A NOTE ON TONELLI LAGRANGIAN SYSTEMS ON $T^2$ WITH POSITIVE TOPOLOGICAL ENTROPY ON A HIGH ENERGY LEVEL Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 4, pp. 625-635. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200407

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37B40, 37J50, 37J99

A Note on Tonelli Lagrangian Systems on T2 with Positive Topological Entropy on a High Energy Level

J. G. Damasceno, J. A. G. Miranda, L. G. Perona

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus T2 = R2/Z2. We prove that the Lagrangian flow restricted to a high energy level E-1 (c) (i.e., c > c0(L)) has positive topological entropy if the flow satisfies the Kupka-Smale property in E-1 (c) (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on E-1(c)). The proof requires the use of well-known results from Aubry-Mather theory.

Keywords: Tonelli Lagrangian system, Aubry-Mather theory, static classes

Received July 08, 2020 Accepted October 21, 2020

Josué G. Damasceno [email protected] Universidade Federal de Ouro Preto

R. Diogo de Vasconcelos, 122, Pilar, 35400-000, Ouro Preto, MG, Brasil

José Antonio G. Miranda [email protected]

Universidade Federal de Minas Gerais

Av. Antonio Carlos 6627, 31270-901, Belo Horizonte, MG, Brasil

Luiz Gustavo Perona Araújo [email protected]

Universidade Federal de Vigosa — Campus Florestal Rodovia LMG 818, km 6, 35.690-000, Florestal, MG, Brasil

1. Introduction

Let T2 = R2/Z2 be endowed with a Riemannian metric {■, ■). A Tonelli Lagrangian on T2 is a smooth function L: TT2 — R that satisfies two conditions:

• convexity: for each fiber TxT2 = R2, the restriction L(x, ■): TxT2 — R has a positive defined Hessian,

L(x,v) 2

• superhneanty: limiLJi = oo, uniformly m x £ T .

11 11 IMI

The action of L over an absolutely continuous curve 7: [a, b] — T2 is defined by

b

Al(y) = j L(y(t),Y(t)) dt.

a

The extremal curves for the action are given by solutions of the Euler - Lagrange equation which in local coordinates can be written as

dL d_dL _ dx dt dv

The Lagrangian flow fat: TT2 — TT2 is defined by fat(x,v) = (Y(t),Y(t)) where 7: R — T2 is the solution of the Euler - Lagrange equation, with the initial conditions 7(0) = x and 7(0) = v. The energy function EL: TT2 — R is defined by

El(x,v) = -L(x,v). (f.f)

The subsets E(c) C TT2 are called energy levels and they are invariant under the Lagrangian flow of L. Note that the superlinearity condition implies that any nonempty energy levels are compact. Therefore, the flow fa is defined for all t G R.

The Lagrangian flow of L is conjugated to a Hamiltonian flow on T*T2, with the canonical symplectic structure, by the Legendre transformation L: TT2 — T*T2 given by

C(x,v) = ^x, ^(x,v)

The corresponding Hamiltonian H: T*T2 — R is

H(x,p) = maxveTxt{p(v) - L(x,v)} and we have the Fenchel inequality

p(v) ^ H(x, p) + L(x, v)

T

with equality if only if (x,p) = C(x, v), or equivalently, p = v) S T*T2. Therefore, by (1.1),

H (x,^(x,v) J = E(x,v).

Given a nonempty energy level E-1(c), the set H-1(c) := L (E-1(c)) C T*T2 is called the Hamiltonian level.

We denote by htop(L,c) the topological entropy of the Lagrangian flow $t\E-i(c), for any

nonempty energy level E-1(c). The topological entropy is an invariant that, roughly speaking, measures the complexity of its orbits structure. The relevant question about the topological entropy is whether it is positive or vanishes. Namely, if 9 e E-1(c) and T,6 > 0, we define the (ö,T)-dynamical ball centered at 9 as

B(9,S,T) = {v e E-1 (c): d(fa(v),fa(9)) < S for all t e [0,T]},

where d is the distance function in E-1 (c). Let N¿ (T) be the minimal quantity of the (S, T^dynamical ball needed to cover E-1(c). The topological entropy is the limit S ^ 0 of the exponential growth rate of Ns(T), that is,

htop(L, c) := lim lim sup ^ log NS(T).

Thus, if htop(L, c) > 0, some dynamical balls must contract exponentially at least in one direction.

For example, if (-, ■) denotes the flat metric and L(x,v) = l/2(v,v), then the corresponding Lagrangian flow is the geodesic flow on the flat torus, which is given by

$t(x,v) = (x + tv mod Z2,v).

It follows from straight computations that htop(L,c) = 0 for all c > 0. In this example, the corresponding Hamiltonian flow is integrable. For an integrable Hamiltonian system on a four-dimensional symplectic manifold under certain regularity assumptions (see [13]), the topological entropy of the Hamiltonian flow restricted to a regular compact energy level vanishes.

In [16], J.P.Schröder obtained a partial answer about the integrability reverse claim of Paternain's theorem[13], which is false in general, as has been known for a long time [6]. Schröder proved that, if the topological entropy of the Lagrangian flow on the level above the Mañé critical value vanishes, then, for all directions Z e S1, there are invariant Lipschitz graphs Tz (Z with irrational slope), T± (Z with rational slope) over T2, contained in {E = e} whose complement of its union is a tubular neighborhood of Tz and the lifted orbits from Tz, T± on the universal cover R2 are going to to, i.e., heteroclinic orbits. He used the gap condition to prove that, if in a strip between two neighboring periodic minimizers no foliation by heteroclinic minimizers exists, then there are instability regions which imply in its turn positive entropy. The Mañé strict critical value of L is the real number co (L) given by

c0(L) = inf{k e R: AL+k(y) ^ 0, for all contractible closed curves on T2}.1 (1.2) Theorem 1. Let L: TT2

^ R be a Tonelli Lagrangian and let c > co(L). Suppose that the Lagrangian flow restricted to an energy level E-1(c) satisfies the following conditions:

1) all closed orbits in E-1(c) are hyperbolic or elliptic, and

2) all heteroclinic intersections in E-1(c) are transverse. Then htop(L, c) > 0.

1 On an arbitrary closed manifold, this equality defines the universal critical value cu(L). The value c0(L) is the infimum value of k e R such that the L + k action is positive on the set of all closed curves that are homologous to zero. Of course, cu(L) < c0(L), and cu(L) = c0(L) for systems on

T2.

Let Cr (T2) be the set of potentials u: T2 — R of class Cr endowed with the Cr-topology. We recall that a subset O C Cr (T2) is called residual if it contains a countable intersection of open and dense subsets. In [12], E. Oliveira proved a version of the Kupka-Smale theorem for the Tonelli Hamiltonian and Lagrangian systems on any closed surfaces. More precisely, it follows from [12] that, for each c G R, there exists a residual set KS(c) C Cr(T2) such that every Hamiltonian Hu = H + u, with u G KS, satisfies the Kupka-Smale property, i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on E-1 (c). See also [15], where L.Rifford and R.Ruggiero proved the Kupka-Smale theorem for Tonelli Lagrangian systems on closed manifolds of any dimension.

So, if we take the residual subset KS(c) C Cr(T2) given by the Kupka-Smale theorem, and by the continuity of the critical values (cf. Lemma 5.1 in [5]) , we have the following corollary.

Corollary 1. Given c > c0(L), there exists a smooth potential u: T2 — R of Cr-norm arbitrarily small (for any r ^ 2) such that htop(L — u,c) > 0.

2. The Mather and Aubry sets

In this section we recall the definitions of the Mather sets and Aubry sets for the case of an (autonomous) Tonelli Lagrangian on the torus. Aubry -Mather theory was introduced by J. Mather in [10, 11] for convex, superlinear and time-periodic Lagrangian systems on any closed manifolds. Details and proofs of the main results of this section can be seen in the original works of J. Mather cited above.

Let us recall the main concepts introduced by J. Mather in [10]. We denote by B(L) the set of all Borel probability measures, with compact support, that are invariant under the Lagrangian flow of L. By duality, given i G B(L), there is a unique homology class p(i) G H1(T2, R) such that

{p(l), M) = J wdi, (2.1)

T t2

for any closed 1-form w on T2.

Then the Mather (3-function is defined by

/3(h) = inf \ j L di: i G B(L) with p(i) = h\.

it t2 )

The function /: H1(M, R) — R is convex and superlinear. A measure i G B(L) that satisfies

J Ldi = 3(p(l))

T t2

is called a p(i)-minimizing measure.

The Mather a-function can be defined as the convex dual (or conjugate) function of /, i.e., a = /*: H 1(M, R) — R is given by the so-called Fenchel transformation

a([w]) = sup {{[w],h) — 3(h)} = — inf < L — w di

heHi(M, r)

<rrM

By convex duality, we have that a is also convex and superlinear, and a* = /. Moreover, a measure /o is p(/o)-minimizing if and only if there is a closed 1-form wo, such that

j L — wo d/o = -a([wo]).

TM

Such a class [wo] G Hl(M, R) is called a subderivative of / at the point p(/o).

We say that / G B(L) is a [w]-minimizing measure of L if

J L — w d/ = min < J L — w dv: v G B(L) > = —a([w]).

T t2 VTt2 /

Let ML([w}) C B(L) be the set of all [w]-minimizing measures (it only depends on the coho-mology class [w]). The ergodic components of a [w]-minimizing measure are also [w]-minimizing measures, so the set M.L([w}) is a simplex whose extremal measures are ergodic [w]-minimizing measures. In particular fflL([w}) is a compact subset of B(L) with the weak*-topology.

For each [w] G Hl(T2, R), we define the Mather set of cohomology class [w] as

Ml([w]) = |J Supp(i). vemuiu ])

We set n(M.L([w])) = M.l([w\), and call it the projected Mather set, where n: TT2 —>• T2 denotes the canonical projection. The celebrated graph theorem proved by J. Mather in [10, Theorem 2], asserts that M.L([w}) is nonempty, compact, invariant under the Euler- Lagrange flow and the map (j^]): M.L([w}) ~^M.L([w}) is a bi-Lipschitz homeomorphism.

Following J.Mather in [11], for t > 0 and x, y G T2, define the action potential for the Lagrangian deformed by a closed 1-form w as

(x, y, t) = inf | J L(y(s),Y(s)) — wl{s)(A/(s)) ds

where the infimum is taken over all absolutely continuous curves 7: [0, t] ^ T2 such that 7(0) = x and y(t) = y. The infimum is in fact a minimum by Tonelli's theorem.

We define the Peierls barrier for the Lagrangian L — w as the function hu: T2 x T2 ^ R given by

hM(x,y) = liminf (x,y,t) + a([w])t} , t—

and the projected Aubry set for the cohomology class [w] G Hl(T2, R) as

Al([w]) = {x G T2 : hu(x,x) = 0}. By symmetrizing hu, we define the semidistance on AL([w}):

&[u](x,y) = K (x,y) + K (y,x). This function is nonnegative and satisfies the triangle inequality.

We define the Aubry set (that is also called static set) as the invariant set

•Al([w]) = {(x,v) G TT2: n o fat(x,v) gAl([w]), V t G R}.

By definition, n(AL([w})) = AL([w]). In [11, Theorem 6.1], J.Mather proved that this set is compact, M.L([w}) C AL([w]) and the extension of the graph theorem to the Aubry set, i.e., the mappings n\jL(jwj): AL([w}) — AL([w}) is a bi-Lipschitz homeomorphism.

Finally, we define the Mane set of cohomology class [w], which we denote by WL(M), as the set of orbits fat(O) = (7(t),i(t)) G TT2 such that, for all a<b G R, the trajectories y: R — T2 satisfy

J L(y(s),Y(s)) — Wy(s)(Y(s)) + a([w]) ds = inf (y(a),i(b),t) + a([w])t}.

a

These curves, which satisfy the above equality, are also called semistatic curves or [w]-minimizing curves.

Let us now state some important properties and results on these invariant sets that we going to use in the proof of Theorem 1.

Using the Mather a-function, we have the following equivalent definition of the Mane strict critical value (1.2) (cf. [4])

co(L) = min {a([w]): [w] G H 1(T2,R)} = —/(0).

In [1], M. J. Carneiro proved that the set ML([w]) is contained in the energy level E-1 (a([w])) and, by the above characterizations (see for example [4]), we have that

Ml([w]) C Al([w]) C NfL([w]) C E-1(a([w]).

By the graph property, we can define an equivalence relation in the set AL([w}): two elements d1 and 62 G AL([w}) are equivalent when (n(d1 ),n(d2)) = 0. The equivalence relation breaks AL([w}) down into classes that are called static classes of L. Let AL([w]) be the set of all static classes. We define a partial order ^ in AL([w]) by: (i) ^ is reflexive and transitive, (ii) if there is d G NL([w}), such that the a-limit set a(d) C Ai and the w-limit set w(d) C Aj, then Ai Aj. The following theorem was proved by G. Contreras and G.Paternain in [5].

Theorem 2. Suppose that the number of static classes is finite. Then, given Ai and Aj in Al([w]), we have that Ai ^ Aj.

Let r C TT2 be an invariant subset. Given e > 0 and T > 0, we say that two points d1,d2 G r are (e,T)-connected by chain in r if there is a finite sequence {({i,ti)}rn=1 C r x R, such that = d1, £n = d2, T < ti and dist(fati({1),{i+1) < e, for i = 1,...,n — 1. We say that the subset r is chain transitive if for all d1,d2 G r and for all e > 0 and T > 0 the points d1 and 62 are (e,T)-connected by chain in r. When this condition holds for d1 = d2, we say that T is chain-recurrent. The proof of the following properties can be seen in [2].

Theorem 3. Al([w\) is chain-recurrent.

The following theorem was proved by Mane in [8]. A proof can be seen also in [2, Theorem IV].

b

Theorem 4. Let i G B(L). Then i G ML([w]) if only if Supp(i) C AL([w]).

Finally, for any closed manifold M, we say that a class h G H\(M, R) is a rational homology if there is A > 0 such that Xh G i*H1(M, Z), where i : H1(M, Z) ^ H1(M, R) denotes the inclusion. The following proposition was proved by D. Massart in [9].

Proposition 1. Let M be a closed and oriented surface and let L be a Tonelli Lagrangian on M. Let i G B(L) be a measure p(i)-minimizing such that p(i) is a rational homology. Then the support of i is a union of closed orbits or fixed points of the Lagrangian flow.

3. Proof of Theorem 1

In this section, we prove Theorem 1.

Let L: TT2 ^ R and c > co (L). We assume that the restricted flow

^e-Hc) : E-l(c) ^ E-l(c)

satisfies two conditions:

(c1) all closed orbits are hyperbolic or elliptic, and (c2) all heteroclinic intersections are transverse.

Lemma 1. Let c > co(L) and let ho G Hl(T2, R) « R2 be a nonzero class. Then there are an invariant probability measure /o and a closed 1-form wo with a([wo]) = c, such that:

(i) p(/o) = Xoho for some Ao G R,

(ii) /o G ML([wo]) and therefore Supp(/o) C E—l(c).

Proof. Since в is superlinear, we have

Г /о 14

lirn = oo. (3.1)

A^^ \Xho|

Let дв: Hi(T2, R) ^ Hi(T2, R)* = H 1(T2, R) be the multivalued function such that to each point h G Hi(T2, R) associates all subderivatives of в at the point h. It is well known that, since в is finite, дв(Ь) is a nonempty convex cone for all h G H1(T2,R), and дв(Ь) is a unique vector if and only if в is differentiable in h (see, for example, [14, Section 23]). We define the subset

5(ho) = U дв(^а). AeR

By (3.1) we have that the subset S(h0) С H 1(T2, R) is not bounded. Since в is continuous, by the above properties of the multivalued function дв, we have that S(h0) is a convex subset. Observe that, if ш G дв(0), then a([wj) = Co(L) = min{a([5|); 5 G H 1(T2, R)} and, by superlinearity of a, the restriction ) is not bounded. Therefore, by the intermediate value theorem, for each с G [c0(L), ж) there is ш0 G дв(Х0h0) С S(h0), for some A0 G R, such that а([ш0]) = с. Therefore, if /л0 G M(L) is a (A0h0)-minimizing measure, then /л0 G Мь([ш0|), and Supp(po) С E-1 (с) (by [1]). □

Let i: Hi(T2, Z) — H^T2, R) be the inclusion. Recall that H i(T2, Z) w Z2 and that H 1(T2, R) w R2. Then {(0,1), (1,0)} c Hi(T2, Z) is a base of Hi(T2, Z). We have that, if a0, a1 are two closed curves in T2, with [a0] = (0,1) and [a1] = (1,0), then ao n a1 = 0.

We fix ho = (0,1) e Hi(T2, Z). By applying Lemma 1 we obtain a closed 1-form W0 and a [w0]-minimizing measure j0 with support into the level E-i(c), for which the rotational vector p(j0) = X0h0 is a rational homology class. Therefore, by Proposition 1, the support of j0 is formed by the union of closed orbits of the flow fatl#-i(c).

Lemma 2. The Mather set M.l([w0\) is the union of a finite number of closed orbits for the Lagrangian flow of L.

Proof. The Mather graph theorem asserts that the map nl^^j) : ML([w0]) — T2 is

injective. Hence, if di, d2: R — T2 are two distinct closed orbits of fat contained in ML([w0]), then Yi(t) = n o di(t) and y2(t) = n o d2(t) must be simple closed curves and [yi] = n[^2] e Hi(T2, Z), because otherwise Yi n y2 = 0.

Since c = a([w0]) > c0(L) = -3(0) and ML([w0]) is a compact set, the continuity of the map p: B(L) Hi(T2, R) (c.f. [10]) implies that there are constants k,l e R such that

0 <k < lp(j)l < l, for all j e Ml ([^0]). (3.2)

By the definition of ML([w0]), we have that Supp(j0) c ML([w0]). Let be the ergodic measure supported in a closed orbit d(t) = (7(t),Y(t)) c Supp(j0). Since p(j0) = A0h0 and by the linearity of the map p, we have that [y] = n0h0 for some 0 = n0 e Z. It follows from (2.1) that

t \no\T

= Jm*)) ds =f ms)) ds =J u,

0 0 y

for any closed 1-form w on T2. Then

Pith) = ^^ = ± —, (3.3)

HKh<' \n0\T T' y '

where T > 0 denotes the minimal period of y. Therefore, the period of any periodic orbit contained in Supp(j0) is bounded.

We fix a simple closed orbit y e n(Supp(j0)). Since [y] = n0h0, we have that the set CY = T2 — {y} defines an open cylinder. Let j = j0 be contained in ML([w0]). The graph property implies that Supp(j) n Supp(j0) = 0. Then n(Supp(j)) c CY and p(j) e i*(Hi(CY, R)) c Hi(T2, R). Therefore, we have that

Ml([W0]) c{j e M(L): p(j) e h)r)}.

Applying Proposition 1 in each measure of the set ML ([w0]), we conclude that the set ML ([w0]) is a union of closed orbits for the Lagrangian flow. Therefore, the same arguments used on the ergodic components of j0 imply that each ergodic measure in ML([w0]) satisfies equation (3.3). Then inequality (3.2) implies that the period of all periodic orbits in M.L([w0}) is uniformly bounded. Therefore, it follows from the compactness of E-i(c) and condition (c1) that there is at most a finite number of closed orbits of fat|E-i(c) in the Mather set ML([w0]). □

Remark 1. It is well known that action minimizing curves do not contain conjugate points2 and a proof of this fact can be seen in [3, § 4]. So, by Proposition A in [3], for each 0 G ML ([wo]) there exists the Green bundle along $t (0), i.e., there is a ^-invariant bi-dimensional subspace E($t (0)) C T^^E-1 (c)= R3 for all t G R. Therefore, the linearized Poincare map on 0 has an invariant one-dimensional subspace, so the periodic orbit $t(0) cannot be elliptic.

By Lemma 2, let Yi : R ^ T2, with i = 1,...,n, be a closed curves such that

n

Ml([wo]) = u Yi.

i=l

Since Supp(^o) cML([wo}), we have that Y] = noho = (0, no) G Hl(T2, Z), for all i G {1,..., n}.

Let AL([wo}) be the Aubry set corresponding to the cohomology class [wo] and let AL([wo]) be the set of all static classes. We recall that

Ml([wo]) CAl([wo]).

We have that either Ml([wo]) = Al([wo}) or Ml([wo]) = Al([wo]).

If ML([wo]) = AL([wo]), for each 0 G AL([wo])\ML([wo]), by the graph property of the Aubry set AL([wo]), the curve Ye := n ◦ $t(0): R ^ T2 has no self-intersection points and Ye nML([wo}) = 0. Moreover, by Theorem 4, we have that the a-limit and w-limit sets of 0 are contained in the Mather set ML([wo}). Since a curve on T2 that accumulates in positive time to more than one closed curve must have self-intersection points, we have that w(0) is a single closed orbit. By the same arguments, we have that a(0) is a single closed orbit. By Remark 1 and condition (c2), the orbit $t(0) is in the transverse intersections of an unstable manifold with a stable manifold of hyperbolic closed orbits, i.e., (0) is a transverse heteroclinic orbit. Certainly, if ML([wo]) is a unique closed orbit, then $t(0) is a transverse homoclinic orbit, which implies htop(L,c) > 0 (see, for example, [7, p. 276]). If the set ML([wo] has more than one orbit, it follows from the recurrence property (Theorem 3) that 0 is an (e,T)-chain connected in AL([wo]) for all e > 0 and T > 0, i.e., there is a finite sequence {((,ti)}k=l C AL([wo]) x R, such that (l = (k = 0, T < ti and dist($ti((0,0+0 < e, for i = 1,...,k — 1. Since the closed orbits in ML([wo]) are isolated on the torus, we have that for e small enough the set

{n((i )}k=

l C AL([wo]) must intersect the interior of each of the cylinders obtained by cutting the torus along the two curves Yi,,Yj G AL([wo]), with 1 ^ i, j ^ n. Therefore, choosing an orientation on AL([wo]) and reordering the indices, we obtain a cycle of transverse heteroclinic orbits. This implies that htop(L,c) > 0.

Now we consider the case of ML([wo]) = AL([wo]). Since each static class is a connected set (Proposition 3.4 in [5]), for each 1 ^ i ^ n, we have that Ai = (Yi,Afi). Hence, the number of static classes is equal to n.

Initially, let us assume that AL([wo] has at least two static classes, i.e., n ^ 2. Let Al and A2 be two distinct static classes. Applying Theorem 2, we have that Al < A2, therefore, there exist classes Al, ..., Aj = A2 and points 0l, ... ,0j_l G TVL([wo]) such that, for all 1 ^ i ^ j — 1, the a-limit set a(0j) C Ai and the w-limit set w(0j) C Ai+l. Also, we have that A2 < Al, then there exist Aj+l,...,Aj+k = Al and points 0j+l,...,0j+k _l g NL([wo]) such that, for

2Two points 01 = 02 are said to be conjugate if 02 = (02) and V(02) n dg1 (V(01)) = {0}, where

V(0) = ker dgn denotes the vertical bundle.

all j + 1 ^ i ^ j + k — 1, we have that a(di) c Ai and w(dj) c Ai+1. By Remark 1, the corresponding orbits (Yi,Afi) = Ai for 1 ^ i ^ j + k are hyperbolic closed orbits of the flow fat\E-i(c), and since the flow fat\E-i(c) satisfies condition (c2), we have that for all 1 ^ i ^ j + k the orbits fat(di) are in the transverse heteroclinic intersections of the unstable manifold of Ai and the stable manifold of Ai+1. Then we obtain a cycle of transverse heteroclinic orbits, in particular, we have that htop(L, c) > 0.

Let us assume now that Al([u0]) contains only one static class A1 = (y1 ,y 1). In this case, we can apply Theorem B in [5], which implies that the stable and unstable manifolds of the hyperbolic closed orbit A1 have transverse homoclinic intersections. Here we will give a proof of this result in our particular setting.

Let r = 2Z x Z be a sublatt.ice of 1? C R2. We denote by T2 the torus defined using the sublatt.ice T, that is, T2 = R2/r. Let p: T2 —>• T2 be the canonical projection. Note that p: T2 —>■ T2 is a double-covering of T2. Let L: T2 —>■ R be the lift of the Lagrangian L to T2. It is well known that. cq(L) = cq(L) and we have that.

AT([ujo})=P-1(Al([UJO}))=P-1(A1)

(cf. [5, Lemma 2.3]). By the construction of the covering p: T2 —>■ T2, the set. p~1(A\) C T2 has two connected components. So, the Aubry set. ^¿([wo]) has two static classes and, as in the above case, there is a heteroclinic orbit connecting these classes. This heteroclinic connection is projected in a transverse homoclinic orbit of the hyperbolic orbit A1 = (y1 ,Y1) c T2. This completes the proof.

We note that the homoclinic orbit that we obtained is not in the Mañé set TVL([woj), however, it. is a projection of an orbit, in the Mañé set. for the corresponding lifted Lagrangian to the covering p: T2 —> T2.

Acknowledgments

We are grateful to Prof. Alexandre Rocha (UFV) for the helpful conversations.

Conflict of Interest

The authors declare that they have no conflict of interest.

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