics in the Jacobi metric of kinetic energy, dσ = (E V )ds, and vice versa. This kinetic energy metric becomes our given metric λds if we set V = 1 λ and take E = 0. The energy relation becomes θ = λ. Thus the trajectory of any solution of (8) with speed λ is a geodesic for the metric λds. Now, since our metric is isotropic the indicatrices are spheres the geodesics are normal to wavefronts, (11) L θ, and the two vectors in (11) point in the same direction. Moreover, the definition of L implies L = λ. Combining this with the energy relation (9), we conclude that L = θ, which together with (11) completes the proof of (10). The support of the bump function β is the union of ε-balls centered at the integer points n; we will refer to such a ball as the lens, denoting it by L n. Let Σ n be the equatorial disk of L n of radius ε 1 3 ; see Figure 5. Remark 3. We choose the radii of sections Σ i so that the minimal geodesic segments γ(p 0,p 1 ) with one of the ends on Σ i does not intersect the corresponding lens sharing the center with Σ i. It turns out that ε 1 3 suffices; see Figure 5, bottom. Let Σ 0 and Σ 1 be two disks of radius ε 1 3 whose planes are parallel to the (x, y)- plane with centers at O 0 =(n 0, 0) and O 1 =(n 1, ), respectively; here n i Z, i =0, 1. We will refer to these disks as sections; see Figure 5. Lemma (existence of geodesic segments). For all ε sufficiently small the following holds. 1. For any p 0 Σ 0 and p 1 Σ 1 there exists a unique z-monotone geodesic γ(p 0,p 1 ) of the metric (5).. γ(p 0,p 1 ) intersects no lenses except possibly L 0 and L 1 centered at O 0 and O 1, respectively. 3. If p i Σ i (i =0or 1), then the geodesic does not intersect L i.

9 710 VADIM KALOSHINAND MARK LEVI p 1 Σ 1 p 0 Σ 0 p Σ Σ L Fig. 5 There is a unique minimal geodesic between any two sections in neighboring z-layers. This geodesic does not intersect any other lenses. Bottom: if p Σ, then the geodesic misses the lens L. p 0 q 0 v 0(q 0)=V 0(q 0) q 0 V 0(q 0,q 1) v 0(q 0) Q 1 p 1 γ 0(q 0,q 1) γ(q 0,q 1) q 1 Q 0 q 0 p 0 Fig. 6 Matching geodesic segments. Proof of Lemma. On the sphere of radius ε 1 3 centered at p 0 Σ 0 we define the spherical cap Q 0 as shown in Figure 6 to consist of points on the sphere whose radius vector forms angle ε 1 3 with the straight line segment p 0 p 1. 6 In a similar way we define the cap Q 1 ; see Figure 6. As the first step we will show that any pair q i Q i (i =0, 1) is connected by a geodesic segment γ 0 (q 0,q 1 ) which avoids all lenses, thus also being a geodesic in the metric λds. Minimal geodesic segments γ(p 0,q 0 ) and γ(q 1,p 1 ) are unique since the sphere s radius ε 1 3 is less than the injectivity radius of 6 The radius ε 1 3 is chosen so that the sphere centered at any p 0 Σ 0 would include the lens L 0.

10 GEOMETRY OF ARNOLD DIFFUSION 711 our metric, which we assure by making ε small enough. We will show that there is a unique choice of q i Q i for which the broken geodesic γ(p 0 q 0 q 1 p 1 ) is a true geodesic by matching tangents at q 0 and q 1, (1) v 0 (q 0 )=V 0 (q 0,q 1 ), v 1 (q 1 )=V 1 (q 0,q 1 ), where v 0 (q 0 ) is the unit tangent at q 0 to γ(p 0,q 0 ) and V 0 (q 0,q 1 ) is the unit tangent at q 0 to γ(q 0,q 1 ), with similar notations for the right end. 7 These are implicit equations for the unknowns q 0, q 1. We will solve (1) using a fixed point argument, by reformulating (1) as the existence of a fixed point of a map from a neighborhood of S S into itself. Heuristically, (1) can be satisfied due to the fact that V 0 (q 0,q 1 ) barely changes with q 0, while v 0 (q 0 ) const. p 0 q 0 is approximately the normalized radius vector of q 0 on the sphere, so that q 0 can be adjusted. The specifics of this are carried out in Step 3 below. We now proceed with the details. Step 1: The existence of the geodesic γ(q 0,q 1 ). Let λ 0 denote the term in parentheses in (5) with β = 0. We first prove this statement in the truncated metric λ 0 ds and then show that the geodesic γ 0 (q 0,q 1 ) does not intersect any of the lenses. This will establish that γ 0 (q 0,q 1 ) is in fact the geodesic in the metric λds. Geodesics in metric λ 0 ds are trajectories of (8) with λ = λ 0 : subject to energy relation (9): ( ) λ ẍ =ÿ =0, z + 0 (z) =0, (13) ẋ +ẏ +ż = λ 0(z). We have to prove the existence of a solution θ(t) =(x(t),y(t),z(t)) with θ(0) = q 0 Q 0, θ(t )=q 1 Q 1 for some T>0, and such that ż>0on0 t T. We start by defining T. Since the (x, y)-motion is free, we must have ẋ +ẏ = (r/t), where r = (x 1 x 0 ) +(y 1 y 0 ). We must also satisfy the energy relation which now becomes ż = λ 0 (r/t), z must satisfy its boundary conditions, z(0) = z 0, z(t )=z 1, and finally we must have ż>0. Due to the latter condition, t is a function of z: z(t) dz z(t) (14) t = z 0 ż = dz z 0 λ0 (z) (r/t). For the boundary conditions for z(t) to hold it is necessary that (15) T = z1 z 0 dz λ0 (z) (r/t). We use relation (15) to define T implicitly. Clearly T is unique, since the two sides in (15) are monotone in T, one increasing and the other decreasing, with the appropriate limits at the ends of the range of T allowed by the right-hand side. With T now defined, we pick (x, y)(t) =(x 0,y 0 )+(x 1 x 0,y 1 y 0 )t/t and define z(t) by (14). It is clear that the resulting solution satisfies the boundary conditions. Uniqueness of 7 At the right end the tangents are assumed to be oriented in the same direction, from p 0 to p 1.

11 71 VADIM KALOSHINAND MARK LEVI ε ż 0 1 z Fig. 7 Phase plane for z. the solution follows from the fact that it was derived unambiguously from necessary properties. Next we show that θ(t), 0 t T, avoids all the lenses and hence is a geodesic in the metric λds. Step : γ 0 (q 0,q 1 ), q i Q i, i = 0, 1, avoids all lenses. Since z(t) from the previous step increases from z(0) = O(ε 0 )toz(t)=+o(ε 0 ), we conclude that the corresponding phase point (z,ż) lies above the separatrix ż =εcos πz in the phase plane (see Figure 7), and hence ż> ε cos πε ε if z < 1. By (13) the horizontal speed of our solution v h = ẋ +ẏ <λ 0 < 1+ε, and thus, as long as z < 1, we have the following bound on the slope of the trajectory: (16) ż ε > v h 1+ε. This slope is steep enough so that θ(t) clears all lenses with centers on z = 0. Indeed, θ(0)=(x 0,y 0,z 0 ) Q 0 is not too low, by the definition of Q 0, z 0 ε 1 3 ε 1 3 = ε 3, which together with the steepness estimate (16) implies that γ(q 0,q 1 ) misses all lenses L with centers on z = 0. Indeed, by (16), z must increase by ε per unit (x, y)- displacement, which would make z(t) z 0 + ε ε 3 + ε >ε, which is too high to meet any lens whose center is on the plane z = 0. The same argument applies near z =. It also follows by the same argument that if p 0 Σ 0, then the geodesic misses L 0 ; see Figures 6 and 8. Indeed, for θ(t) to reach L 0 the projection point (x(t),y(t)) must move the distance >ε 1 3 ε and by the slope estimate (16) z(t) must increase by > (ε 1 3 ε) ε >ε, thus missing the lens. We have thus shown that γ 0(q 0,q 1 )= γ(q 0,q 1 ) is a geodesic in the full metric λds. Step 3: Matching the geodesic segments γ(p 0,q 0 ), γ(q 0,q 1 ), and γ(q 1,p 1 ). Our goal now is to prove that there exist points q 0,q 1 satisfying the matching conditions (1). We will reformulate (1) as a condition for the existence of a fixed point of a map on S S. The points on S will be the unit normal to the sphere at q 0 and q 1. If ε is sufficiently small, then the radius ε 1 3 of the spheres in Figure 6 is less than the injectivity radius of our metric λds. Since q 0 p 0 =ε 1 3, there is a unique

12 GEOMETRY OF ARNOLD DIFFUSION 713 z Geodesic misses the lens ε 1 ε Σ 0 p 0 Σ 0 ε 1 3 L 0 Fig. 8 If p 0 Σ 0, then the geodesic misses L 0. minimal geodesic segment γ(p 0,q 0 ). The unit tangent vector v(q 0 )toγ(p 0,q 0 )atq 0 satisfies v 0 (q 0 )=n 0 + εg 0 (p 0,q 0,ε), where n 0 = q0 p0 q 0 p 0 and g 0 is bounded in C 1 -norm as a function of q 0. Expressing q 0 = p 0 + ε 1 3 n 0 we obtain v 0 as a function of n 0, rather than of q 0 ; by a slight abuse of notation, (17) v 0 (n 0 )=n 0 + εg 0 (p 0,n 0,ε), where g 0 is bounded in C 1 -norm as a function of n 0 S. Consider now the geodesic segment γ(q 0,q 1 ) whose existence was proven in Steps 1 and. The unit tangent V 0 (q 0,q 1 )atq 0 to γ(q 0,q 1 ) satisfies V 0 (q 0,q 1 )=e 01 + where e 01 = p1 p0 p 1 p 0 and (18) h 0 C 1 (Q 0 Q 1) c, ε 1 3 q 1 q 0 h 0(q 0,q 1,ε), (q 0,q 1 ) Q 0 Q 1, as follows from the above proof of Steps 1 and. We again express q 0 in terms of n 0 and q 1 via n 1 = p1 q1 p 1 q 1 ; again, by some abuse of notation we have (19) V 0 (n 0,n 1 )=e 01 + ε 1 3 p 1 p 0 h 0(n 0,n 1,ε), (n 0,n 1 ) S S. Similar statements hold for the unit tangent vectors v 1 (n 1 ), V 1 (n 1,n 0 )atq 1 (see Figure 6): (0) v 1 (n 1 )=n 1 + εg 1 (n 1,p 1,ε) and ε 1 3 (1) V 1 (n 0,n 1 )=e 01 + p 1 p 0 h 1(n 0,n 1,ε), (n 0,n 1 ) S S, with similar estimates on g 1, h 1. Each n i, i =0, 1, ranges over a spherical cap: Q = {n : n =1, (n, p 1 p 0 ) ε 1 3 }. We will complete the proof of the lemma once

13 714 VADIM KALOSHINAND MARK LEVI we show that (17) (1) imply the existence of a unique pair (q 0,q 1 ) Q 0 Q 1 such that (1) holds. Using the above expressions for the unit tangent vectors, we rewrite the matching conditions v 0 = V 0 and v 1 = V 1, obtaining () n 0 = e 01 + R 0 (n 0,n 1,ε), where ( ) ε 1 3 (3) R 0 C1 (C 0 C 1) c ε + p 1 p 0 and (4) n 1 = e 01 + R 1 (n 0,n 1,ε), with a similar estimate for R 1. We now combine (), (4) into (5) n = f(n, ε), where n =(n 0,n 1 ), e =(e 01,e 01 ), f = e + R(n, ε), and R =(R 0,R 1 ). It now remains to prove that f : Q Q S S has a fixed point. By the definition, Q is the ε 1 3 -neighborhood on the unit sphere of e 01, and thus f maps Q Q into itself provided (6) R 0, R 1 ε 1 3. The last two estimates certainly hold according to (3), provided p 1 p 0 >Cfor a sufficiently large C and provided ε is sufficiently small. Now f has a fixed point by the Bohl Brouwer theorem, and hence (5) has a solution. The lemma is proven. In fact, we have proved that the solution of (5) is unique and depends on p i smoothly; indeed, R is C 1 -small and hence f is a contraction. The solution is a smooth function of parameters p 0,p 1 since f also has that property. We describe now the setup for the next lemma. Consider two sections Σ 0 and Σ centered at (0, 0) and (n, ), respectively, where n is an even integer vector: n Z. Consider the metric λ obtained by removing that bump of β which is concentric with Σ; this device of removing the bump turns out to avoid a significant number of technicalities. Fixing any p 0 Σ 0, let L(p) = L(p 0,p) be the geodesic distance from p 0 to p Σ in the metric λds. p Σ is treated from now on as a two-dimensional variable. That is, we identify p =(x, y, ) with p =(x, y), by dropping the last coordinate; accordingly, L R is a two-vector. Lemma 3 (geodesic length). With the above notation for large n we have (7) p L(p0,p)= n n 1 cε 3 + δ, δ < n + e c ε n, where c and c are positive constants independent of ε. Proof of Lemma 3. By Lemma 1 projected onto the (x, y)-plane we have (8) p L(p0,p)=(ẋ, ẏ), where (x(t),y(t),z(t)) = θ(t) is the solution of (8) satisfying (9) and connecting p 0 with p. By Lemma, θ(t) intersects no lenses between the time t of exit from L and the time t p of arrival at p. Thus (ẋ, ẏ) v = const. for t (t,t p ),

14 GEOMETRY OF ARNOLD DIFFUSION 715 i.e., the (x, y)-projection of the trajectory is a straight line; since the radii of Σ and Σ are ε 1 3 and the distance between their centers projected onto the (x, y)-plane is n, we have ( p L(p0,p), n) < cε 1 3 n, where c is a constant independent of ε (any c > works). This establishes the angular part of (7). It remains to show that 0 < 1 v <e c n. The idea is to show that almost all the energy is in the horizontal motion. More precisely, for t (t,t p ) the energy relation gives where v = v, and hence (9) t p t = v +ż = λ 0(z), z(t ) dζ λ 0 (ζ) v. It is intuitively clear that if the above integral is large, then v must be close to min λ 0 = 1, which is just what we are trying to prove. We now make this idea more precise. First, t p t is large if n is large: dθ t p t = θ which in combination with (9) gives θ(t p) θ(t ) max λ 0 > 1 n, (30) dζ λ 0 (ζ) v 1 n, where we have replaced the lower limit of integration z(t )=O(ε) by, obviously increasing the integral. Note that λ 0 (1) = 1, implying v<1. Using λ 0 =1+ε(z 1) + o((z 1) ) we obtain dζ λ 0 (ζ) v c 1 1 ln ε 1 v. Combining this with (30) implies 0 < 1 v < e c n and thus completes the proof. Let Σ, Σ, and Σ + be any three sections with respective centers at ( n, ), (0, 0), and (n +, ), where n ± Z, 0 =(0, 0) Z ; see Figures 5 and 3. For arbitrary points p ± Σ ±, consider the sum (31) S(p) =L(p,p)+L(p, p + ), p Σ, of geodesic lengths of the minimal geodesics 8 from p to p and from p to p +. Here p ± will be treated as fixed and p as a variable. By Lemma S(p) is well defined for all p Σ. 8 These geodesics exist by Lemma.

15 716 VADIM KALOSHINAND MARK LEVI Lemma 4. There exists ε 0 > 0 such that for all 0 <ε<ε 0 the following holds. If the sections Σ, Σ, and Σ + described above are sufficiently far apart, (3) n, n + >ε (k+), and are sufficiently well aligned in the (x, y)-direction, (33) n + n + n n <εk+ 5, then for any p ± Σ ± the sum S(p) =L(p,p)+L(p, p + ) has a minimum in the interior of Σ. If S(p) has a minimum at p = p min Σ, then the broken geodesic γ(p p min p + ) is a true geodesic. The last lemma will be used to shadow pseudogeodesics by true geodesics. Proof of Lemma 4. We first consider the modification Ŝ(p) ofsums(p) obtained by removing the bump β 0 from the metric. We will first show that this truncated action Ŝ(p) is nearly constant on Σ (thanks to the removal of the bump). Following that, we will show that restoring the bump β 0 creates a well inside Σ for the true action S(p), thus completing the proof of the lemma. For any p Σ,p Σ, Lemma 3 gives (34) p L(p,p)= n n + δ, p L(p, p+ )= n + n + + δ +, where δ ± < cε 1 3 n + ± e c ε n ±. Adding these two equations, we obtain (35) Ŝ(p) = n n n ( + n + + cε n + 1 ) + e c ε n + e c ε n +. n + Using (33) and (3) in (35) we obtain (36) Ŝ(p) <εk+ 5 + cε 1 3 ε k+ + exp. small < 3cε k The last estimate shows that Ŝ is nearly constant. More precisely, for all p Σ, Ŝ(0) Ŝ(p) + sup p S 0 (q) Ŝ(p)+ε 1 3 3cε k+ 1 3 < Ŝ(p)+ε k+.5. q Σ If p Σ, then Ŝ(p) =S(p) from Lemma, and the last inequality gives (37) Ŝ(0) S(p)+ε k+.5,p Σ. We will now show that restoring the bump β at (0, 0) decreases S by a larger amount than the preceding error ε k+.5 : (38) S(0) < Ŝ(0) εk+. Once the latter inequality has been shown, the proof of the lemma will be complete; indeed, for all p Σ, S(0) (38) < Ŝ(0) εk+ (37) < S(p) ε k+ + ε k+.5 <S(p),

16 GEOMETRY OF ARNOLD DIFFUSION 717 showing that the minimum is not on the boundary and is thus in the interior of Σ, thus completing the proof of the lemma modulo (38). For γ = γ(p p) γ(pp + )wehave (39) S(0) = γ λds (A) γ λds = ( λds ε k+1 β 0 )ds = Ŝ(0) εk+1 γ γ β 0 ds, where (A) follows from the minimality of γ(p p) and γ(pp + ) in the metric λds, γ similarly corresponds to the metric λds, and β 0 is the bump centered at the origin (i.e., the summand in (6) corresponding to n = 0). It remains to observe that β 0 ds β 0 ds (A) = ds (B) ε, γ γ B ε/ γ B ε/ where (A) follows from the fact that β 0 = 1 inside the ε/-ball B ε/ centered at 0 and where (B) follows from the fact that the curve γ passes through the center of the ball and hence is at least as long as the ball s diameter. The last estimate on the effect of the bump used in (39) proves (38). 4. Proofs of Theorems 1,, and 3. The turning Theorem 1 is a trivial consequence of the shadowing Theorem, and it suffices to prove the latter. Proof of Theorem. We choose 0 <ε<ε 0, where ε 0 is as in Lemma 4. Let us pick an arbitrary finite sequence of integer vectors n i Z satisfying the conditions (7) of the theorem. Pick also an arbitrary sequence σ i = ±, i Z. We now consider an associated sequence of sections Σ i obtained by consecutive displacements of the section Σ 0 := {(x, y, 0) : x + y ε 1 3 } by the vectors (n i,σ i ). By Lemma the geodesic distance L(p i,p i+1 ) is a smooth function for any p i Σ i,p i+1 Σ i+1, and by Lemma 4 the sum S i (p) =L(p i 1,p)+L(p, p i+1 ) has a minimum inside Σ i for any fixed p i 1 and p i+1 (i =1,...,N 1). This implies that the minimum of the length function (40) S(p 1,p 1,...,p N )= N 1 k=0 L(p k,p k+1 ) lies in the interior of its domain Σ =Σ 1 Σ N 1. Indeed, assume that a minimizer (p 1,...,p N ) Σ, so that p k Σ k for some 1 k N. But by moving p k to the interior of Σ k we can decrease the sum S k (p k )=L(p k 1,p k )+L(p k,p k+1 ) without affecting other terms in S, thus decreasing S, in contradiction with the assumption. An interior minimizer of S corresponds to a geodesic. 9 Proof of Theorem 1. Let us choose a sequence of n i Z,0 i N, with N>πε (k+ 5 ) subject to the conditions of Theorem and, in addition, let arg e i+1 arg e i 1 εk+ 5, ei = n i n i. 9 Without extra assumptions on the bump we cannot claim that the geodesic is unique. In fact, uniqueness may fail if the function η defining the shape of the bump oscillates.

17 718 VADIM KALOSHINAND MARK LEVI R R r Arbitrary choice of n i. n i are chosen to return to inner cylinder. Fig. 9 With R = Cε (k+ 9 ) (with an appropriately large C) there is enough room for the geodesic to wind in the (x, y)-direction while staying in the cylinder (Theorem 1). By Theorem there exists a geodesic with the itinerary given by (n i,σ i ); this geodesic turns through the angle ( ) ( ) 1 α N >N εk+ 5 > πε (k+ 5 1 ) εk+ 5 = π. Proof of the corollary to Theorem. To estimate the fastest possible speed of Arnold diffusion we make order of magnitude transitions as small as the lower bounds allow; specifically, we impose the upper bounds n i < ε (k+) consistent with the preexisting conditions on n i. Similarly, we impose an upper bound on N: N < πε (k+ 5 ) + 1. Then the total length of the geodesic with N corners is of the order of ni <N ε (k+) < 3πε (k+ 5 ) ε (k+) =3πε (k+4 1 ). This shows that the speed of diffusion is O(ε k+4 1 ). Proof of Theorem 3. We first show that any unit vector r R 3 is a rotation vector for some geodesic. The idea is similar to the one used to prove that rotation numbers in a van der Pol type equation form an interval. We choose a sequence of (n i,σ i ) with σ i = and n i satisfying the conditions of Theorem, and such that (41) lim k k i=1 (n i,σ i ) k i=1 (n i,σ i ) = r. One specific way to make such a choice is sketched in Figure 9. Once the itinerary is chosen, the corresponding geodesic exists by Theorem ; this geodesic s rotation number is r, according to (41). It remains to show that there exists a geodesic whose rotation set is S. The problem reduces to finding an admissible sequence (n i,σ i ) for which the closure of the set {u k } of approximate directions k i=1 u k = (n i,σ i ) k i=1 (n i,σ i )

18 GEOMETRY OF ARNOLD DIFFUSION 719 is S. The idea is the same as that in extracting a diagonal subsequence. More specifically, let r m be a dense sequence on S. By the above argument we can find an itinerary {(ni, σ i )} such that for some sequence of integers k 1 <k < we have u k1 r 1 1, (4) u k r 1 1, u k 3 r 1, u k4 r 1 1 3, u k 5 r 1 3, u k 6 r 1 3,... With this choice of {(ni, σ i )} the set {u k } is dense in S. But u k is a good approximation of the direction of a long geodesic segment: u k γ(t ( ) k) ε γ(t k ) = O. γ(t k ) Now the sequence u k is dense, while γ(t k ) as k. Thus the sequence γ(t k ) γ(t k ) is dense as well. We have produced a geodesic whose rotation set is the entire S. Acknowledgment. The authors would like to thank Patrick Bernard for bringing Bessi s work to their attention and for helpful comments. The first author would like to thank his Ph.D. advisor John Mather for teaching him the Aubry Mather theory. REFERENCES [A] V. Arnold, Mathematical Methods of Classical Mechanics, nd ed., Grad. Texts in Math. 60, Springer-Verlag, New York, [A1] V. Arnold, Instabilities in dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5 (1964), pp [AKN] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, Translated from the 1985 Russian original by A. Iacob. [B] P. Bernard, The dynamics of pseudographs in convexhamiltonian systems, J. Amer. Math. Soc., 1 (008), pp [BC] P. Bernard and G. Contreras, A generic property of families of Lagrangian systems, Ann. of Math. (), 167 (008), pp [BB] M. Berti and Ph. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (00), pp [Bs] U. Bessi, An approach to Arnold s diffusion through the calculus of variations, Nonlinear Anal., 6 (1996), pp [BCV] U. Bessi, L. Chierchia, and E. Valdinoci, Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl., 80 (001), pp [BK] J. Bourgain, and V. Kaloshin, On diffusion in high-dimensional Hamiltonian systems, J. Funct. Anal., 9 (005), pp [CY1] C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (004), pp [CY] C.-Q. Cheng and J. Yan, Arnold Diffusion in Hamiltonian Systems: The A Priori Unstable Case, preprint. [CI] G. Contrerasand R. Iturriaga, Global Minimizers of Autonomous Lagrangians, Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, Brazil, [DLS] A. Delshams, R. de la Llave, and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (006). [D] R. de la Llave, A tutorial on KAM theory, in Smooth Ergodic Theory and Its Applications (Seattle, 1999), Proc. Sympos. Pure Math. 69, AMS, Providence, RI, 001, pp [Do] R. Douady, Stabilité ou instabilité des point fixes elliptiques, Ann. Sci. École Norm. Sup. (4), 1 (1988), pp

19 70 VADIM KALOSHINAND MARK LEVI [Fa] A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Stud. Adv. Math. 88, Cambridge University Press, Cambridge, UK, 003. [Fe] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 1 (1971), pp [H] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l anneau Vol. II, Astérisque, (1986), p [HPS] M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, Berlin, [KL] V. Kaloshin and M. Levi, An example of Arnold diffusion for near-integrable Hamiltonians, Bull. Amer. Math. Soc. (N.S.), 45 (008), pp [Le] M. Levi, Shadowing property of geodesics in Hedlund s metric, Ergodic Theory Dynam. Systems, 17 (1997), pp [Lo] P. Lochak, Canonical perturbation theory: An approach based on joint approximations, Uspekhi Mat. Nauk, 47 (199), pp (in Russian); translation in Russian Math. Surveys, 47 (199), pp [LM] P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Cent. Eur. J. Math., 3 (005), pp [MS] J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convexnearintegrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., 96 (00), pp [Ma1] J. Mather, Action minimizing invariant measures for positive definite Lagrangians, Math. Z., 07 (1991), pp [Ma] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier, 43 (1993), pp [Ma3] J. Mather, Modulus of continuity of Peierls s barrier, in Periodic Solutions Hamiltonian Systems and Related Topics, P. Rabinowitz, ed., NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 09, Reidel, Dordrecht, The Netherlands, 1987, pp [Ma4] J. Mather, Arnold diffusion. I: Announcement of results, J. Math. Sci. (N.Y.), 14 (004), pp [Ma5] J. Mather, Total disconnectedness of the quotient Aubry set in low dimensions, Comm. Pure Appl. Math., 56 (003), pp [Ma6] J. Mather, Graduate Class , Princeton, 00. [Ma7] J. Mather, Arnold Diffusion. II, preprint, 006. [Mo] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Math.-Phys. Kl. IIa, 196, pp [MZ] J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lect. Notes Math. 1, New York University, Courant Institute of Mathematical Sciences, New York, 005. [Ne] N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspekhi Math. Nauk, 3 (1977), pp [Ni] L. Niederman, Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, 7 (007), pp [P] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (198), pp [S] K. Siburg, The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Math. 1844, Springer-Verlag, Berlin, 004. [T1] D. Treschev, Multidimensional symplectic separatrixmaps, J. Nonlinear Sci., 1 (00), pp [T] D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (004), pp [X] Z. Xia, Arnold diffusion: A variational construction, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 1998, pp