Positive metric entropy arises in some nondegenerate nearly integrable systems

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^\infty$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent, namely, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori.


Introduction
Already in the early 50's the study of nearly integrable Hamiltonian systems has drawn the attention of many outstanding mathematicians such as Arnol'd, Kolmogorov and Moser. Indeed, for any integrable Hamiltonian system the whole phase space is foliated by invariant Lagrangian submanifolds that are diffeomorphic to tori, generally called KAM tori, and on which the dynamics is conjugated to a rigid rotation. Therefore, it is natural to ask what happens to such a foliation and to these stable motions once the system is slightly perturbed. In 1954 Kolmogorov [10] -and later Arnol'd [1] and Moser [11] in different contexts -proved that, for small perturbations of an integrable system it is still possible to find a big measure set of KAM tori. This result, commonly referred to as KAM theorem, contributed to raise new interesting questions, for instance about the destiny of the stable motions that are destroyed by effect of the perturbation (in other words, about the dynamics outside KAM tori). In this context, Arnol'd [2] constructed an example of a perturbed integrable system, in which some orbits outside KAM tori have a wide range in action variables (even though the rate of change of action variables is exponentially small [12]). This striking phenomenon, known as Arnol'd diffusion and still quite far from being fully understood, shows the presence of some randomness in the dynamics outside KAM tori. The question we address in the present paper is therefore the following: how much random can the motion outside KAM tori be?" It is well-known that, C 2 -generically the Hamiltonian flow has positive topological entropy (cf. [13], see also [6] for an analogous statement for Riemannian geodesic flows). Once we turn our attention to metric entropy, the problem becomes more challenging and one cannot simply derive positive metric entropy from positive topological entropy. In fact, Bolsinov and Taimanov [3] built an example of a Riemannian manifold on which the geodesic flow has positive topological entropy but zero metric entropy.
Recently Burago and Ivanov [4] used dual lens map to construct a reversible Finsler metric C ∞ -close to the standard metric on S n , n ≥ 4, such that its geodesic flow has positive metric entropy. However the geodesic flow on the sphere is degenerate, hence it does not lie in the realm of KAM theory.
Unlike the case of spheres, the geodesic flow on flat tori are nondegenrate. In this paper we therefore provide an example analogous to Burago-Ivanov's one on T 3 . More precisely, we prove the following: Main Theorem. For every ǫ > 0 there exists a reversible Finsler metric on T 3 which is ǫ-close to the Euclidean metric in the C ∞ -sense and such that the associated geodesic flow has positive metric entropy.
Our theorem shows that in the complement of KAM tori, the behavior of nearly integrable Hamiltonian flows can be quite stochastic. However our example does not possess Arnold diffusion. For details, see Remark 4. In order to prove the main theorem we start with perturbing the return map associated with the standard geodesic flow on a specific section to get positive metric entropy. To prove positiveness of metric entropy we use Maupertuis principle and Donnay-Burns-Gerber cap [5][7] to perturb the kinetic Hamiltonian. This method was also used by Donnay and Liverani [8]. By pulling back via the standard projection, we can perturb the return map to get positive metric entropy. Now, using Burago-Ivanov dual lens map theory [4], from the perturbed return map we get a (reversible) Finsler metric on T 3 satisfying all the requirements of the main theorem. We shall however notice that, by upper semicontinuity (see [14]), the metric entropy we get is microscopic.

Preliminaries
Let M be a smooth n-dimensional manifold, T * M its cotangent bundle, and ω the standard symplectic form on T * M. To the pair (H, ω) we can associate a unique vector field X H by ω(X H , V ) = dH(V ) for any smooth vector field V on T * M, which is called the Hamiltonian vector field. The flow Φ t H on T * M defined by X H is called the Hamiltonian flow of H.
A typical example of a Hamiltonian flow is the geodesic flow on a Finsler manifold. Let ϕ be a Finsler metric on M, i.e. a smooth family of quadratically convex norms ϕ(x, ·) on each tangent space T Denote with UT M its unit tangent bundle; the Finsler metric ϕ defines a dual norm on the cotangent bundle T * M by The geodesic flow g t on (M, ϕ) is defined to be the Hamiltonian flow on T * M with Hamiltonian (ϕ * ) 2 /2. Recall that the geodesic flow can also be viewed as the Euler-Lagrange flow on T M associated with the 2-homogeneous Lagrangian ϕ 2 /2. One can easily see that Φ t H is a symplectomorphism (i.e. preserves ω) and hence volume-preserving. Once we fix a level set H −1 (c), we can define a conditional measure on this level set from the volume form. Such conditional measure is invariant under Φ t H and it is called the Liouville measure. For any point x in (M, ϕ), the unit ball B x in T x M is a convex body. By F. John [9], among all ellipsoids contained in B x , there exists a unique ellipsoid E x with maximum volume. E x is the unit sphere of some quadratic form on T x M. In this way we can define quadratic forms on each tangent spaces and these forms are close to Finsler norms. In this way we can associate with the Finsler metric ϕ a Riemannian metric g ϕ , from which UT M inherits a Riemannian structure (see [16] for details). This metric is called the Sasaki metric. For each vector ζ ∈ T v UT M we define the Lyapunov exponent by For our purpose, there is no need to recall the precise definition of the metric entropy h µ for the Liouville measure µ on UT M. Indeed, it is enough to know that Pesin's inequality [15] provides a lower bound of metric entropy. Indeed, this formula tells us that the metric entropy is no less than the mean of upper Lyapunov exponent.
The Gaussian curvature is positive on {r ≤ r 0 }, negative at C r 1 , and strictly decreasing from center to boundary.
If a torus contains a non-ergodic DBG cap and outside the cap the Gaussian curvature is nonpositive, then we call it a non-ergodic DBG torus. Sketch of proof. The proof is similar to the proof of Theorem 1.1 in [5]. By virtue of Clairaut's integral, any geodesic entering the cap C will go out of the cap. Let c : [−T 1 , T 1 ] → C be an arc-length parametrized geodesic with endpoints in C r 1 such that c(0) is the point of c closest to the origin; suppose furthermore that c(±T 2 ) lie in C r 0 , for some 0 < T 2 < T 1 . Let J S , J C be two Jacobi fields on c with be the Gaussian curvature at c(t). Then both u S and u C satisfy the Riccati equation: By imitating the proofs of Lemma 2.5 and Lemma 2.6 in [5], we get This means the graph of u must lie above that of u S . By (A) and (B) we have u(T 1 ) ≥ 0. So the cone J ′ J ≥ 0 is preserved by the cap. Figure 2. Graphs of u S , u C and u By Poincaré recurrence theorem, almost every vector in UT C will come back infinitely many times. For any geodesic c entering the cap C at time t 0 , when it returns to the cap again, say at time t 1 > t 0 , the image of the cone {J ′ (t 0 )J(t 0 ) ≥ 0} under the translation will lie strictly in the interior of {J ′ (t 1 )J(t 1 ) ≥ 0}. By Wojkowski's cone field theory [17], the vectors with non-zero Lyapunov exponents form a set with positive Liouville measure. By Pesin's inequality (1) the geodesic flow has positive metric entropy.

Construction of a Non-ergodic DBG torus
In this section we construct a conformal metric on [−1, 1] × [−1, 1] which is flat outside a disc and centrally symmetric inside the disc. More precisely we want to build a function g : [0, 2] → (0, 1] such that the torus with conformal metric is a non-ergodic DBG torus.
In order to get such a function g we change our coordinate system to geodesic polar coordinates. However before doing this we need some preliminary.
Proof. Since Both sides of ( * ) have singularity at 0. Since ρ is odd at 0, ρ ′ (0) = 1, for small l we whereρ is a smooth function that is even at 0. We integrate both sides of ( * ) regarding r as a function of l with r(0) = 0. Then we get Therefore lim l→0 ln(r/l) = 0 and ln(r/l) is even at 0. By direct computation, it is now easy to see that r/l is even at 0. This implies that r is odd at 0. From ( * ) we have Therefore ln r(l) is strictly increasing and smooth, so is r(l). By the Inverse Function Theorem there exists a smooth l : R ≥0 → R ≥0 which is the inverse function of r(l). (v) There exists l 2 > l 1 such that K(l) is negative on [l 1 , l 2 ) and ρ ′ (l) = 1 for l ≥ l 2 .
Indeed once we have such a function ρ, by Lemma 4.2 we have smooth functions g, l : R ≥0 → R ≥0 with ρ(l(r)) = rg(r) and l(r) = r 0 g(t)dt. Consider the metric defined by (2). By changing the coordinate system to geodesic polar coordinates, the metric becomes ds 2 = dl 2 + ρ(l) 2 dθ 2 .
(v) guarantees the metric is negatively curved on the annulus {r 1 < r < r 2 } and is flat outside {r = r 2 }. So once ρ satisfies (i)-(v), the torus with metric (3) will be a non-ergodic DBG torus.
Here is the construction of ρ(l): For any a > 0, let λ 1 : R ≥0 → [0, 1] be a C ∞ function with the properties that a . The last part to be verified is (iv). Since we only need to verify that ρρ ′′′ − ρ ′ ρ ′′ = 100a 2 l 3 (1 + 12al 2 − 40a 2 l 4 ) is positive on (0, 1 √ 5a ]. This can be done by direct calculation. This finishes the construction. Remark 1. The function g constructed in this way is strictly decreasing on [0, r 2 ] and constant for r ≥ r 2 since dρ dl = dρ dr a . So the supremum of g is g(0) = 1. From Lemma 4.2 we know that the lower bound is positive. Figure 3. Graph of ρ Remark 2. If g satisfies the condition that a torus with metric g(r) 2 (dx 2 + dy 2 ) is non-ergodic DBG, we can find a constant δ 0 such that for all δ ∈ (−δ 0 , δ 0 ), a torus with metric (g(r) 2 + δ)(dx 2 + dy 2 ) is also non-ergodic DBG. This follows from the fact that being a non-ergodic DBG torus is an open condition.

Perturbation of the Hamiltonian H 0
Suppose the fundamental domain of the deck group on the universal cover of our torus T 2 ∼ = R 2 /Z 2 is {−1 < x, y < 1}. We use α, β to denote the coordinates in the cotangent space and denote B * T 2 := {(x, y, α, β) ∈ T * T 2 : α 2 + β 2 < 1}. In this section we want to perturb the kinetic Hamiltonian Since g is positive and 0 ≤ 1 − g 2 < 1 (by Remark 1), we have Notice that if H ǫ < 1/6 then α 2 + β 2 < 1/3, therefore ξ ≡ 1 whenever the total energy is small. By the Maupertuis principle, the Hamiltonian flow Φ t Hǫ on the level set {H ǫ = ǫ} is a time change of the geodesic flow on T 2 with metric This metric has positive metric entropy since, by Lemma 3.2, the metric ds 2 = g(r) 2 (dx 2 + dy 2 ) does.

Perturbation ofH 0
In this section we prove that a smooth perturbation of can be derived from a suitable perturbation of H 0 . Since this result holds for all degrees of freedom, we use (q, p) to denote the coordinates instead of (x, y, α, β).
Suppose T n = R n /Z n has coordinates q = (q 1 , ..., q n ) and let p = (p 1 , ..., p n ) be the coordinates in the cotangent bundle. Denote B * T n = {(q, p) : Let V (q, p) be a C 2 -smooth function on B * T n . We perturb H 0 andH 0 by V in the following way: Then we have Lemma 6.1. If suppV ⊆ { p 2 i ≤ C < 1} for some C ∈ R + , then for every δ, m, T > 0, there exists ǫ = ǫ(V, δ, m, T ) > 0 such that for each 0 ≤ T ≤ T we have (q, p) by (∆q, ∆p) as they usually do this in calculus books. Put (q(t), p(t)) := Φ tH ǫ (q, p). Suppose that H ǫ (q, p) = E. Theṅ If p 2 i > C, thenṗ(t) ≡ 0, hence ∆p = 0. Consider the trajectory (q(t), p(t)), V p vanishes along it, hence ∆q = 0. Therefore we only need to consider the case p 2 i ≤ C. Since V is compactly supported we may assume that ǫ is small enough so that p 2 i + 2ǫV < (1 + C)/2 < 1. In this case

POSITIVE METRIC ENTROPY ARISES IN SOME NONDEGENERATE NEARLY INTEGRABLE SYSTEMS 11
We can see from the above calculation that since p 2 i + 2ǫV < (1 + C)/2 < 1, (∆q, ∆p) converges to 0 uniformly in C m as ǫ → 0.

The Burago-Ivanov Theorem
Here we use the notions and definitions from [4]. A Finsler metric ϕ on an n-dimensional disc D is called simple if it satisfies the following three conditions: (S1) Every pair of points in D is connected by a unique geodesic.
(S2) Geodesics depend smoothly on their endpoints. (S3) The boundary is strictly convex, that is, geodesics never touch it at their interior points.
Once (D, ϕ) is simple, denote by U in , U out the set of inward, outward pointing unit tangent vectors with base points in ∂D respectively. With any vector ν ∈ U in , we can associate a unique vector β(ν) ∈ U out , namely the tangent vector of the (unique) geodesic with initial velocity ν at its next intersection point with ∂D. This defines a map β : U in → U out , which is called the lens map of ϕ. If ϕ is reversible, then the lens map is reversible in the following sense: −β(−β(ν)) = ν for every ν ∈ U in .
We denote by UT * D the unit sphere bundle with respect to the dual norm ϕ * . Let L : T D → T * D be the Legendre transform of the Lagrangian ϕ 2 /2. It maps UT D to UT * D. For a tangent vector ν ∈ UT x D, its Legendre transform L (ν) is the unique covector χ ∈ U * x D such that χ(ν) = 1. Then consider subsets U * in = L (U in ) and U * out = L (U out ) of UT * D. The dual lens map of ϕ is the map σ : U * in → U * out given by σ := L • β • L −1 where β is the lens map of ϕ. If ϕ is reversible then σ is symmetric in the sense that −σ(−σ(χ)) = χ for all χ ∈ U * in . Note that U * in and U * out are (2n − 2)-dimensional submanifolds of T * D. The restriction of the canonical symplectic 2-form of T * D to U * in and U * out determines the symplectic structure. And the dual lens map σ is symplectic. In [4], Burago and Ivanov proved the following theorem: Theorem 7.1 (Burago-Ivanov [4]). Assume that n ≥ 3. Let ϕ be a simple metric on D = D n and σ its dual lens map. Let W be the complement of a compact set in U * in .
Then every sufficiently small symplectic perturbationσ of σ such thatσ| W = σ| W is realized by the dual lens map of a simple metricφ which coincides with ϕ in some neighborhood of ∂D.
The choice ofφ can be made in such a way thatφ converges to ϕ wheneverσ converges to σ (in C ∞ ). In addition, if ϕ is a reversible Finsler metric andσ is symmetric thenφ can be chosen reversible as well.

Perturbation of flat metric
Let ϕ 0 be the Euclidean metric on T 3 . We regard T 3 as the cube [−1, 1] 3 with sides identified. Let T 0 := [−1, 1] 2 × {−1} be the 2-torus on T 3 given by the "bottom face" of T 3 , and we use x, y, α, β to denote the coordinates in its cotangent bundle. Let z be the vertical coordinate of T 3 and γ the corresponding coordinate in the cotangent space.
Observe that Π is a symplectic bijection between Γ 0 and B * T 0 . Let By a simple calculation we know that the map R : B * R 3 → B * R 3 defined by R(x, y, α, β) := x + α is a lift of R 1 to the universal cover. Define a functionH 0 on B * T 0 bỹ It is not hard to see that R 1 = Φ 1H 0 .
Let R ǫ be a lift of R ǫ 1 to the universal cover. Define a dual lens map σ ǫ : U * in → U * out by σ ǫ (χ) = otherwise.
It is clear that σ ǫ is symmetric and coincides with σ outside a compact set. Moreover σ ǫ → σ in C ∞ as R ǫ → R in C ∞ . The map σ ǫ is a symplectic perturbation of σ 0 and σ ǫ = σ 0 outside a compact set in U * in . By Theorem 7.1, there exists a reversible Finsler metric ϕ ǫ in D 3 that agrees with ϕ 0 in a neighborhood of the boundary ∂D 3 and such that the dual lens map for (D 3 , ϕ ǫ ) is σ ǫ . Now extend ϕ ǫ to the whole T T 3 by setting it equal to ϕ 0 outside D 3 . It has positive metric entropy since the return map does. As ǫ → 0, we have ϕ ǫ → ϕ 0 in C ∞ . Remark 4. The example we construct in the main theorem does not have Arnold diffusion. In fact, since ϕ is close to the flat metric, we have only to prove that the return map on Γ 0 cannot have large range in action variables α, β. This is clear, since for ǫ > 0 small enough and for α 2 + β 2 ≥ 1 the return map coincides withF hence the ranges of action variables are uniformly bounded.

Acknowledgments
The author thanks Dmitri Burago for numerous helpful conversations. In particular he suggested this topic and offered help on this problem. The author is grateful to Moisey Guysinsky, Sergei Ivanov, Anatole Katok, Mark Levi, Federico Rodriguez Hertz and Yakov Sinai for useful discussions. The author thanks the anonymous referee for the helpful comments in revising this paper.