An entropic characterization of the flat metrics on the two torus

The geodesic flow of the flat metric on a torus is minimizing the polynomial entropy among all geodesic flows on this torus. We prove here that this properties characterises the flat metric on the two torus.


Introduction
There are several classes of hyperbolic manifolds on which the metrics with constant curvature are characterized by the fact that their geodesic flow is minimizing the topological entropy, see [6,16] for example. The situation is different on tori. Flat metrics have zero entropy, but other metrics also have zero entropy, such as the tori of revolution. In order to characterize the flat metrics, it is therefore useful to consider a finer dynamical invariant of the geodesic flow, such as the polynomial entropy, introduced in [22]. f n (we will write (n, ε)separated). Recall that a set E is said to be ε-separated for a metric d if for all (x, y) in E 2 , d(x, y) ≥ ε. Denote by S f n (ε) the maximal cardinal of a (n, ε)-separeted set contained in X.
Observing that S One easily checks that if φ 1 is the time-one map of φ, h pol (φ) = h pol (φ 1 ).
The following properties of the polynomial entropy are proved in [22].

Property 2.1 1. h pol is a C 0 conjugacy invariant, and does not depend on the choice of topologically equivalent metrics on X. 2. If A is an f-invariant subset of X, then h pol
We conclude this section with the following useful result which relates the polynomial entropy of a flow with that of a Poincaré map. Proposition 2.1 Let M be a smooth manifold, d a distance on M associated with a Riemannian metric, and X a C 1 complete vector field on M with flow φ = (φ t ) t∈R . Let A a be compact φ-invariant subset of M and let be a C 1 codimension 1 embeddded submanifold of X such that: • for any a ∈ A, there exists t > 0 such that φ t (a) ∈ .
• for any a ∈ A ∩ , X(a) is transverse to .
Then the Poincaré return map ϕ : A ∩ → A ∩ is well defined, continuous and satisfies Proof Let τ : A ∩ → R * + : a → τ a be the first return time map of ϕ. Since the function τ is continuous on the compact set A ∩ , we have T := max{τ a | a ∈ A ∩ } < ∞. Let d be the distance induced by d on .
There exists τ * > 0 and a neighborhood V of A ∩ in such that the map is a C 1 -diffeomorphism onto its image. Its inverse is thus locally Lipschitz, hence its restriction to the compact set K := ([−τ * , τ * ]×(A∩ )) is Lipschitz. As a consequence, there exists δ > 0 such that for any t, t ∈ [−τ * , τ * ] and a, a ∈ A ∩ Note that τ * < 1 4 min{τ a | a ∈ A∩ }. Since the compact sets A∩ and A\ (]−τ * , τ * [×V ) are disjoint, the constant δ can be chosen such that d(a, x) δτ * for each a ∈ ∩ A and x ∈ A\K .
Let τ k x be the successive return times of the point x, so that ϕ k (x) = φ τ k x (x). Note that τ 1 x = τ x , and τ k+1 kT for all x ∈ A ∩ . We will now prove that two points x and y of A ∩ which are (n, ε)-separated by ϕ are (nT, δε) separated by φ for ε < τ * . There exists m ∈ {0, . . . , n} such that d (ϕ m (x), ϕ m (y)) ε. Let us assume for definiteness that τ m (2), hence x and y are (τ m x , δε)separated by φ.

Some definitions from weak KAM theory
We work on the d-dimensional torus T := R d /Z d , and will mostly consider the case d = 2. A Tonelli Hamiltonian on T is a C 2 Hamiltonian function H (q, p) : T × R d (=T * T) −→ R such that, for each q ∈ T, the function p −→ H (q, p) is convex with positive definite Hessian and superlinear. The Hamiltonian vectorfield on T * T is given by It generates a complete flow ϕ t H which preserves the function H.
is a C 2 field of positive definite symmetric matrices, we associate the Hamiltonian It is well-known that the Hamiltonian flow of H is conjugated to the geodesic flow by In other words, geodesics are the projections of Hamiltonian orbits.
Returning to the general case of a Tonelli Hamiltonian, the α function of Mather is defined on where the infimum and the minimum are taken respectively on the set of smooth functions on T and on the set of C 1 functions with Lipschitz differential. It was proved in [4] that the minimum exists on the set of C 1,1 functions, see also [11]. A C 1,1 function satisfying the inequality at each point q ∈ M is called a c-critical subsolution (as we just recalled, such functions exist). There may exist several c-critical subsolutions. At least one of them, w, has the property that for all critical subsolutions u. We define where the intersection is taken on all c-critical subsolutions u. This is a non-empty compact set, called the projected Aubry set. In view of the strict convexity of H in p, the differential du(q) of a c-subsolution u at a point q ∈ A(c) does not depend on the c-critical subsolution u. We define for each c-critical subsolution u. This set is called the Aubry set, it is invariant under the flow of H, compact, and not empty. It is moreover contained in the graph of the Lipschitz closed form c + du for each c-critical subsolution u. A consequence of the invariance of A * (c) is that the projected Aubry A(c) set is invariant under the vectorfield on T for each c-critical subsolution u. The special c-critical subsolution w introduced above has the property that the strict inequality where μ is the projection of μ * on T. So it is just the rotation number of μ seen as a χ-invariant probability measure on T.
There is an interesting relation between the function c −→ α(c) and the rotation numbers of Mather measures, which was discovered by Mather [26]: The function α is convex and superlinear on H 1 (T, R). Moreover, its subdifferential ∂α(c) ∈ H 1 (T, R) in the sense of convex analysis is precisely the set of rotation numbers of Mather measures at cohomology c.
It is well-known (and easy to prove using the convexity of H) that the inequality holds for each compactly supported invariant probability measure μ * . Moreover, the invariant probability measures achieving equality are precisely the Mather measures at cohomology c. This explains the c-minimizing terminology. Our presentation however is not completely standard because we work on T * T instead of T T. If everything is sent back on T T using the Legendre transform, then the function p · ∂ p H (q, p) − H (q, p) is sent to the more familiar Lagrangian function. Following Mather, we denote by β(h) : In the geodesic case, where H is quadratic in the fibers, the functions α and β are homogeneous of degree 2. The function √ β, which is homogeneous of degree one, is called the stable norm.

Flat metrics and proof of the Corollary 1.1
Let us discuss the case of flat metrics on T = T d , which form an easy class of examples. A flat metric is isometric to a constant metric of the form is a fixed positive definite symmetric matrix. The corresponding Hamiltonian function is For each non resonant v ∈ , the torus T × {v} is invariant, topologically transitive, and d-dimensional, hence it is sent by ϕ to a torus with the same properties, which must be a torus of the form T × {w}, for some w ∈˜ . By density of the non resonant vectors in , we conclude that the map ϕ is of the form ϕ(q, v) = (ϕ 1 (q, v), φ(v)). Since ϕ is a homeomorphism, so is φ : −→˜ . In the case d 3 the map ϕ induces a homology map

Proposition 3.1 If the Hamiltonian flows of two Hamiltonians H
is upper triangular by blocks, and the left upper bloc is a matrix A ∈ Gl 2 (Z) which describes the action on homology of the restriction ϕ |T×{v} for each v ∈ . For each v ∈ ∩ (RZ d ), each orbit of T × {v} is periodic and gives rise to an oriented closed curve whose homology is the only indivisible element h of R + v ∩ Z d . Moreover, the minimal period of this orbit is the positive real number T such that T v = h. The image by ϕ of such a periodic orbit is a periodic orbit of minimal period T and homologyh = Ah. Since this orbit belongs to T × {φ(v)}, we conclude thath = T φ(v), hence that φ(v) = Av. By continuity, this equality holds for each v, hence˜ This implies thatñ • A = n, hence that G = A tG A.
Proof of Corollary 1.1 If the geodesic flow of the metric g is topologically conjugated to the geodesic flow of the flat metric g, then the polynomial entropy of g is equal to the polynomial entropy of g, which is equal to 1. Theorem 1 implies that g is flat, and Proposition 3.1 then implies that g and g are isometric.

The special case of dimension two, the main statement in the Tonelli case
In this section, we work on the two-dimensional torus T = R 2 /Z 2 . We first recall from [12] some useful facts on rotation sets of flows on T.
Let us consider a flow on T and lift it to the flow ψ t on the cover R 2 [identified with The rotation set of our flow is the set of limits of sequences of the form where x k is a sequence in R 2 and t k −→ ∞. This is a compact and convex subset of R) which is not empty. If the flow is generated by a vectorfield ν, then its rotation set is also the set of the rotation numbers νdμ of all invariant probability measures μ. The rotation set of a flow on the two torus T is a compact interval contained in a straight line through the origin of R 2 = H 1 (T, R). Moreover, see [12]: For each e > min α, the set is a compact and convex set, whose interior is {α < e} and whose boundary is α The following is well known, see [2,23,25], but since we give the statement in a way which is not obviously equivalent to those of these papers, we will provide a proof in Sect. 3.4. We recall that the Mather set M * (c) is the union of the supports of c-minimizing measures, and that the projected Mather set M(c) ⊂ T is its projection. If H is the Hamiltonian associated to a Riemaniann metric, then this implies that the metric is flat, in view of the Theorem of Hopf, see also [15]. As a consequence, Theorem 1 follows from: Theorem 2 Let e > min α be a given energy level. If there exists a cohomology c ∈ α −1 (e) in case 3, then the polynomial entropy of the Hamiltonian flow restricted to the energy level {H = e} is not less than 2. In other words, if the polynomial entropy of the flow restricted to the energy level {H = e} is less than two, the Aubry sets A * (c), c ∈ α −1 (e) are Lipschitz invariant graphs which partition the energy level.
We will make use in the proof of two important properties of the Aubry sets:

Property 3.1 The set-valed map c −→ A * (c) is outer semi-continuous. It means that each open set U ⊂ T * T containing A * (c), also contains A * (c ) for c close to c.
We recall the definition of the vectorfield χ(q) := ∂ p H (q, c + dw(q)) on T.

Property 3.2 For each c ∈ α −1 (e), there exists a global curve of section of A(c). More precisely, there exists a cooriented C 1 embedded circle ⊂ T such that χ(q) is transverse to on A(c) and respects the coorientation. Moreover, each half orbit of A(c) intersects . The flow of A(c) thus induces a homeomorphism ψ of ∩ A(c) which preserves the cyclic order of .
Proof of Property 3.1 It is proved in [5] using the content of [10].
Proof of Property 3.2 Let us consider a cohomology c 0 such that α(c 0 ) < e and such that c − c 0 ∈ H 1 (T, Q). Since ρ(c) is the outer normal to ∂ A(e) at c and c 0 belongs to the interior of A(e), we have We consider a c-critical subsolution w of class C 1,1 and strict outside the Aubry set. We also consider a c 0 -critical subsolution u 0 . Let l ∈ N be such that l(c − c 0 ) ∈ H 1 (T, Z). Let us consider the C 1,1 functionˆ on R 2 defined by Let us consider a regular value θ of . Such a value exists by fine versions of Sard's Theorem (see [3]) since is C 1,1 . The preimage −1 (θ ) is a 1-dimensional cooriented submanifold of T. It can be seen as an intersection cocycle of cohomology l(c − c 0 ). It is a finite union of embedded cooriented circles i each of which is a cocycle of cohomology σ i , with σ i = l(c − c 0 ). Since ρ(c) · l(c − c 0 ) > 0, there exists j such that ρ(c) · σ j > 0. We denote by the cooriented circle j . Since d (q) · χ(q) > 0, the orbits of A(c) are transverse to , and intersect it according to the coorientation.
Finally, each half orbit of A(c) has a rotation number contained in ρ(c). We have seen that σ · ρ(c) > 0, where σ is the cohomology of the intersection cocycle associated to . Each half orbit of A(c) thus intersects . As a consequence, the flow of A(c) generates a Poincaré map which is a bi-Lipschitz homeomorphism preserving the cyclic order on the circle . This implies that ψ can be extended to a homeomorphism of preserving the cyclic order.

Faces of the balls of α on the two torus
We take d = 2 and fix an energy level e > min α. We study the affine parts of the ball α −1 (e) and prove Propositions 3.2 and 3.3. The following is a variant of a Lemma of Massart [23]: The following Lemma also comes from Massart [23]:

Then the Aubry set A * (c) does not change when c varies in
We will see that, unlike the Mather set, the Aubry set can be bigger at the boundary points.

Proof Let us consider a point c = ac
1} be a c i -critical subsolution strict outside the Aubry set. Then, w c := aw 0 + (1 − a)w 1 is a ccritical subsolution. Using the strict convexity of H in p, we observe that the strict inequality H (q, c + sw c (q)) < e holds outside of the set where H (q, c 0 + dw 0 (q)) = e and H (q, c 1 + dw 1 (q)) = e and c 0 + dw 0 = c 1 + dw 1  We recall that F(c) is defined as the largest segment of α −1 (e) containing c. The following was first proved by Bangert, see [2]:

Corollary 3.2 If ρ(c) has an irrational direction, then F(c) = c.
Proof As above, let us consider a cohomology c satisfying the hypothesis of the Corollary, and a cohomology c such that [c, c ] ∈ α −1 (e), hence M * (c ) = M * (c), by Lemma 3.1. We will prove that c = c, which implies the Corollary. Note that (c − c) · ρ(c) = 0, so that it is enough to prove that (c − c) · [ ] = 0, where [ ] ∈ H 1 (T, Z) is the homology of the section given by Property 3.2 (equipped with an orientation). Let w and w be c and c -critical subsolutions. We consider the closed Lipschitz form η = c − c + dw − dw, whose cohomology is c − c and prove that η = 0.
Since M * (c ) = M * (c), we have η = 0 on M(c). It is thus enough to prove that I η = 0 for each connected component I of the complement of M(c) ∩ in . We first observe that where ψ is a homeomorphism of extending the return map of ∩ A(c). To prove this equality, we integrate η on the contractible closed curve made of the interval I = ]q − , q + [, followed by the orbit of q + until its return ψ(q + ), followed by the interval −ψ(I ) = ]ψ(q + ), ψ(q − )[ followed by the piece of orbit of q − in negative time direction from ψ(q − ) to q + .
Since the intervals ψ k (I ) are two by two disjoint in , their lengh is converging to zero. Since the form η is bounded, this implies that ψ k (I ) η −→ 0, hence that I η = 0.
In view of these corollaries there are three cases: Let us study more precisely the last case. We denote by [c − , c + ] the face F(c). We assume for definiteness that c is an interior point of this face, which means that either c ∈ ]c − , c + [ or c − = c = c + .
We consider the cooriented section given by Property 3.2. We orient in such a way that is contained in such an annulus U, is α-asymptotic to one of its boundaries, and is ω-asymptotic to its other boundary. We say that such an orbit is positive if it crosses the annulus U according to the orientation of , and that it is negative if it crosses in the other direction. In other words, the heteroclinic orbit is positive if the sequence of its successive intersections with the interval I is increasing. The following implies Proposition 3.2: , and x m < ψ τ c m (x m ). At the limit, using the semi-continuity of the Aubry set, we find a point x ∈ J ∩ A(c + ) such that ψ τ (x) x, hence ψ τ (x) > x. We conclude that the heteroclinics are positive.
For the convenience of the reader, and because our statement is not exactly the one of [24], Theorem 3, we now prove Proposition 3.3, following [24]: We consider an energy level e > min α such that the curve α −1 (e) does not contain any non-trivial segment, which is equivalent to saying that M(c) = T for each c such that ρ(c) is rational. Note then that the map ρ : α −1 (e) −→ S H 1 (T, R) is continuous and bijective, hence it is a homeomorphism. Since the set S H 1 (T, Z) of rational directions is dense in For each point c in this set, we have A(c) = T. In view of the semi-continuity of the Aubry set, we deduce that A(c) = T for each c ∈ α −1 (e). As a consequence, there exists a unique (up to the addition of a constant) c-critical subsolution w c , which is actually a solution, and the Aubry set A * (c) is the graph of c + dw c . Moreover, the functions dw c , c ∈ α −1 (e) are equi-Lipschitz. The semicontinuity of the Aubry set A * implies that the map c −→ c + dw c (q) is continuous for each q ∈ T.
The orbits of A * (c) all have a forward rotation number in ρ(c). For c = c, the orbits of A * (c ) all have a forward rotation number in ρ(c ), and, since ρ(c ) = ρ(c), the sets A * (c) and A * (c ) are disjoint. As a consequence, for each q ∈ T, the map c −→ c + dw c (q) is one to one on α −1 (e), hence it has degree ±1 as a circle map into { p ∈ T q T : H (q, p) = e}. It is thus onto, which implies that the Aubry sets fill the energy level.

Lower bound for the polynomial entropy
We prove Theorem 2. We consider an energy level e > min α, assume that the ball α −1 (e) contains a non-trivial face [c − , c + ], and prove that the entropy of the Hamiltonian flow on the energy level H −1 (e) is at least two. The proof have similarities with the ones of [19,21,22]. We work with the section of A(c + ) given in Property 3.2. We fix a parameterisation R/Z −→ , and put on the distance such that this parameterisation is isometric. This distance is Lipschitz equivalent to the restriction of the distance on T.
We denote by * the set of points of the energy surface H −1 (e) which project on , we endow it with a distance which satisfies d((q, p), (q , p )) d(q, q ). We consider the compact invariant set of the Hamiltonian flow (on the energy level) defined by The surface * is a transverse section of the flow on this invariant set, as required in Proposition 2.1. We denote by the corresponding return map of A ∩ * . The restriction of to A * (c m ) ∩ * is conjugated to ψ c m by the projection. In view of Proposition 2.1, it is enough to bound from below the polynomial entropy of on A ∩ * . We exhibit a sufficiently large separated set using the orbits  is ( 0 , 4m)-separated by τ , hence ( 0 , 4mτ )-separated by (for m m 0 ).
Since the cardinal of this union is more than m 2 , we conclude that the polynomial entropy of is at least two. By Proposition 2.1 the polynomial entropy of the Hamiltoinan flow on the energy surface is at least two.