Generic hyperbolicity of Aubry sets on surfaces

Abstract

Given a Tonelli Hamiltonian of class $C^2$ on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the $C^2$ topology for which the Aubry set is hyperbolic in its energy level.

1 Introduction

Let $M$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 2$, and $H:T^*M \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$. As shown by Mather [32], one can construct a compact invariant subset of $T^*M$ which enjoys several variational properties and has the distinguished feature of being a Lipschitz graph over a part of $M$. This set, called the Aubry set associated to $H$ and denoted by $\tilde{\mathcal {A}}(H)$, captures many important features of the Hamiltonian dynamics.

Fathi [18] established a bridge between the Aubry-Mather theory and the properties of viscosity solutions/subsolutions of the critical Hamilton–Jacobi equation associated with $H$, giving rise to the weak KAM theory. The differentials of critical (viscosity) subsolutions are uniquely determined on the projection of $\tilde{\mathcal {A}}(H)$ onto $M$ [denoted by $\mathcal {A}(H)$], and all critical subsolutions are indeed $C^{1,1}$ on the projected Aubry set $\mathcal {A}(H)$. We refer the reader to Sect. 2.1 below for a precise definition of the Aubry set and more details about weak KAM theory.

A famous open problem concerning the structure of $\tilde{\mathcal {A}}(H)$ is the so-called “Mañé conjecture” [29] which states that, for a generic Hamiltonian, the Aubry set is either a hyperbolic equilibrium or a hyperbolic periodic orbit. In [21, 22], the second and third author obtained several results in the direction of proving the validity of the Mañé conjecture. However, all those results heavily rely on the assumption of the existence of a sufficiently smooth critical (sub-)solution. The goal of this paper is to combine some of the techniques developed in [21, 22] with tools from dynamical systems and new regularity estimates for viscosity solutions, to answer in low dimension an open problem proposed by Herman during the ICM in 1998 [25, Section 6.2, Question 2] (in the context of twist maps on $\mathbb T^1$, this question was posed by A. Katok, and positively solved by Le Calvez [27]):

Is it true that generically the Aubry set is hyperbolic?

As mentioned by Herman at the beginning of [25, Section 6], the subject of the instabilities of Hamiltonian flows and the problem of topological stability “lacks any non-trivial result”. Our main theorem solves in the affirmative Herman’s problem on surfaces for the $C^2$-topology.

Theorem 1.1

Let $H:T^*M \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$, and assume that $\dim M=2$. Then there is an open dense set of potentials $\mathcal {V} \subset C^2(M)$ such that, for every $V\in \mathcal {V}$, the Aubry set associated to the Hamiltonian $H+V$ is hyperbolic in its energy level.

The proof of Theorem 1.1 relies on the properties of Green bundles which can be associated with each orbit of the Aubry set. The basic idea is based on the following dichotomy for Green bundles: either they are always transverse, in which case one gets hyperbolicity of the Aubry set; or the Green bundles coincide along a given orbit of the Aubry set, and in this latter case, elaborating on previous works by Arnaud [4, 5], we show that the restriction to the projected Aubry set of any critical solution is $C^2$ along the projected orbit. This additional regularity property is not enough to apply the techniques which were introduced in [21, 22], since there the authors had to require the existence of a critical solution which is $C^{1,1}$ in a neighborhood of the projected orbit and $C^2$ along it. In our case, we do not have any regularity property outside the projected orbit, and critical solutions may be merely Lipschitz in any neighborhood of the projected orbit. Still, by some new refined estimates on the regularity of a critical solution near a point where the Green bundles coincide, we are able to exploit the techniques used in [21, 22] to conclude the argument and prove our theorem.

Our proof together with the shadowing lemma (see [26]) yields the following closing-type result:

Theorem 1.2

Let $H:T^*M \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$, and assume that $\dim M=2$. Then, for every open set $\mathcal {U}\subset T^*M$ containing $\tilde{\mathcal {A}}(H)$, and any neighborhood $\mathcal {V}$ of $0$ in $C^2(M)$, there exist $\theta \in \mathcal {U}$ and $V \in \mathcal {V}$ such that the orbit with respect to the Hamiltonian $H+V$ passing through $\theta$ is periodic and hyperbolic.

We notice that a similar statement could be deduced as a direct consequence of the results in [9, 19, 29]: more precisely, by [19, Theorem 1.5] and [9, Theorems 1 and 2] the Aubry set is upper-semicontinuous on surfaces, so [29, Theorem F]¹ implies that generically in $C^\infty$ topology one can find a periodic orbit close to the Aubry set. However, in contrast with Theorem 1.2 above, this orbit may not be hyperbolic (even if one introduces an additional small perturbation by a potential, see [37]). Therefore, if we work in the $C^2$ topology, Theorem 1.1 allows us to say that the Aubry set of $H+V$ is hyperbolic, which in turn implies the hyperbolicity of the sequence of periodic orbits approaching the Aubry set (see Proposition 2.18). All in all, we get the following refinement of [29, Theorem F] in two dimensions and $C^2$ topology:

Theorem 1.3

Let $H:T^*M \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$, and assume that $\dim M=2$. Then there is a residual set of potentials $\mathcal {G} \subset C^2(M)$ such that, for every $V \in \mathcal {G}$, the Lagrangian associated with $H+V$ admits a unique minimizing measure, which is indeed a strong limit of a sequence of probability measures supported on hyperbolic periodic orbits.

The paper is structured as follows. First, we collect several preliminary results which are fundamental for the proof of Theorem 1.1: Sect. 2.1 is concerned with reminders in weak KAM theory; Sect. 2.2 contains a result on connecting trajectories; Sects. 2.3, 2.4, and 2.5 are devoted to the constructions of Green bundles, paratingent cones, and Arnaud-type results; Sect. 2.6 contains reminders on hyperbolicity and quasi-hyperbolicity; finally, Sects. 2.7 and 2.8 contain material on semiconcave and BV functions and a lemma from harmonic analysis, which play a major role in the proof of Theorem 1.1. Section 3 is concerned with the proof of Theorem 1.1, which is split into a stability and a density part. Finally, in Sect. 4 we present some examples of Tonelli Hamiltonians on surfaces of positive genus whose Aubry set is a non-trivial minimal hyperbolic set.

2 Preliminary results

2.1 Reminders in weak KAM theory

Recall that a Tonelli Hamiltonian $H:T^*M \rightarrow \mathbb {R}$ of class $C^2$ is a Hamiltonian satisfying the two following properties:

• (H1) Superlinear growth: For every $K\ge 0$ there is a finite constant $C^*(K) \in \mathbb {R}$ such that

  $$\begin{aligned} H(x,p) \ge K\Vert p\Vert _x + C^*(K)\quad \forall \, (x,p)\in T^*M. \end{aligned}$$

• (H2) Uniform convexity: For every $(x,p) \in T^*M$, the second derivative along the fibers $\frac{\partial ^2 H}{\partial p^2} (x,p)$ is positive definite.

The critical value $\mathbf {c}[H] \in \mathbb {R}$ of $H$ may be defined as the infimum of the values $c\in \mathbb {R}$ for which there exists a smooth function $u:M \rightarrow \mathbb {R}$ satisfying

$$\begin{aligned} H(x,du(x)) \le c \quad \forall \, x\in M. \end{aligned}$$

A Lipschitz function $u:M\rightarrow \mathbb {R}$ is called a critical subsolution (for $H$) if

$$\begin{aligned} H\bigl ( x,du(x)\bigr ) \le \mathbf {c}[H] \quad \text{ for } \text{ a.e. } x \in M. \end{aligned}$$

(2.1)

It can be shown that the set of critical subsolutions, denoted by $\mathcal {SS}$, is a nonempty compact convex subset of $C^0(M;\mathbb {R})$ [18, 36]. Fathi and Siconolfi [20] proved that the set of critical subsolutions of class $C^1$ (denoted by $\mathcal {SS}^1$) is nonempty, and their result has been improved later by Bernard [8] who showed the existence of critical subsolutions of class $C^{1,1}$. Also, they proved that the so-called Aubry set can be seen as the nonempty compact subset of $T^*M$ defined by

$$\begin{aligned} \tilde{\mathcal {A}} (H) := \bigcap _{u\in \mathcal {SS}^1} \Bigl \{ (x,du(x)) \, \vert \, x \in M \text{ s.t. } H\bigl (x,du(x)\bigr )=\mathbf {c}[H] \Bigr \}. \end{aligned}$$

This set is invariant under the Hamiltonian flow, and by Mather’s graph property it is a Lipschitz graph over the projected Aubry set defined as

$$\begin{aligned} \mathcal {A}(H) := \pi ^*\bigl (\tilde{\mathcal {A}} (H) \bigr )\subset M, \end{aligned}$$

where $\pi ^*: T^*M \rightarrow M$ denotes the canonical projection map (see for instance [18, 36]).

The Lagrangian $L:TM \rightarrow \mathbb {R}$ associated with $H$ by Legendre–Fenchel duality is defined by

$$\begin{aligned} L(x,v) := \max _{p\in T_x^*M} \Bigl \{ \langle p,v\rangle - H(x,p) \Bigr \} \quad \forall \, (x,v)\in TM. \end{aligned}$$

Thanks to (H1)–(H2), it is easy to see that $L$ is a Tonelli Lagrangian of class $C^2$, that is a Lagrangian satisfying both properties of superlinear growth and uniform convexity (see [10, 18]). Critical subsolutions have important variational properties, and for instance they can be characterized as follows (see [18, 36]):

Proposition 2.1

A function $u:M \rightarrow \mathbb {R}$ is a critical subsolution if and only if

$$\begin{aligned} u\bigl ( \gamma (b)\bigr ) - u\bigl ( \gamma (a)\bigr ) \le \int \limits _a^b L\bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds + \mathbf {c}[H] \, (b-a) \end{aligned}$$

(2.2)

for any Lipschitz curve $\gamma : [a,b] \rightarrow M$.

The (backward) Lax–Oleinik semigroup

$$\begin{aligned} \{\mathcal {T}^-_t\}_{t\ge 0}: C^0(M;\mathbb {R}) \longrightarrow C^0(M;\mathbb {R}) \end{aligned}$$

associated with $L$ is defined as follows: for every $t\ge 0$ and $u\in C^0(M;\mathbb {R})$, the function $\mathcal {T}^-_t u := \mathcal {T}^-_t(u)$ is given by

$$\begin{aligned} \mathcal {T}^-_t u (x) := \inf \left\{ u\bigl ( \gamma (-t)\bigr ) + \int \limits _{-t}^0 L\bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds\right\} \quad \forall \,x \in M, \end{aligned}$$

(2.3)

where the infimum is taken over all Lipschitz curves $\gamma : [-t,0] \rightarrow M$ such that $\gamma (0)=x$. The set of critical subsolutions $\mathcal {SS}$ is invariant under $\{\mathcal {T}^-_t\}_{t\ge 0}$. A critical subsolution $u:M\rightarrow \mathbb {R}$ is called a critical solution or a weak KAM solution if,

$$\begin{aligned} \mathcal {T}^-_t u = u - \mathbf {c}[H]\, t \quad \forall \,t \ge 0. \end{aligned}$$

(2.4)

Critical solutions may be characterized in several ways (see for instance [18, 36]):

Proposition 2.2

Let $u \in C^0(M;\mathbb {R})$. The following properties are equivalent:

• (i) $u$ is a critical solution.

• (ii) $u\in \mathcal {SS}$ and, for every $x\in M$, there exists a Lipschitz curve $\gamma _x : (-\infty ,0] \rightarrow M$ with $\gamma _x(0)=x$ such that

  $$\begin{aligned} u\bigl ( \gamma _x(b)\bigr ) - u\bigl ( \gamma _x(a) \bigr )&= \int \limits _a^b L \bigl ( \gamma _x(s),\dot{\gamma }_x(s)\bigr ) \, ds\nonumber \\&\quad +\, \mathbf {c}[H] \, (b-a) \quad \forall \,a<b \le 0. \end{aligned}$$

  (2.5)

• (iii) $u\in \mathcal {SS}$ and for every smooth function $\phi :M \rightarrow \mathbb {R}$ with $\phi \le u$ and all $x\in M$,

  $$\begin{aligned} \phi (x) = u(x) \Longrightarrow H\bigl (x,d\phi (x)\bigr ) \ge \mathbf {c}[H]. \end{aligned}$$

As shown in [35], critical solutions enjoy some regularity properties. One of them is the fact that critical solutions are semiconcave. Recall that, given an open set $\Omega \subset \mathbb {R}^n$, a function $v :\Omega \rightarrow \mathbb {R}^n$ is said to be locally semiconcave in $\Omega$ if, for every $x\in \Omega$, there are $C_x\ge 0$ and a ball $\mathcal {B}_x \subset \Omega$ containing $x$ such that the function $y\mapsto v(y) - C_x|y|^2$ is concave on $\mathcal {B}_x$. A function $v:M \rightarrow \mathbb {R}$ is called locally semiconcave if it is locally semiconcave in charts, that is, if for every $x\in M$ there are an open neighborhood $\mathcal {V}_x$ of $x$ and a smooth diffeomorphism $\phi _x : \mathcal {V}_x \rightarrow \phi _x(\mathcal {V}_x) \subset \mathbb {R}^n$ such that $v\circ \phi _x^{-1}$ is locally semiconcave on $\phi _x(\mathcal {V}_x) \subset \mathbb {R}^n$. Of course, if the manifold $M$ is compact then the constant $C_x$ can be chosen independent of the point, and we say that the function is semiconcave.

Proposition 2.3

Any critical solution is semiconcave on $M$.

Let $u:M \rightarrow \mathbb {R}$ be a critical solution for $H$ and $x\in M$ be fixed. By assertion (ii) in Proposition 2.2 above, there exists a curve $\gamma _x : (-\infty ,0] \rightarrow M$ with $\gamma _x(0)=x$ satisfying (2.5). Since $u$ is a critical subsolution (see Proposition 2.1), we infer that for every $T>0$ the restriction of $\gamma _x$ to $[-T,0]$ minimizes the quantity

$$\begin{aligned} u \bigl ( \gamma (-T)\bigr ) + \int \limits _{-T}^0 L \bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds + \mathbf {c}[H] \, T, \end{aligned}$$

among all Lipschitz curves $\gamma : [-T,0] \rightarrow M$ such that $\gamma (0)=x$. In particular, $\gamma _x$ is the projection of a Hamiltonian trajectory, and whenever $u$ is differentiable at $\gamma _x(-T)$, by the first variation formula one gets

$$\begin{aligned} d u \bigl (\gamma _x(-T)\bigr ) = \frac{\partial L}{\partial v} \bigl ( \gamma _x(-T),\dot{\gamma }_x(-T)\bigr ). \end{aligned}$$

We call limiting differential of $u$ at $x\in M$, and we denote it by $D^*_xu$, the set of $p\in T_x^*M$ such that there is a sequence $\{x_k\}_k$ of points converging to $x$ such that $u$ is differentiable at $x_k$ and $p=\lim _{k\rightarrow \infty } du (x_k)$. Note that, by the Lipschitz regularity of $u$, the graph of the multivalued mapping $D^*u$ is a compact subset of $T^*M$. As shown in [35], by the above discussion one can prove that there is a one-to-one correspondence between the limiting differentials and the curves satisfying (2.5):

Proposition 2.4

Let $u: M \rightarrow \mathbb {R}$ be a critical solution and $x\in M$. For every $p\in D_x^*u$ the curve $\gamma _x:(- \infty ,0] \rightarrow M$ defined by

$$\begin{aligned} \gamma _x (-t) := \pi ^* \left( \phi ^H_{-t} (x,p) \right) \quad \forall \,t \ge 0, \end{aligned}$$

(2.6)

satisfies $\gamma _x(0)=x$, (2.5), and

$$\begin{aligned} \left( \gamma _x (-t), D^*_{\gamma _x(-t)} u\right) = \left\{ \phi ^H_{-t} (x,p) \right\} \quad \forall \,t \ge 0. \end{aligned}$$

(2.7)

In particular $u$ is differentiable at $\gamma _x(-t)$ for any $t>0$. Moreover, for every curve $\gamma _x:(- \infty ,0] \rightarrow M$ satisfying $\gamma _x(0)=x$ and (2.5), there is $p \in D_x^*u$ such that (2.6) holds.

A curve of the form $\gamma _x:(- \infty ,0] \rightarrow M$ satisfying (2.5) is called a semi-calibrated curve. A curve defined on $\mathbb {R}$ satisfying (2.5) for any $a,b \in \mathbb {R}$ is called calibrated. As we said previously, the Aubry set $\tilde{\mathcal {A}}(H)$ is invariant under the Hamiltonian flow, and it is a Lipschitz graph over $\mathcal {A}(H)$. Fathi and Siconolfi [20] proved that, for every point of $\mathcal {A}(H)$, the limiting differential of a critical solution is a singleton there. In particular, since $\tilde{\mathcal {A}}(H)$ is a Lipschitz graph over $\mathcal {A}(H)$, this means that every critical solution $u$ is differentiable on $\mathcal {A}(H)$, its differential is independent of $u$, and $x \mapsto du(x)$ is Lipschitz on the Aubry set. In addition, for any $(x,p) \in \tilde{\mathcal {A}}(H)$, the curve (2.6) is calibrated. All these facts are summarized in the following:

Proposition 2.5

Let $u: M \rightarrow \mathbb {R}$ be a critical solution and $x\in \mathcal {A}(H)$. Then $u$ is differentiable at $x$, $du(x)$ does not depend on $u$, $D_x^*u = \{du(x)\}$, and the calibrated curve $\gamma _x : \mathbb {R}\rightarrow M$ defined by

$$\begin{aligned} \gamma _x (t) := \pi ^* \left( \phi ^H_{t} (x,du(x)) \right) \quad \forall \,t \in \mathbb {R}, \end{aligned}$$

(2.8)

satisfies $\gamma _x(0)=x$,

$$\begin{aligned} u\bigl ( \gamma _x(b)\bigr ) - u\bigl ( \gamma _x(a) \bigr ) = \int \limits _a^b L \bigl ( \gamma _x(s),\dot{\gamma }_x(s)\bigr ) \, ds + \mathbf {c}[H] \, (b-a) \quad \forall \, a<b, \end{aligned}$$

(2.9)

$\gamma _x(t) \in \mathcal {A}(H)$ for all $t\in \mathbb {R}$, and

$$\begin{aligned} \left( \gamma _x(t),D^*_{\gamma _x(t)} u\right) = \Bigl \{ \phi ^H_{t} (x,du(x)) \Bigr \} \quad \forall \,t \in \mathbb {R}. \end{aligned}$$

(2.10)

Finally, the mapping $\mathcal {A}(H) \ni x \mapsto du(x)$ is Lipschitz.

We refer the reader to [21, 23] for a more detailed introduction to weak KAM theory, to the notes [36] for the proofs of the above results, and to [18] for further details.

2.2 The Dirichlet problem and the connection of trajectories

Let $H:\mathbb {R}^n\times (\mathbb {R}^n)^* \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$, $V:\mathbb {R}^n \rightarrow \mathbb {R}$ a $C^2$ function, and denote by $H_V$ the Hamiltonian $H+V$. We split $\mathbb {R}^n$ as $\mathbb {R}\times \mathbb {R}^{n-1}$ and we define the $(n-1)$-dimensional disks

$$\begin{aligned} \Pi ^{\tau }_r := \{\tau \}\times B^{n-1}(0,r) \quad \forall \, \tau \in \mathbb {R}, \, \forall \,r>0, \end{aligned}$$

where $B^{n-1}(0,r)\subset \mathbb {R}^{n-1}$ denotes the $(n-1)$-dimensional open ball of radius $r$ centered at the origin. Denoting by $\pi ^*:\mathbb {R}^n\times (\mathbb {R}^n)^*\rightarrow \mathbb {R}^n$ the projection onto the space variable, we define the following Poincaré-type maps:

Given $\tau >0$ small, $\tau _1,\tau _2 \in [0, \tau ]$, and $(x^0,p^0) \in \Pi _{1/2}^{\tau _1}\times \mathbb {R}^n$ such that $[-2\tau ,2\tau ]\ni t \mapsto \pi ^*\bigl (\phi _{t}^{{H}_V}(x^0,p^0)\bigr )$ intersects $\Pi _{1}^{\tau _2}$ transversally, we define the maps

$$\begin{aligned}&\mathcal P_{\tau _1,\tau _2}^*(x^0,p^0):=\phi _{\mathcal T_{\tau _1,\tau _2}^*(x^0,p^0)}^{{H}}(x^0,p^0),\\&\mathcal P_{\tau _1,\tau _2}(x^0,p^0):=\pi ^*\Bigl (\mathcal P_{\tau _1,\tau _2}^*(x^0,p^0)\Bigr ), \end{aligned}$$

where $\mathcal T_{\tau _1,\tau _2}^*(x^0,p^0)\in [-2\tau ,2\tau ]$ is the first time (positive if $\tau _1<\tau _2$, negative if $\tau _1>\tau _2$) such that $\mathcal P_{\tau _1,\tau _2}(x^0,p^0) \in \Pi _1^{\tau _2}$.

As shown in [22, Lemma 5.1], the following holds (we denote by $e_1$ the first vector in the canonical basis of $\mathbb {R}^n=\mathbb {R}\times \mathbb {R}^{n-1}$):

Lemma 2.6

Let $\bar{u}:B^n(0,1) \rightarrow \mathbb {R}$ be a $C^{1,1}$ function such that

$$\begin{aligned} \frac{d}{dt} \left( \pi ^* \Bigl ( \phi _t^{{H}}(x^0, d \bar{u}(x^0)) \Bigr )\right) _{|{t=0}} \cdot e_1 \ge \frac{1}{2} \quad \forall \,x^0 \in \Pi _1^0. \end{aligned}$$

Then there exists $\bar{\tau }>0$ small such that the following properties are satisfied:

• (i) For every $\tau \in (0,5\bar{\tau }],$ the Poincaré time mapping $\mathcal {T}_{0,\tau }^{d\bar{u}} : \Pi _{1/2}^0 \rightarrow \mathbb {R}$ defined by

  $$\begin{aligned} \mathcal {T}_{0,\tau }^{d\bar{u}} \bigl ( x_0\bigr ) := \mathcal {T}_{0,\tau }^* \bigl (x^0,d\bar{u}(x^0)\bigr ) \quad \forall \,x^0 \in \Pi _{1/2}^0, \end{aligned}$$

  is well-defined and is Lipschitz;

• (ii) for every $\tau \in (0,5\bar{\tau }],$ the Poincaré mapping $\mathcal {P}_{0,\tau }^{d\bar{u}}:\Pi _{1/2}^0 \rightarrow \Pi _1^{\tau }$ defined by

  $$\begin{aligned} \mathcal {P}_{0,\tau }^{d\bar{u}} := \mathcal {P}_{0,\tau } \bigl (x^0,d\bar{u}(x^0)\bigr ) \quad \forall \,x^0 \in \Pi _{1/2}^0, \end{aligned}$$

  is $2$-Lipschitz;

• (iii) the following inclusion holds for every $\tau \in (0,5\bar{\tau }]$:

  $$\begin{aligned}&\Bigl \{ \pi ^* \Bigl ( \phi _t^{{H}}(x^0, d\bar{u}(x^0)) \Bigr ) \, \vert \, x^0 \in \Pi ^0_{3/8}, \, t \in \bigl [0,\mathcal {T}_{0,\tau }(x^0)\bigr ]\Bigr \}\\&\quad \subset [0,\tau ] \times B^{n-1}(0,1/2); \end{aligned}$$

• (iv) the viscosity solution $\bar{u}_0$ to the Dirichlet problem

  $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {H} \bigl ( z, d \bar{u}_{0} (z)\bigr ) =0 &{}\text {in } [0,5\bar{\tau }]\times B^{n-1}(0,1/2),\\ \bar{u}_{0} = \bar{u} &{}\text {on } \Pi _1^0, \end{array} \right. \end{aligned}$$

  is of class $C^{1,1}$.

We now define the cylinder

$$\begin{aligned} \mathcal {C} \Bigl (\bigl (x^0, p^0\bigr );t;r \Bigr ) := \Bigl \{ \pi ^* \Bigl ( \phi _s^{{H}}(x^0,p^0) \Bigr ) + (0,\hat{y}) \, \vert \, s\in [0,t],\, |\hat{y}| < r \Bigr \}, \end{aligned}$$

and the action

$$\begin{aligned} \mathbb {A}_V \bigl ( (x^0,p^0);\tau \bigr )&:= \int \limits _0^\tau {L}_V \left( \pi ^* \Bigl ( \phi _t^{{H}_V}(x^0,p^0)\Bigr ), \frac{d}{dt} \left( \pi ^* \Bigl ( \phi _t^{{H}_V}(x^0,p^0) \Bigr )\right) \right) \,dt \\&= \int \limits _0^\tau {L} \left( \pi ^* \Bigl ( \phi _t^{{H}_V}(x^0,p^0)\Bigr ), \frac{d}{dt} \left( \pi ^* \Bigl ( \phi _t^{{H}_V}(x^0,p^0) \Bigr )\right) \right) \\&\quad -\, V\left( \pi ^* \Bigl ( \phi _t^{{H}_V}(x^0,p^0) \Bigr )\right) \, dt,\nonumber \end{aligned}$$

where $\phi _t^{H_V}$ denotes the Hamiltonian flows associated to $H_V$. Then the following holds:

Proposition 2.7

Let $u:B^n(0,1)\rightarrow \mathbb {R}$ be a viscosity solution of $H(x, du(x))=\mathbf {c}[H]$ and assume that

$$\begin{aligned}&\frac{d}{dt} \left( \pi ^* \Bigl ( \phi _t^{{H}}(x^0,p^0) \Bigr )\right) _{|{t=0}} \cdot e_1 \ge \frac{1}{2} \quad \forall \,x^0 \in \Pi _1^0,\, \forall \,p^0\in D_{x^0}^*u,\nonumber \\&\langle p_0,\dot{\gamma }_{0,p_0}(0)\rangle \ge -c_0 \quad \forall \,p_0 \in D_0^* u,\quad \text {where} \quad \gamma _{0,p_0}(t):=\pi ^* \bigl ( \phi _t^{{H}}(0,p_0) \bigr )\nonumber \\ \end{aligned}$$

(2.11)

for some positive constant $c_0$. Then, provided $c_0$ is sufficiently small (the smallness depending only on $H$), for any $\bar{\tau }>0$ sufficiently small there are $\bar{\delta }, \bar{r}, \bar{\epsilon } \in (0,1/4)$ and $K>0$ such that the following property holds: for any $r \in (0,\bar{r}), \hat{\epsilon } \in (0,\bar{\epsilon })$, $x^0 \in \Pi _1^0, x^f \in \Pi _1^{\bar{\tau }}$, $p^0 \in D_{x^0}^*u$, $p^f \in D_{x^f}^*u$, and $\sigma \in \mathbb {R}$ satisfying

$$\begin{aligned} |x^0| <\bar{\delta } \end{aligned}$$

(2.12)

and

$$\begin{aligned} \bigl |(x^f,p^f)- \mathcal {P}_{0,\bar{\tau }}^*(x^0,p^0)\bigr | < r \hat{\epsilon }, \quad |\sigma | < r^2 \hat{\epsilon }, \end{aligned}$$

(2.13)

there exist a time $T^f>0$ and a potential $V:\mathbb {R}^n \rightarrow \mathbb {R}$ of class $C^2$ such that:

• (i) $\mathrm{Supp}(V) \subset \mathcal {C} \Bigl (\bigl (x^0, p^0\bigr );\mathcal {T}_{0,\bar{\tau }}^*(x^0,p^0);r \Bigr )$;

• (ii) $\Vert V\Vert _{C^2} < K\hat{\epsilon }$;

• (iii) $\bigl | T^f-\mathcal T_{0,\bar{\tau }}^*(x^0,p^0) \bigr | < K r \hat{\epsilon }$;

• (iv) $\phi ^{H_V}_{T^f} \bigl (x^0,p^0\bigr ) = \bigl (x^f,p^f\bigr )$;

• (v) $\mathbb {A}_V \bigl ( (x^0,p^0);T^f\bigr ) = \mathbb {A}\bigl ( (x^0,p^0);\mathcal T_{0,\bar{\tau }}^*(x^0,p^0)\bigr )+ \bigl \langle d {u} \bigl (\mathcal {P}_{0,\bar{\tau }}(x^0,p^0)\bigr ), x^f-\mathcal {P}_{0,\bar{\tau }}(x^0,p^0)\bigr \rangle +\sigma$.

Proof of Proposition 2.7

First of all, it follows by (2.13) and Lemma 2.6(ii) that, provided $\bar{\tau }$ is sufficiently small (the smallness being independent of $r$ and $\hat{\epsilon }$),

$$\begin{aligned} \Bigl |\mathcal P_{\bar{\tau },\bar{\tau }/2}^*\bigl (x^f,p^f\bigr )- \mathcal P_{0,\bar{\tau }/2}^*\bigl (x^0,p^0\bigr ) \Bigr | < 2 r \hat{\epsilon }. \end{aligned}$$

Hence, we first apply [21, Proposition 3.1] on $[0,\bar{\tau }/2]$ to connect $(x^0,p^0)$ to $\mathcal P_{\bar{\tau },\bar{\tau }/2}^*\bigl (x^f,p^f\bigr )$ in a time $T^f_1 \sim \bar{\tau }/2$ with a “default” of action bounded by $Kr^2\hat{\epsilon }^2$. Then, thanks to (2.11), we see that assumption (A4) in [21, Proposition 4.1] is satisfied provided $c_0$ is small enough. Thus, if $\bar{\epsilon }$ is sufficiently small we can apply [21, Proposition 4.1] on $[\bar{\tau }/2,\bar{\tau }]$ to “compensate” the default of action so that (v) above holds. Moreover it is easily seen that also all the other properties are satisfied. We leave the details to the reader. $\square$

2.3 Green bundles and reduced Green bundles

Let us endow the cotangent bundle $T^*M$ with its standard symplectic structure $\omega$, and denote by $V_{\theta }:=\text{ ker }(d_{\theta }\pi ^*)$ the vertical space in $T_{\theta }(T^*M)$ at any $\theta \in T^*M$ (recall that $\pi ^*:T^*M\rightarrow M$ denotes the canonical projection). A subspace $E\subset T_{\theta }(T^*M)$ is called Lagrangian if it is a $n$-dimensional vector subspace where the symplectic bilinear form $\omega _{\theta }:T_{\theta }(T^*M) \times T_{\theta }(T^*M) \rightarrow \mathbb {R}$ vanishes. As an example, vertical spaces are Lagrangian. If we fix a symplectic set of local coordinates, we can identify $T_{\theta }(T^*M)$ with $T_{x}M \times T_x^*M$ and $V_\theta$ with $\{0\}\times T_x^*M$. Then, any $n$-dimensional vector subspace $E \subset T_{\theta }(T^*M)$ which is transversal to $V_{\theta }$ (i.e. $E\cap V_{\theta }=\{0\}$) can be written as the graph of some linear map $S:T_{x}M \rightarrow T_{x}^*M$, and it can be checked that $E$ is Lagrangian if and only if $S$ is represented by a symmetric matrix.

Given a Hamiltonian $H:T^*M \rightarrow \mathbb {R}$ of class $C^2$, the Hamiltonian vector field $X_H$ on $T^*M$ is defined by $\omega _{\theta } \bigl (X_H(\theta ),\cdot \bigr ) = -d_{\theta }H$ for any $\theta \in T^*M$. In a symplectic set of local coordinates, the Hamiltonian equations (i.e., the equations satisfied by any solution of the ODE $(\dot{x},\dot{p})=X_H\bigl ((x,p)\bigr )$) are given by $\dot{x}=\frac{\partial H}{\partial p}, \dot{p}=-\frac{\partial H}{\partial x}$. Finally, we recall that the Hamiltonian flow $\phi ^H_t$ of $X_H$ preserves the symplectic form $\omega$. In particular, the image of a Lagrangian space $E\subset T_{\theta }(T^*M)$ by $D_{\theta }\phi ^H_t$ is Lagrangian in $T_{\phi ^H_t(\theta }(T^*M)$. We refer the reader to [1, 11] for more details about the notions of symplectic geometry introduced above.

We recall now the construction and properties of Green bundles and reduced Green bundles along orbits of the Hamiltonian flow without conjugate points. We refer the reader to [4, 6, 14] for further details and historical accounts. For every $\theta \in T^*M$ and every $t\in \mathbb {R}$, we define the Lagrangian subspace $G_{\theta }^t \subset T_{\theta }(T^*M)$ as the pushforward of the vertical distribution at $\phi _{-t}^H(\theta )$ by $\phi _t^H$, that is

$$\begin{aligned} G_{\theta }^t := \left( \phi _t^H \right) _* \left( V_{\phi _{-t}^H(\theta )} \right) = D_{ \phi _{-t}^H(\theta )} \phi ^H_t \left( V_{\phi _{-t}^H(\theta )} \right) \quad \forall \, \theta \in T^*M. \end{aligned}$$

(2.14)

The orbit of $\theta \in T^*M$ is said to be without conjugate points if for any $t, t'\in \mathbb {R}$,

$$\begin{aligned} t\ne t' \, \Longrightarrow \, \left[ D_{ \phi _t^H (\theta )} \phi ^H_{t'-t} \left( V_{\phi _{t}^H(\theta )}\right) \right] \cap V_{ \phi _{t'}^H (\theta )} = \{0\}. \end{aligned}$$

We denote by $\mathcal {D}$ the set of $\theta \in T^*M$ whose orbit has no conjugate point, and we assume that $\mathcal {D}$ is nonempty. Given $\theta \in \mathcal {D} \subset T^*M$, we fix a symplectic set of local coordinates around $\theta =(x,p)$. Then, for every $t\in \mathbb {R}\setminus \{0\}$, the Lagrangian subspace $G^t_{\theta }$ is transverse to the vertical subspace $V_{\theta }$ in $T_{\theta }(T^*M)\simeq T_{x}M \times T_x^*M$. Hence, there is a linear operator $K^t_{\theta } : T_xM \rightarrow T_x^*M$ such that

$$\begin{aligned} G^t_{\theta } = \Bigl \{ \bigl ( h,K^t_{\theta } h \bigr )\in T_xM \times T_x^*M \ \vert \ h \in T_xM\Bigr \}. \end{aligned}$$

Since $G^t_{\theta }$ is Lagrangian, the linear operator $K_{\theta }^t$ can be represented by a symmetric matrix in our symplectic set of local coordinates. There is a natural partial order for the Lagrangian subspaces which are transverse to the vertical, that simply corresponds to the usual order for symmetric operators. Later on, given two Lagrangian subspaces $E, E' \in T_{\theta } (T^*M)$ which are transverse to $V_{\theta }$, we shall write $E \prec E'$ (resp. $E\preceq E'$) if the corresponding symmetric operators $K, K'$ are such that $K'-K$ is positive definite (resp. nonnegative definite). The following property is a consequence of the uniform convexity of $H$ in the fibers (see [4, Proposition 3.7] and [14, Proposition 1.4]):

Proposition 2.8

Let $\theta \in \mathcal {D}$. The following properties hold:

• (i) For every $t'> t >0$, $G^{t'}_{\theta } \prec G^t_{\theta }$.

• (ii) For every $t'< t <0$, $G^{t}_{\theta } \prec G^{t'}_{\theta }$.

• (iii) For every $t<0<t'$, $G^t_{\theta } \prec G^{t'}_{\theta }$.

As a consequence, for every $\theta \in \mathcal {D}$, the sequence of Lagrangian subspaces $(0,+\infty )\ni t\mapsto G^t_{\theta }$ (resp. $(0,+\infty )\ni t\mapsto G^{-t}_{\theta }$) is decreasing (resp. increasing) and bounded from below by $G^{-1}_{\theta }$ (resp. bounded from above by $G^{1}_{\theta }$). Hence, both limits as $t \rightarrow \pm \infty$ exist, which leads to the following definition:

Definition 2.9

For every $\theta \in \mathcal {D}$, we define the positive and negative Green bundles at $\theta$ as

$$\begin{aligned} G^+_{\theta } := \lim _{t\rightarrow +\infty } G^t_{\theta } \quad \text{ and } \quad G_{\theta }^- := \lim _{t\rightarrow -\infty } G^t_{\theta }. \end{aligned}$$

We shall keep in mind that the positive Green bundle $G^+_{\theta }$ depends on the behavior of the Hamiltonian flow along the orbit of $\theta$ for large negative times, while the negative Green bundle $G^-_{\theta }$ depends on what happens for large positive times. By construction, we also have the following result (see [4], Corollaire 3.8 and Proposition 3.9] and [14, Proposition 1.4 (d)]):

Proposition 2.10

Let $\theta \in \mathcal {D}$. The following properties hold:

• (i) $G_{\theta }^- \preceq G_{\theta }^+$.

• (ii) $D_{\theta } \phi ^H_t \left( G_{\theta }^-\right) = G_{\phi ^H_t (\theta )}^-$ and $D_{\theta } \phi ^H_t \left( G_{\theta }^+ \right) = G_{ \phi ^H_t (\theta )}^+$ for all $t\in \mathbb {R}$.

Moreover, the function $\theta \mapsto G_\theta ^+$ is upper-semicontinuous on $\mathcal {D}$, and $\theta \mapsto G_\theta ^-$ is lower-semicontinuous on $\mathcal {D}$. Thus, if $G_{\theta }^+=G_{\theta }^-$ for some $\theta \in \mathcal {D}$ then both of them are continuous at $\theta$.

The following result, which first appeared in [14], plays a major role in recent works by Arnaud [4–6] (see [4, Proposition 3.12], [6, Proposition 1], and [14, Proposition 1.11]):

Proposition 2.11

Let $\theta \in \mathcal {D}$ and $\psi \in T_{\theta }(T^*M)$. Then the following properties hold:

• (i) $\psi \notin G_{\theta }^- \, \Longrightarrow \, \lim _{t \rightarrow +\infty } \left\| D_{\theta } \bigl ( \pi ^* \circ \phi ^H_t \bigr ) (\psi ) \right\| = + \infty$.

• (ii) $\psi \notin G_{\theta }^+ \, \Longrightarrow \, \lim _{t \rightarrow -\infty } \left\| D_{\theta } \bigl ( \pi ^* \circ \phi ^H_t \bigr ) (\psi ) \right\| = + \infty$.

For every $\theta =(x,p) \in T^*M$, denote by $\Sigma _{\theta } \subset T^*M$ the energy level

$$\begin{aligned} \Sigma _{\theta } := \Bigl \{ \theta ' =(x',p') \in T^*M \, \vert \, H(x',p')=H(x,p) \Bigr \}. \end{aligned}$$

From the previous result one easily gets the following conclusion (see [4, Example 2 page 17] and [14, Corollary 1.12]):

Proposition 2.12

Let $\theta \in \mathcal {D}$ be such that $X_H(\theta ) \ne 0$. Then

$$\begin{aligned} X_H(\theta ) \in G^-_{\theta } \cap G_{\theta }^+ \quad \text{ and } \quad G^-_{\theta } \cup G_{\theta }^+ \subset T_{\theta } \Sigma _{\theta }. \end{aligned}$$

Let $\Sigma \subset T^*M$ be a regular energy level of $H$, that is an energy level satisfying $\frac{\partial H}{\partial p} (x,p) \ne 0$ for every $\theta =(x,p) \in \Sigma$. By superlinear growth (H1) and uniform convexity (H2) of $H$, the hypersurface $\Sigma$ is compact and, for every $\theta =(x,p) \in \Sigma$, the fiber $\Sigma \cap T_x^*M$ is the boundary of a uniformly convex set in $T_x^*M$. For every $\theta \in \Sigma$ we define the subspace $N_{\theta } \subset T_{\theta }\Sigma$ by

$$\begin{aligned} N_{\theta } := \Bigl \{ \psi \in T_{\theta }\Sigma \, \vert \, \left\langle D_{\theta }\pi ^* (\psi ), D_{\theta }\pi ^* \bigl (X_H(\theta )\bigr ) \right\rangle _{\pi ^*(\theta )} =0\Bigr \}, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the Riemannian metric on $M$. By construction, we have

$$\begin{aligned} T_{\theta }\Sigma = N_{\theta } \oplus \mathbb {R}X_H(\theta ) \quad \forall \,\theta \in \Sigma . \end{aligned}$$

For every $\theta \in \mathcal {D} \cap \Sigma$, we define the reduced Green bundles $\hat{G}^-_{\theta }$ and $\hat{G}^+_{\theta }$ as

$$\begin{aligned} \hat{G}_{\theta }^- := G_{\theta }^- \cap N_{\theta } \quad \text{ and } \quad \hat{G}_{\theta }^+ := G^+_{\theta } \cap N_{\theta }. \end{aligned}$$

As shown in [4], the reduced Green bundles can be seen as the Green bundles associated with a specific symplectic bundle over the orbit of $\theta$; they satisfy some of the properties of the Green bundles, in particular Proposition 2.10 [except (ii)]. If $M$ has dimension two, then, for every $\theta \in \mathcal {D} \cap \Sigma$, the reduced Green bundles $\hat{G}_{\theta }^+$ and $\hat{G}_{\theta }^-$ should be seen as lines in the plane $N_{\theta } \simeq T_{\theta }\Sigma / \mathbb {R}X_H(\theta )$. Finally we observe that, since $G^+_{\theta }$ depends on the behavior of the Hamiltonian flow near $\phi ^H_t(\theta )$ for large negative times, its construction can be performed as soon as the orbit of $\theta \in T^*M$ has no conjugate points in negative time. In particular, this can be done for any semi-calibrated curve (see Proposition 2.4).

2.4 Paratingent cones and Green bundles

The present section is mainly inspired by ideas and techniques developed by Arnaud [4–7] to study in particular the link between Green bundles and regularity of weak KAM solutions. Before presenting Arnaud-type results, we first recall a result from [22]. Let $S \subset \mathbb {R}^k$ be a compact set which has the origin as a cluster point. The paratingent cone to $S$ at $0$ is the cone defined as

$$\begin{aligned} C_0(S)&:= \left\{ \lambda \, \lim _{i\rightarrow \infty } \frac{x_i - y_i}{|x_i - y_i|} \ \vert \ \lambda \in \mathbb {R}, \lim _{i \rightarrow \infty } x_i = \lim _{i \rightarrow \infty } y_i = 0,\right. \\&\quad \quad \quad \left. x_i \in S,\, y_i \in S\ x_i \ne y_i\, \,\forall \, i ,\right\} , \end{aligned}$$

and the paratingent space of $S$ at $0$ is the vector space generated by $C_0(S)$:

$$\begin{aligned} \Pi _0(S) := \text{ Span } \Bigl \{ C_0(S)\Bigr \}. \end{aligned}$$

As shown in [22], the set $S$ is contained locally in the graph of a function from $\Pi :=\Pi _0(S)$ onto its orthogonal complement $\Pi ^\perp$. Let $d$ be the dimension of $\Pi$, denote by $\text{ Proj }_\Pi$ the orthogonal projection onto the space $\Pi$ in $\mathbb {R}^k$, and set $\mathcal H_{S} := \text{ Proj }_\Pi (S)$. Finally, for any $r,\nu >0$ we define the cylinder

$$\begin{aligned} C(r,\nu ) := \Bigl \{ (h,v) \in \Pi \times \Pi ^{\perp } \ \vert \ |h|<r, |v| <\nu \Bigr \}, \end{aligned}$$

where $|\cdot |$ denotes the Euclidean norm. Also, we set $B_{r}:=B(0,r)$. The following result holds (see [22, Lemma 3.3]).

Lemma 2.13

There exist $r_{S}>0$ and a Lipschitz function $\Psi _S: \Pi \cap \bar{B}_{r_{S}} \rightarrow \Pi ^{\perp }$ such that the following properties hold:

• (i) $S\cap C \left( r_{S},r_{S}\right) \subset \mathrm{graph}(\Psi _{S})|_{B_{r_{S}}}:=\bigl \{ h + \Psi _{S}(h) \ \vert \ h\in \Pi \cap B_{r_{S}}\bigr \}$;

• (ii) $h+\Psi _{S} (h)$ belongs to $S \cap C \left( r_{S},r_{S}\right)$ for every $h\in \mathcal H_{S} \cap B_{r_{S}}$;

• (iii) For any $r\in (0,r_{S})$, let $\ell (r)>0$ denote the Lipschitz constant of $\Psi _{S}$ on $\Pi \cap {B}_{r}$. Then $\lim _{r \downarrow 0} \ell (r) =0$.

In particular $\Psi _{S}(0)=0$, $\Psi _{S}$ is $C^1$ at $0$, and $\nabla \Psi _{S} (0)=0$.

By Proposition 2.5, through each point $\theta =(x,p)$ of the Aubry set $\tilde{\mathcal {A}}(H)$ passes a calibrated curve [defined by (2.8)] which corresponds to the projection of its orbit under the Hamiltonian flow, and whose restriction to any subinterval is always minimizing the action between its endpoints. Being minimizing, such a curve has necessarily no conjugate points, hence $\theta \in \mathcal {D}$ (see any textbook on the classical theory of calculus of variations, for example [13]). We also observe that, since the Aubry set is invariant under the Hamiltonian flow,

$$\begin{aligned} D_{\theta } \phi _t^H \left( C_{\theta }\left( \tilde{\mathcal {A}} (H)\right) \right) = C_{\phi _t^H(\theta )} \left( \tilde{\mathcal {A}} (H)\right) \quad \forall \,t \in \mathbb {R}\end{aligned}$$

(2.15)

and $X_H (\theta )$ belongs to the paratingent cone to $\tilde{\mathcal {A}} (H)$ at $\theta$, that is

$$\begin{aligned} X_H (\theta ) \in C_{\theta }\left( \tilde{\mathcal {A}} (H)\right) \subset T_{\theta }\Sigma _H \quad \forall \,\theta \in \tilde{\mathcal {A}}(H), \end{aligned}$$

(2.16)

where $\Sigma _H := \bigl \{H=\mathbf {c}[H]\bigr \}$ is a regular energy level of $H$. Given $\theta =(x,p) \in \tilde{\mathcal {A}}(H)$ with $X_H(\theta )\ne 0$, we define the reduced paratingent cone to the Aubry set as

$$\begin{aligned} \hat{C}_{\theta } := C_{\theta }\left( \tilde{\mathcal {A}} (H)\right) \cap N_{\theta }, \end{aligned}$$

where $N_{\theta }$ has been defined in Sect. 2.3. If $M$ has dimension two, $\hat{C}_{\theta }$ is a collection of lines in the plane $N_{\theta }$. All those lines can be compared with other lines in this plane. The following proposition is a variant of Arnaud’s results (compare with [4, Proposition 3.11], [4, Proposition 3.16 (3)], [6, Theorem 9]), and it follows from the Lipschitz graph property of the Aubry set.

Proposition 2.14

Assume that $\dim M =2$ and that $\theta \in \tilde{\mathcal {A}}(H)$ is not an equilibrium of $X_H$. Then

$$\begin{aligned} \hat{G}_{\theta }^- \preceq \hat{C}_{\theta } \preceq \hat{G}_{\theta }^+. \end{aligned}$$

In other terms, any line in $\hat{C}_{\theta }$ is squeezed between $\hat{G}_{\theta }^-$ and $\hat{G}_{\theta }^+$.

Proof of Proposition 2.14

Since $\tilde{\mathcal {A}}(H)$ is a Lipschitz graph, its paratingent cones cannot intersect the vertical bundle, hence taking $\epsilon >0$ small enough yields

$$\begin{aligned} G_{\theta }^{-\epsilon } \prec C_{\theta } := C_{\theta }\left( \tilde{\mathcal {A}} (H)\right) \prec G_{\theta }^{\epsilon }. \end{aligned}$$

(2.17)

To explain the meaning of the above formula notice that, for every $t\ne 0$, the Lagrangian space $G_{\theta }^t$ is transverse to $V_{\theta }$, it does not contain $X_H(\theta )$, and it is contained in $T_{\theta }\Sigma _H$. Hence its intersection with $N_{\theta }$ is a line in the plane $N_{\theta }$, and (2.17) means that the intersection of $C_{\theta }$ with $N_{\theta }$ is a collection of vector lines which are squeezed between the lines $G_{\theta }^{-\epsilon }\cap N_{\theta }$ and $G_{\theta }^{\epsilon } \cap N_{\theta }$. Thus, thanks to this discussion, to prove the result it is sufficient to show that no line $G_{\theta }^{t}\cap N_{\theta }$ with $t\in \mathbb {R}\setminus [-\epsilon ,\epsilon ]$ is contained in $C_{\theta } \cap N_{\theta }$. Argue by contradiction and assume that there is $\bar{t}>\epsilon$ (the other case is left to the reader) such that

$$\begin{aligned} G_{\theta }^{-\bar{t}} \cap N_{\theta } \subset C_{\theta } \cap N_{\theta }. \end{aligned}$$

By (2.14)–(2.15), this means that $V_{\phi ^H_{\bar{t}}(\theta )}$ and $D_{\theta }\phi _{\bar{t}}^H\left( C_{\theta }\right) = C_{\phi ^H_{\bar{t}}(\theta )}$ do intersect, which contradicts the Lipschitz graph property of the Aubry set. $\square$

As an application of Proposition 2.14 and Lemma 2.13, we deduce that if $\dim M =2$ and the positive and negative Green bundles coincide for some $\theta =(x,p) \in \tilde{\mathcal {A}}(H)$ with $X_H(\theta ) \ne 0$, then the Aubry set is locally contained in the graph of a Lipschitz $1$-form which is $C^1$ at $x$. It will be convenient to extend the $1$-form along a piece of projected orbit of the Aubry set.

Corollary 2.15

Assume that $\dim M =2$ and that $\theta =(x,p) \in \tilde{\mathcal {A}}(H)$ with $X_H(\theta ) \ne 0$ satisfies

$$\begin{aligned} G^-_{\theta } = G_{\theta }^+. \end{aligned}$$

Assume moreover that $\theta$ is not on a periodic orbit and let $\gamma (t):= \pi ^*\bigl (\phi _t^H(\theta )\bigr )$ for any $t\in \mathbb {R}$. Then, for every $T>0$ there are an open neighborhood $\mathcal {V}$ of $\gamma ([-T,T])$ in $M$ and a function $f:\mathcal {V} \rightarrow \mathbb {R}$ of class $C^{1,1}$ which is $C^2$ along $\gamma ([-T,T])$ such that

$$\begin{aligned} \tilde{\mathcal {A}} (H) \cap T^*\mathcal {V} \subset \text{ Graph } (df), \end{aligned}$$

and for every $t\in [-T,T]$, $G^-_{\phi _t^H(\theta )}=G^+_{\phi _t^H(\theta )}$ is the graph of $D_{\gamma (t)}^2f$ (in a symplectic set of local coordinates in $T^*\mathcal {V}$).

Proof of Corollary 2.15

By Proposition 2.12, (2.16) and Proposition 2.14, if the two Green bundles coincide, the paratingent cone $C_{\theta } := C_{\theta }\left( \tilde{\mathcal {A}} (H)\right)$ is a line which is transverse to the vertical subspace $V_{\theta }$. Then, working in a symplectic set of local coordinates, by Lemma 2.13 we deduce that there are an open neighborhood $\mathcal {U}$ of $x$, and a Lipschitz $1$-form $\Psi$ on $\mathcal {U}$ which is $C^1$ at $x$, such that

$$\begin{aligned} \tilde{\mathcal {A}} (H) \cap T^*\mathcal {U} \subset \text{ Graph } (\Psi ), \end{aligned}$$

and the Lagrangian plane $C_{\theta }$ coincides with the graph of $d_{x}\Psi$. Since $\frac{\partial H}{\partial p}(\theta )\ne 0$ (because $X_H(\theta ) \ne 0$ and the Aubry set is a Lipschitz graph), the set of $\theta ' \in T^*M$ with $H(\theta ')=\mathbf {c}[H]$ is locally (in a neighborhood of $\theta$) a submanifold of dimension $3$ of class $C^2$. Then up to compose $\Psi$ with a retraction $r$ of class at least $C^1$ onto the set $\{H=\mathbf {c}[H]\}$, we may assume that $\Psi$ is a Lipschitz $1$-form satisfying

$$\begin{aligned} H \bigl ( \Psi (x) \bigr ) = \mathbf {c}[H] \quad \forall \,x \in \mathcal {U}. \end{aligned}$$

Let $S\subset \mathcal {U}$ be a local section (that is, a smooth curve) which is transverse to $\gamma$ at $x$. By the properties of $\Psi$, the map $\Phi : [-2T,2T] \times S \rightarrow M$ defined by

$$\begin{aligned} \Phi (t,y) := \pi ^* \left( \phi ^H_t \bigl (\Psi (y)\bigr )\right) \quad \forall \,t \in [-2T,2T], \, \forall \,y \in S, \end{aligned}$$

is Lipschitz, and it is $C^1$ along the segment $[-T,T] \times \{x\}$. Moreover, since $C_{\theta }=G^-_{\theta }=G^+_{\theta }$, the differential of $\Phi$ is invertible at $(t,0)$ for every $t\in [-T,T]$. Therefore, by the Clarke Lipschitz inverse function theorem (see [12, Theorem 5.1.1]), $\Phi$ admits a Lipschitz inverse $\Phi ^{-1}= (\tau ,\epsilon ) : \mathcal {V} \rightarrow [-2T,2T] \times S$ in a simply connected neighborhood $\mathcal {V}$ of $\gamma ([-T,T])$ (remember that $\gamma$ is not periodic) which is $C^1$ along $\gamma ([-T,T])$. By construction, the $1$-form $\alpha$ on $\mathcal {V}$ defined by

$$\begin{aligned} \alpha (x) := \phi ^H_{\tau (x)} \bigl (\Psi (\epsilon (x))\bigr ) \quad \forall \,x \in \mathcal {V}, \end{aligned}$$

is a closed Lipschitz $1$-form which is $C^1$ along the curve $\gamma ([-T,T])$. By the Poincaré lemma, we get a function satisfying the conclusions of Corollary 2.15.

We notice that an alternative way to perform the above construction is to approach $\Psi$ by a sequence of $1$-form of class $C^1$, to construct a sequence of functions of class $C^2$ by the method of characteristics (see [18]) and to get the $C^{1,1}$ function $f$ by taking the limit. Such an approach can be found in [17]. $\square$

2.5 Hessians and positive Green bundles

As shown by Alexandrov (see for instance [16, 39]), locally semiconcave functions are two times differentiable almost everywhere.

Theorem 2.16

Let $U$ be an open subset of $\mathbb {R}^n$ and $u:U \rightarrow \mathbb {R}$ be a function which is locally semiconcave on $U$. Then, for a.e. $x\in U$, $u$ is differentiable at $x$ and there exists a symmetric operator $A(x):\mathbb {R}^n \rightarrow \mathbb {R}^n$ such that the following property is satisfied:

$$\begin{aligned} \lim _{t\downarrow 0} \frac{u(x+tv) -u(x) -t du(x)\cdot v - \frac{t^2}{2} \langle A(x)\cdot v,v\rangle }{t^2}=0 \quad \forall v \in \mathbb {R}^n. \end{aligned}$$

Moreover, $x \mapsto du(x)$ is differentiable a.e. in $U$ (that is for a.e. $x\in U$, any section of $z \mapsto D^*_zu$ is differentiable at $x$), and its differential is given by $A(x)$.

We infer that if $u:M \rightarrow \mathbb {R}$ is semiconcave then, for almost every $x\in M$, $u$ is differentiable at $x$, $D^*_xu$ is a singleton, $du$ is differentiable at $x$ and the graph of its differential is a Lagrangian subspace $D^2_xu \subset T_{(x,du(x))}(T^*M)$. Notice that if $u:M \rightarrow \mathbb {R}$ is a critical solution, then by Proposition 2.4 regularity properties of $u$ propagate in negative time. That is, for every $x\in M$ such that $u$ is two times differentiable at $x$, the function $u$ is two times differentiable along the semi-calibrated curve $\gamma _x : (-\infty ,0] \rightarrow M$ given by (2.6). Moreover we have

$$\begin{aligned} D_{(x,du(x))} \phi _{-t}^H \left( D^2_xu \right) = D^2_{\gamma _x(-t)}u \quad \forall \,t \ge 0. \end{aligned}$$

(2.18)

Recall that for every $\theta = (x,p) \in D^*u$, the Hamiltonian trajectory starting at $\theta$ at time zero has no conjugate points in negative times (see Proposition 2.4), which allows us to construct $G^+_{\theta }$ at any such points. Then, proceeding as in the proof of Proposition 2.14 [replacing (2.15) by (2.18)] we obtain the following one-sided estimate (notice that, since $D^2_xu$ is a Lagrangian subspace, the assumption on the dimension of $M$ could be dropped, see the proof of Proposition 2.14 and [4, Proposition 3.11]):

Proposition 2.17

Assume that $\dim M =2$, let $u:M \rightarrow \mathbb {R}$ be a critical solution, and let $x\in M$ be such that $du(x)$ and $D^2u(x)$ exist and $X_H(x,du(x)) \ne 0$. Then

$$\begin{aligned} D^2_xu \preceq G_{(x,du(x))}^+. \end{aligned}$$

Later on, in the proof of Theorem 1.1, the above result together with the upper semicontinuity of the positive Green bundle will allow us to obtain a local bound from above for $D^2u$ in a neighborhood of a given point of the projected Aubry set [see (3.13)].

2.6 Reminders on hyperbolicity

We recall here basic facts in hyperbolic dynamics, we refer the reader to the Katok–Hasselblatt monograph [26] for further details. Recall that $\phi _t^H$ denotes the Hamiltonian flow in $T^*M$. A compact $\phi ^H_t$-invariant set $\Lambda \subset T^*M$ is called hyperbolic in its energy level $\Sigma$ with respect to the Hamiltonian flow if the following properties are satisfied:

• (h1) At each point $\theta \in \Lambda$, the tangent space $T_{\theta }\Sigma$ is the direct sum of three subspaces $E_{\theta }^s, E_{\theta }^u,$ and $E_{\theta }^0=\mathbb {R}X_H(\theta )$.

• (h2) At each point $\theta \in \Lambda$, we have $D_{\theta }\phi ^H_t \left( E_{\theta }^s\right) = E_{\phi ^H_t(\theta )}^s$ and $D_{\theta }\phi ^H_t \left( E_{\theta }^u\right) = E_{\phi ^H_t(\theta )}^u$, for any $t\in \mathbb {R}$.

• (h3) There are a Riemannian metric in an open neighborhood of $\Lambda$, and constants $C\ge 1$ and $\mu >0$ such that, for each $\theta \in \Lambda$, $\psi ^s \in E_{\theta }^s$, and $\psi ^u \in E_{\theta }^u$, we have

  $$\begin{aligned} \left\| D_{\theta }\phi ^H_t (\psi ^s) \right\|&\le C e^{-\mu t} \Vert \psi ^s \Vert ,\\ \left\| D_{\theta }\phi ^H_{-t} (\psi ^u) \right\|&\le C e^{-\mu t} \Vert \psi ^u \Vert , \end{aligned}$$

  for all $t> 0$.

Note that, as a consequence of hyperbolicity, the splitting $T_{\theta }\Sigma = E_{\theta }^s \oplus E_{\theta }^u \oplus E_{\theta }^0$ defined for $\theta \in \Lambda$ is continuous. Let us extend it into a continuous (not necessarily invariant) splitting $T_{\theta }\Sigma _{\theta } = E_{\theta }^s \oplus E_{\theta }^u \oplus E_{\theta }^0$ with $E_{\theta }^0=\mathbb {R}X_H(\theta )$ defined for all $\theta$ in an open neighborhood $\mathcal {V}$ of $\Lambda$. Then, for every $\theta \in \mathcal {V}$ and any $\rho \in (0,1)$, we define the family of horizontal and vertical cones $\{H_{\theta }^{\rho }\}$ and $\{V_{\theta }^{\rho }\}$ as

$$\begin{aligned} H_{\theta }^{\rho }&:= \Bigl \{ \xi + \eta \, \vert \, \xi \in E_{\theta }^u, \, \eta \in E_{\theta }^s, \Vert \eta \Vert \le \rho \Vert \xi \Vert \Bigr \},\\ V_{\theta }^{\rho }&:= \Bigl \{ \xi + \eta \, \vert \, \xi \in E_{\theta }^u, \, \eta \in E_{\theta }^s, \Vert \xi \Vert \le \rho \Vert \eta \Vert \Bigr \}. \end{aligned}$$

By (h2)–(h3), for every $\theta \in \Lambda$, $\rho \in (0,1)$, and $t>0$, we have

$$\begin{aligned} D_{\theta }\phi ^H_t \left( H_{\theta }^{\rho } \right)&\subset H_{\phi ^H_t(\theta )}^{e^{-2\mu t} C^{2} \rho }, \quad D_{\theta }\phi ^H_{-t} \left( V_{\theta }^{\rho } \right) \subset V_{\phi ^H_{-t} (\theta )}^{e^{-2\mu t} C^2 \rho },\\ \left\| D_{\theta }\phi ^H_t (\psi ) \right\|&\ge \frac{e^{\mu t}}{C'(1+\rho )}\, \Vert \psi \Vert \quad \forall \, \psi \in H_{\theta }^{\rho },\\ \left\| D_{\theta }\phi ^H_{-t} (\psi ) \right\|&\ge \frac{ e^{\mu t}}{C'(1+\rho )}\, \Vert \psi \Vert \quad \forall \, \psi \in V_{\theta }^{\rho }. \end{aligned}$$

Hence, by continuity and compactness, we can find $T>0$, $\mu ' >0$, an open neighborhood $\mathcal {V}'\subset \mathcal {V}$, and continuous disjoint cones ${\mathbb S}_{\theta }$, ${\mathbb U}_{\theta }\subset E^s_{\theta }\oplus E^u_{\theta }$ containing $E^s_{\theta }$, $E^u_{\theta }$ respectively, such that, for every $\theta \in \mathcal {V}'$,

$$\begin{aligned} D_\theta \phi ^H_T \bigl ({\mathbb U}_{\theta }\bigr )\subset \text{ Int } \left( {\mathbb U}_{\phi _T^H(\theta )} \right) , \quad D_\theta \phi ^H_{-T} \bigl ({\mathbb S}_{\theta }\bigr )\subset \text{ Int } \left( {\mathbb S}_{\phi _{-T}^H(\theta )} \right) , \end{aligned}$$

and

$$\begin{aligned} \left\| D_{\theta }\phi ^H_T (\psi ) \right\|&\ge e^{\mu ' T} \, \Vert \psi \Vert \quad \forall \, \psi \in {\mathbb U}_{\theta },\\ \left\| D_{\theta }\phi ^H_{-T} (\psi ) \right\|&\ge e^{\mu ' T} \, \Vert \psi \Vert \quad \forall \, \psi \in {\mathbb S}_{\theta }. \end{aligned}$$

This shows that any $\phi _t^H$-invariant compact set sufficiently close to $\Lambda$ will satisfy the Alekseev cone criterion, which provides an alternative more handy characterization for hyperbolicity (see [26]). This criterion is also robust under perturbation of the dynamics, and allows us to obtain that following:

Proposition 2.18

Let $\Lambda \subset T^*M$ be a compact $\phi ^H_t$-invariant set which is hyperbolic in its energy level with respect to the Hamiltonian flow. Then there exists an open neighborhood $\mathcal {V}$ of $0$ in $C^2(M)$ and an open neighborhood $\mathcal {O}$ of $\Lambda$ such that, for every potential $V\in \mathcal {V}$, any compact set $\Lambda ' \subset \mathcal {O}$ which is $\phi _t^{H+V}$-invariant is hyperbolic in its energy level with respect to $\phi _t^{H+V}$.

The above result will be useful to show the stability part (that is, openness) of Theorem 1.1. As shown in [14, 28], a way to obtain hyperbolicity is to show quasi-hyperbolicity properties. Let $B$ be a compact metric space and $\pi :E \rightarrow B$ a vector bundle equipped with a continuous norm $|\cdot |_p$ on each fiber $\pi ^{-1}(p)$. Let $\Psi$ be a continuous $\mathbb {R}$-action $\Psi _t : \mathbb {R}\rightarrow \text{ Isom }(E)$ such that $\Psi _{s+t}= \Psi _s \circ \Psi _t$. We say that $\Psi$ is quasi-hyperbolic if

$$\begin{aligned} \sup _{t\in \mathbb {R}} \Bigl \{ \left| \Psi _t (\xi ) \right| \Bigr \} = +\infty \quad \forall \,\xi \in E \setminus \{0\}. \end{aligned}$$

The following result holds (see [14, Theorem 0.2], and compare with [5, §3] and [6, Theorem 2] where it is shown that the non-wandering assumption can indeed be dropped):

Proposition 2.19

Assume that any point in $B$ is non-wandering and that $\Psi$ is quasi-hyperbolic. Then $\Psi$ is hyperbolic.

In the proof of Theorem 1.1, the above result allows us to obtain the hyperbolicity of the Aubry set in the case when the Green bundles are always transverse. Such an approach is nowadays classical.

2.7 Some properties of semiconcave and BV functions

2.7.1 Derivatives of semiconcave functions

Let $v:\mathbb {R}^n\rightarrow \mathbb {R}$ be a semiconcave function, i.e., $v$ can be written as the sum of a concave function and a smooth function. Since second distributional derivatives of convex functions are nonnegative Radon measures (see [16], sect. 6.3]), the Radon–Nikodým theorem [2, Theorem 1.28] allows us to write $D^2v$ as the sum of an absolutely continuous matrix-valued measure and a singular matrix-valued measure:

$$\begin{aligned} D^2v = \nabla ^2v\,dx + D_{\mathcal {S}}^2v, \end{aligned}$$

where $\nabla ^2v\in L^1_{loc}$ is the pointwise Hessian of $v$ (which exists almost everywhere by Alexandrov’s theorem), and $D_{\mathcal {S}}^2v$ is a singular measure (with respect to the Lebesgue measure). Also, by semiconcavity we have that $D^2 v$ is locally bounded from above (as a measure): for any $R>0$ there exists a constant $C_R>0$ such that

$$\begin{aligned} \int \limits _E \langle D^2 v\cdot e,e\rangle \le C_R |E| \quad \forall \,E\subset B_R \text { Borel}, \, \forall \,e \in \mathbb {R}^n \text{ with } |e|=1, \end{aligned}$$

where $B_R:=B^n(0,R)$ denotes the $n$-dimensional ball of radius $R$ centered at the origin. In particular, choosing $E$ of measure zero we get

$$\begin{aligned} \int \limits _E \langle D_{\mathcal {S}}^2 v\cdot e,e\rangle \le 0 \quad \forall \,E\subset \mathbb {R}^n \text { Borel with |E|=0}, \forall \, e \in \mathbb {R}^n \text{ with } |e|=1. \end{aligned}$$

(2.19)

Hence, since the measure $D_{\mathcal {S}}^2v$ is singular with respect to the Lebesgue measure, by the arbitrariness of $E$ we deduce that $\langle D_{\mathcal {S}}^2 v\cdot e,e\rangle$ is a negative singular measure for any vector $e\in \mathbb {R}^n$.

Since the distributional derivative of $\nabla v$ is equal to the measure $D^2v$, by definition $\nabla v:\mathbb {R}^n\rightarrow \mathbb {R}^n$ is a function of bounded variation (see [2], sect. 3]). Given $x' \in \mathbb {R}^{n-1}$, let us consider the function $w_{x'}:\mathbb {R}\rightarrow \mathbb {R}^n$ defined by

$$\begin{aligned} w_{x'}(s):=\nabla v(x',s) \quad \text{ for } \text{ a.e. } s \in \mathbb {R}. \end{aligned}$$

(2.20)

Note that, since $v$ is differentiable almost everywhere, by Fubini’s theorem the function $w_{x'}$ is defined for almost every $x' \in \mathbb {R}^{n-1}$. It is well-known that the functions $w_{x'}$ are of bounded variation on $\mathbb {R}$ for almost every $x'\in \mathbb {R}^{n-1}$ as well (see [2, Theorem 3.103] and the subsequent discussion), so their distributional derivative on $\mathbb {R}$ is a measure which can be decomposed as the sum of an absolutely continuous and a singular part:

$$\begin{aligned} Dw_{x'}= \nabla w_{x'}\,ds+ D_{\mathcal S} w_{x'}, \end{aligned}$$

where here $D$ is the distributional derivative on $\mathbb {R}$, $\nabla w_{x'}\in L^1_{loc}$ is the pointwise derivative of $w_{x'}$ which exists almost everywhere [2, Theorem 3.28(c)], and $D_{\mathcal S} w_{x'}$ is singular with respect to the one-dimensional Lebesgue measure. Also, the fundamental theorem of calculus holds between every couple of points where $\nabla v$ exists [2, Theorem 3.28]:

$$\begin{aligned} \nabla v(x',s_2) - \nabla v(x',s_1)&= w_{x'}(s_2) - w_{x'}(s_1)\nonumber \\&= \int \limits _{s_1}^{s_2} \nabla w_{x'}(s)\,ds + \int \limits _{s_1}^{s_2}d\bigl (D_{\mathcal S} w_{x'}\bigr )(s), \end{aligned}$$

(2.21)

for every $s_1<s_2$ in $\mathbb {R}$.

Let us recall that, given a vector-valued measure $\mu$, one denotes by $|\mu |$ its total variation, which is defined as

$$\begin{aligned} |\mu |(E)&:= \sup \left\{ \sum _{h=0}^\infty |\mu (E_h)|\,:\,E_h \text { disjoint Borel sets s.t. }\right. \\&\quad \qquad \qquad \left. E=\bigcup _{h=0}^\infty E_h \right\} \quad \forall \,E\text { Borel}. \end{aligned}$$

It is easy to check that, with this definition, $\left| \int _E d\mu \right| \le \int _Ed|\mu |$. Hence, it follows from (2.21) that

$$\begin{aligned} \bigl |\nabla v(x',s_2) - \nabla v(x',s_1) \bigr |\le \int \limits _{s_1}^{s_2} \bigl |\nabla w_{x'}(s)\bigr |\,ds + \int \limits _{s_1}^{s_2}d\bigl |D_{\mathcal S} w_{x'}\bigr |(s). \end{aligned}$$

Finally, we recall that the derivative of $w_{x'}$ is related to $D^2v$: if we define the family of lines $\ell _{x'}:=\{(x',s)\,:\,s \in \mathbb {R}\}$, it follows from [2, Theorem 3.107] that

$$\begin{aligned} \int \limits _E \langle D^2v\cdot e_n,e\rangle = \int \limits _{\mathbb {R}^{n-1}} \, dx' \, \int \limits _{E\cap \ell _{x'}} D w_{x'}\cdot e \quad \forall \,E\subset \mathbb {R}^n \text { Borel}, \, \forall \,e \in \mathbb {R}^n. \end{aligned}$$

(2.22)

This has the following useful consequences: since the measures

$$\begin{aligned} \left\langle D_{\mathcal S}^2v\cdot e_n,e\right\rangle -D_{\mathcal S}w_{x'}\cdot e\quad \text {and}\quad \bigl (\langle \nabla ^2v\cdot e_n,e\rangle -\nabla _{\mathcal S}w_{x'}\cdot e\bigr )\,dx \end{aligned}$$

are mutually singular, we deduce that (2.22) holds with $\nabla ^2v\cdot e_n$ and $\nabla w_{x'}$ (resp., with $D^2_{\mathcal S}v\cdot e_n$ and $D_{\mathcal S}w_{x'}$) in place of $D^2v\cdot e_n$ and $D w_{x'}$. Hence

$$\begin{aligned} \nabla w_{x'}(s)&= 7\nabla ^2v(x',s)\cdot e_n\quad \text {for a.e. } (x',s) \in \mathbb {R}^n,\end{aligned}$$

(2.23)

$$\begin{aligned} \int \limits _E \langle D_{\mathcal S}^2v\cdot e_n,e\rangle&= \int \limits _{\mathbb {R}^{n-1}} \, dx' \, \int \limits _{E\cap \ell _{x'}} D_{\mathcal S} w_{x'}\cdot e \quad \forall \,E\subset \mathbb {R}^n \text { Borel}, \, \forall \,e \in \mathbb {R}^n. \end{aligned}$$

(2.24)

(see also [2, Theorem 3.107]). In particular, (2.19) and (2.24) imply that

$$\begin{aligned} D_{\mathcal S} w_{x'}\cdot e_n \text { is a negative measure for a.e. } x' \in \mathbb {R}^{n-1}. \end{aligned}$$

(2.25)

2.7.2 The case of a critical solution

We now gather some extra properties when $v=u$ solves the Hamilton–Jacobi equation. Let us assume that $u:B_2 = B^n(0,2) \rightarrow \mathbb {R}$ is a semiconcave function satisfying

$$\begin{aligned} H(x,\nabla u(x))=\mathbf {c}[H] \quad \text{ for } \text{ a.e. } x \in B_2. \end{aligned}$$

(2.26)

Then $w:=\nabla u$ is a function of bounded variation, and since semiconcave functions are locally Lipschitz, $w$ is locally bounded inside $B_2$. Let us consider the family of bounded Borel functions $a_h:B_1=B^n(0,1)\rightarrow \mathbb {R}$, $h\in (0,1/2)$, defined as

$$\begin{aligned} a_h(x):=\int \limits _0^1\frac{\partial H}{\partial x_n}\bigl (x+\tau he_n,\tau \nabla u(x+he_n)+ (1-\tau )\nabla u(x)\bigr )\,d\tau \end{aligned}$$

and the family of bounded Borel vector fields $\xi _h:B_1\rightarrow \mathbb {R}^n$, $h\in (0,1/2)$, given by

$$\begin{aligned} \xi _h(x):=\int \limits _0^1\frac{\partial H}{\partial p}\bigl (x+\tau he_n,\tau \nabla u(x+he_n) +(1-\tau )\nabla u(x)\bigr )\,d\tau . \end{aligned}$$

Let us recall that, since $w \in BV_{loc}(B_2)$, the following bound holds:

$$\begin{aligned}&\int \limits _{B_r} \frac{|w(x+he_n) - w(x)|}{h}\,dx\nonumber \\&\quad \le \int \limits _{B_{r+h}}|Dw\cdot e_n|(dx)<\infty \quad \forall \,r \in (0,1),\,h \in (0,1/2) \end{aligned}$$

(2.27)

(for smooth functions the above estimate follows from the fundamental theorem of calculus, and for the general case one argues by approximation using [2, Theorem 3.9]). Hence the measures

$$\begin{aligned} \mu _h:=\frac{w(x+he_n) - w(x)}{h}\,dx \end{aligned}$$

satisfy

$$\begin{aligned} \int \limits _{B_1}|\mu _h|(dx)\le C, \end{aligned}$$

which implies that, up to a subsequence, $\mu _h$ (resp. $|\mu _h|$) converge weakly$^*$ to a finite measure $\mu$ (resp. $\nu$) as $h\rightarrow 0$. Also, there exists $\bar{a}:B_1\rightarrow [0,+\infty )$ bounded such that $|a_h| \rightharpoonup ^* \bar{a}$ in $L^\infty (B_1)$.

It is easy to show that $\mu =Dw\cdot e_n=D^2u\cdot e_n$. Furthermore, it follows from [2, Example 1.63] and (2.27) that

$$\begin{aligned} \nu (B_r) \le \liminf _{h \rightarrow 0} |\mu _h |(B_r) \le \int \limits _{\overline{B}_r}|Dw\cdot e_n|=|\mu |(\overline{B}_r) \quad \forall \,r \in (0,1), \end{aligned}$$

so letting $r \nearrow 1$ we obtain $\nu (B_1)\le |\mu |(B_1)$. This information combined with the bound $|\mu | \le \nu$ (see [2, Proposition 1.62(b)]) implies that $|\mu |=\nu$, thus

$$\begin{aligned} |\mu _h| \rightharpoonup ^* |\mu |. \end{aligned}$$

(2.28)

We now exploit the fact that $u$ solves the Hamilton–Jacobi equation (2.26). Since

$$\begin{aligned} 0&= H(x+he_n,\nabla u(x+he_n)) - H(x,\nabla u(x))\\&= h\int \limits _0^1\frac{\partial H}{\partial x_n}\bigl (x+\tau he_n,\tau \nabla u(x+he_n) + (1-\tau )\nabla u(x)\bigr )\,d\tau \\&+ \left( \,\int \limits _0^1\frac{\partial H}{\partial p}\bigl (x+\tau he_n,\tau \nabla u(x+he_n) + (1-\tau )\nabla u(x)\bigr )\,d\tau \right) \\&\quad \times \bigl (\nabla u(x+he_n)- \nabla u(x)\bigr )\\&= h\,a_h(x)+\xi _h(x)\cdot \bigl (w(x+he_n) - w(x)\bigr ), \end{aligned}$$

we have

$$\begin{aligned} a_h+\xi _h\cdot \mu _h\equiv 0. \end{aligned}$$

Let $\Omega \subset B_1$ be an open set and assume that there exist $h_0>0$ and a continuous vector field $\Xi :\Omega \rightarrow \mathbb {R}^n$ such that

$$\begin{aligned} |\Xi (x)- \xi _h(x)|\le \frac{1}{2}|\Xi (x)| \quad \forall \,x \in \Omega , \, \forall \,h \in (0,h_0). \end{aligned}$$

(2.29)

Then

$$\begin{aligned} 0=a_h+\xi _h\cdot \mu _h=a_h+(\xi _h-\Xi )\cdot \mu _h + \Xi \cdot \mu _h, \end{aligned}$$

so that, thanks to [2, Proposition 1.62(b)] and (2.28)–(2.29), letting $h \rightarrow 0$ we obtain

$$\begin{aligned} |\Xi \cdot \mu |\le \liminf _{h \rightarrow 0} |\Xi \cdot \mu _h|&\le \liminf _{h\rightarrow 0} \frac{1}{2}|\Xi |\,|\mu _h|+|a_h|\\&= \frac{1}{2}|\Xi |\,|\mu |+\bar{a}\quad \text { inside } \Omega , \end{aligned}$$

where $|\Xi \cdot \mu |$ denotes the total-variation of the measure $\Xi \cdot \mu$ (and analogously for $\mu _h$), and $|\Xi |$ denotes the continuous function $x \mapsto |\Xi (x)|$.

We now recall that, as observed above, the measure $\mu$ coincides with the measure $D^2u\cdot e_n$, hence

$$\begin{aligned} |\langle D^2u\cdot e_n,\Xi \rangle |\le \frac{1}{2}|\Xi |\,|D^2u\cdot e_n|+\bar{a}\quad \text {inside }\Omega . \end{aligned}$$

In particular, if we restrict this inequality to the singular part of $D^2u$, since $\bar{a}$ is a bounded function we get

$$\begin{aligned} |\langle D_{\mathcal S}^2u\cdot e_n,\Xi \rangle |\le \frac{1}{2}|\Xi |\,|D^2_{\mathcal S}u\cdot e_n|\quad \text {inside } \Omega , \end{aligned}$$

which by (2.22) can be written as a superposition of the measures $Dw_{x'}$:

$$\begin{aligned} |\Xi \cdot D_{\mathcal S} w_{x'}| \le \frac{1}{2}|\Xi |\,|D_{\mathcal S} w_{x'}|\quad \text {inside }\Omega , \hbox { for a.e. } x'. \end{aligned}$$

Using the polar decomposition theorem [2, Corollary 1.29], we can write $D_{\mathcal S}w_{x'}=\theta |D_{\mathcal S}w_{x'}|$, where $\theta :B_1\rightarrow \mathbb S^{n-1}$ is a $|D_{\mathcal S}w_{x'}| \otimes dx'$-measurable function. Hence the above equation can be rewritten as

$$\begin{aligned} |\Xi \cdot \theta | \le \frac{1}{2}|\Xi |\quad \text {inside } \Omega , |D_{\mathcal S} w_{x'}| \otimes dx'\hbox {-a.e.}. \end{aligned}$$

(2.30)

This information is particularly useful when $n=2$ and $\Xi$ never vanishes: indeed, assuming for instance that $\Xi \equiv e_1$, then (2.30) implies that

$$\begin{aligned} |\theta | \le 2|\theta \cdot e_2|\quad \text {inside } \Omega , |D_{\mathcal S} w_{x'}| \otimes dx' \hbox { -a.e.}, \end{aligned}$$

from which we get

$$\begin{aligned} |D_{\mathcal S}w_{x'}|\le 2| D_{\mathcal S}w_{x'}\cdot e_2| \quad \text {inside } \Omega , \hbox { for a.e. } x'. \end{aligned}$$

(2.31)

This means that $|D_{\mathcal S}^2u\cdot e_2|$ is controlled by $|\langle D_{\mathcal S}^2u\cdot e_2,e_2\rangle |$, or equivalently, since $D_{\mathcal S}^2u\cdot e_2$ is a vector-valued measure of components $\langle D_{\mathcal S}^2u\cdot e_2,e_1\rangle$ and $\langle D_{\mathcal S}^2u\cdot e_2,e_2\rangle$, the measure $|\langle D_{\mathcal S}^2u\cdot e_2,e_1\rangle |$ is controlled by $|\langle D_{\mathcal S}^2u\cdot e_2,e_2\rangle |$. Hence, the size of the pure second derivatives in the $e_2$ direction controls the size of the mixed second derivatives in $e_1,e_2$ in the region where the Hessian is singular (that is, roughly speaking, where $\nabla u$ has a jump).

2.8 A lemma from harmonic analysis

In this section we recall a classical result from harmonic analysis (see [38]), and we show its simple proof for the convenience of the reader. We denote by $|A|$ the Lebesgue measure of a set $A\subset \mathbb {R}^n$.

Lemma 2.20

Let $f \in L^1(\mathbb {R}^n)$, and define the maximal function

$$\begin{aligned} Mf(x):=\sup _{x \in \overline{B}\,, B\, \text {open ball}} \left\{ \frac{1}{|B|} \int \limits _{B}|f(y)|\,dy\right\} \quad \forall \,x \in \mathbb {R}^n. \end{aligned}$$

There exists a dimensional constant $C_n>0$ such that

$$\begin{aligned} \left| \Bigl \{x \in \mathbb {R}^n:Mf(x) >\delta \Bigr \}\right| \le \frac{C_n}{\delta }\Vert f\Vert _{L^1(\mathbb {R}^n)}\quad \forall \,\delta >0. \end{aligned}$$

Proof of Lemma 2.20

Let $K\subset \{Mf>\delta \}$ be any compact subset. By the definition of $Mf$, for any $x \in K$ there exists an open ball $B_x$ such that

$$\begin{aligned} x \in \overline{B}_x ,\quad |B_x| \le \frac{1}{\delta } \int \limits _{B_x}|f(y)|\,dy. \end{aligned}$$

Let $\rho B$ denote the dilation of a ball $B$ by a factor $\rho >0$ with respect to its center. Since $x \in \overline{B}_x\subset 2B_x$, the family of open balls $\{2B_x\}_{x \in K}$ covers $K$. So, by compactness we can find a finite collection of these balls which still covers $K$, and by Vitali’s Lemma [16], sect. 1.5.1, Theorem 1] we can select a disjoint subcollection $\{2B_{x_1},\ldots ,2B_{x_m}\}$ such that $K \subset \cup _{j=1}^m 10 B_{x_j}$. Hence

$$\begin{aligned} |K|\le 10^n \sum _{j=1}^m|B_{x_j}| \le \frac{10^n}{\delta } \sum _{j=1}^m\int \limits _{B_{r_j}(x_j)}|f(y)|\,dy \le \frac{10^n}{\delta }\Vert f\Vert _{L^1(\mathbb {R}^n)}, \end{aligned}$$

and the result follows by the arbitrariness of $K$. $\square$

3 Proof of Theorem 1.1

Let $H:T^*M \rightarrow \mathbb {R}$ be a Tonelli Hamiltonian of class $C^2$, and denote by $L:TM\rightarrow \mathbb {R}$ its associated Lagrangian. We want to show that the set of potentials $V\in C^2(M)$ such that the Aubry set $\tilde{\mathcal {A}}(H+V)$ is hyperbolic contains an open dense set. Hence we need to prove a stability result (the openness) and a density result.

We proceed as follows: first, in Sect. 3.1 we show that if the Aubry set $\tilde{\mathcal {A}}(H)$ is minimal and hyperbolic, then all Aubry sets $\tilde{\mathcal {A}}(H+V)$ associated with potentials $V\in C^2(M)$ which are sufficiently small in $C^2$ topology are hyperbolic. Then, in Sect. 3.2 we show that the set of potentials $V\in C^2(M)$ such that the Aubry set of $H+V$ is minimal and hyperbolic is dense. We recall that a nonempty compact $\phi _t^H$-invariant set $\Lambda \subset T^*M$ is called minimal if any orbit of $\phi _t^H$ contained in $\Lambda$ is dense inside $\Lambda$. By Zorn’s Lemma, any nonempty compact $\phi _t^H$-invariant set contains a minimal subset.

3.1 The stability part

Recall that the Peierls barrier is the function $h:M \times M \rightarrow \mathbb {R}$ defined as

$$\begin{aligned} h(x,y) := \liminf _{t \rightarrow +\infty } \Bigl \{ h_t(x,y)+ \mathbf {c}[H]t\Bigr \} \quad \forall \,x,y \in M, \end{aligned}$$

(3.1)

where

$$\begin{aligned} h_t(x,y) := \inf \int \limits _0^t L\bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds \end{aligned}$$

(3.2)

and the infimum is taken over all Lipschitz curves $\gamma : [0,t] \rightarrow M$ such that $\gamma (0)=x$ and $\gamma (t)=y$ (we refer the reader to [18, 20, 36] for further details). By construction $h$ is Lipschitz on $M\times M$ (see for instance [18, Corollary 5.3.3]) and any critical subsolution $u$ satisfies

$$\begin{aligned} u(y) - u(x) \le h(x,y) \quad \forall \,x,y \in M \end{aligned}$$

(3.3)

(this fact follows easily from Proposition 2.1). Moreover, it can be checked that (see [18, Proposition 5.3.8], [20, 36])

$$\begin{aligned} \mathcal {A}(H) = \Bigl \{ x\in M \, \vert \, h(x,x)=0\Bigr \}. \end{aligned}$$

(3.4)

Following Mather [32], the function $\delta _M : M \times M \rightarrow \mathbb {R}$ given by

$$\begin{aligned} \delta _M(x,y) := h(x,y) + h(y,x) \quad \forall \,x,y \in M \end{aligned}$$

is a semi-distance (sometimes called the Mather semi-distance).

Lemma 3.1

Assume that $\tilde{\mathcal {A}}(H)$ is minimal. Then $H$ admits a unique weak KAM solution (up to a constant) and $\delta _M(x,y)=0$ for any $x,y \in \mathcal {A}(H)$.

Proof of Lemma 3.1

Let $u_1, u_2 : M \rightarrow \mathbb {R}$ be two weak KAM solutions. Since their differentials coincide along any orbit of the Aubry set (see Proposition 2.5) and in addition all the orbits are dense in $\mathcal {A}(H)$, there is a constant $a\in \mathbb {R}$ such that $u_1-u_2=a$ on $\mathcal {A}(H)$. By Fathi’s comparison theorem (see [18, Theorem 8.5.5]), we infer that $u_1$ and $u_2$ differ by a constant on the whole $M$. The second assertion follows from the fact that the pointed functions $\{h(z,\cdot )\}_{z \in M}$ are weak KAM solutions (see [18, Theorem 5.3.6] or [20, Proposition 4.1]) and from the equality [using (3.4)]

$$\begin{aligned} \delta _M(x,y) = \Bigl ( h(x,y)-h(x,x) \Bigr ) - \Bigl ( h(y,y)-h(y,x) \Bigr ) \quad \forall \,x,y \in \mathcal {A}(H). \end{aligned}$$

$\square$

As shown in [15, Theorem C], by the uniqueness of weak KAM solutions (or equivalently the uniqueness of static classes) one obtains the upper-semicontinuity of the mapping $V\mapsto \tilde{\mathcal {A}}(H+V)$ at $V=0$ (compare with [9], corollary 5]), from which the stability of the hyperbolicity of Aubry sets follows:

Lemma 3.2

Assume that $\tilde{\mathcal {A}}(H)$ is minimal and hyperbolic. Then there is an open neighborhood ${\mathcal V}$ of $0$ in $C^2(M)$ such that, for every $V \in {\mathcal V}$, $\tilde{\mathcal {A}}(H+V)$ is hyperbolic.

Proof of Lemma 3.2

We first show that, since $H$ admits a unique weak KAM solution (which follows from the previous lemma), the mapping $V \in C^2(M) \mapsto \tilde{\mathcal {A}}(H+V) \subset T^*M$ is upper semicontinuous with respect to the Hausdorff topology, that is, for every open set $\mathcal {O} \in T^*M$ containing $\tilde{\mathcal {A}}(H)$ there is an open neighborhood $\mathcal {V}$ of $0$ in $C^2(M)$ such that, for every $V \in C^2(M)$,

$$\begin{aligned} V \in \mathcal {V} \, \Longrightarrow \, \tilde{\mathcal {A}}(H+V) \subset \mathcal {O}. \end{aligned}$$

Without loss of generality, up to adding a constant to $H$ we can assume that $\mathbf {c}[H]=0$.

We argue by contradiction and assume that there are an open neighborhood $\mathcal {O}$ of $\tilde{\mathcal {A}}(H)$, a sequence of potentials $\{V_k\}_k$ which tends to zero in the $C^2$ topology, and a sequence $\{\theta _k\}_k\subset T^*M$ satisfying $\theta _k \in \tilde{\mathcal {A}}(H+V_k)\setminus \mathcal {O}$ for all $k$. For every $k$, we pick a critical solution $u_k$ for the Hamiltonian $H+V_k$, and we define the calibrated curves $\gamma _k(t):=\pi ^*\bigl (\phi _t^{H+V_k}(\theta _k)\bigr )$. Because critical solutions (resp. calibrated curves) are uniformly Lipschitz (resp. $C^{1,1}$) [18], taking subsequences if necessary, we may assume that $\{u_k\}_k$ converge uniformly to a weak KAM solution $u$ for $H$, and $\{\gamma _k\}_k$ converge in $C^1$ topology to a calibrated (with respect to $u$) curve $\gamma : \mathbb {R}\rightarrow M$ with $\gamma (0) \notin \mathcal {A}(H)$, that is,

$$\begin{aligned} u\bigl ( \gamma (b)\bigr ) - u\bigl ( \gamma (a) \bigr ) = \int \limits _a^b L \bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds=h_{b-a}(\gamma (a),\gamma (b)) \quad \forall \, a<b. \end{aligned}$$

(3.5)

It can be shown that $\omega$-limit and $\alpha$-limit sets of any calibrated curves are contained in the Aubry set (see [36, Proposition 4.1]). Hence, there is a sequence $\{T_l\}_l \uparrow +\infty$ such that $\gamma (T_l)$ and $\gamma (-T_l)$ tend to $\mathcal {A}(H)$ as $l$ tends to $+\infty$. Let us denote by $d$ a Riemannian distance on $M$, and by $K$ a Lipschitz constant for $h$.

Given $\eta >0$ we choose $l$ large enough and $\alpha _l, \beta _l \in \mathcal {A}(H)$ such that

$$\begin{aligned} d\bigl (\gamma (-T_l),\alpha _l \bigr )+ d\bigl (\gamma (T_l),\beta _l \bigr ) < \eta . \end{aligned}$$

Set $x:=\gamma (0)$. Then, using the definition of $h$ (3.1), the fact that $h_{t+s}(x,y)\le h_t(x,z)+h_s(z,y)$, (3.5), and that $\delta _M(\alpha _l,\beta _l)=0$ (which follows from Lemma 3.4), we get

$$\begin{aligned} h(x,x)&\le h_{T_l}\bigl (x,\gamma (T_l)\bigr ) + h \bigl ( \gamma (T_l),\gamma (-T_l)\bigr ) + h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) \\&= h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) + h_{T_l}\bigl (x,\gamma (T_l)\bigr ) + h \bigl (\beta _l,\alpha _l\bigr )\\&\quad +\, h \bigl ( \gamma (T_l),\gamma (-T_l)\bigr ) - h \bigl (\beta _l,\alpha _l\bigr ) \\&\le h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) + h_{T_l}\bigl (x,\gamma (T_l)\bigr ) + h \bigl (\beta _l,\alpha _l\bigr ) + K \bigl [d \bigl ( \gamma (T_l),\beta _l\bigr ) \\&\quad +\, d\bigl ( \gamma (-T_l),\alpha _l\bigr )\bigr ] \\&\le h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) + h_{T_l}\bigl (x,\gamma (T_l)\bigr ) + h \bigl (\beta _l,\alpha _l\bigr ) + K\eta \\&= h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) + h_{T_l}\bigl (x,\gamma (T_l)\bigr ) - h \bigl (\alpha _l,\beta _l\bigr ) + K\eta \\&\le h_{T_l} \bigl ( \gamma (-T_l),x\bigr ) + h_{T_l}\bigl (x,\gamma (T_l)\bigr ) - h \bigl ( \gamma (-T_l),\gamma (T_l)\bigr ) + 2K\eta \\&= \left( u(x) - u\bigl ( \gamma (-T_l)\bigr ) \right) + \left( u\bigl ( \gamma (T_l)\bigr ) - u(x)\right) - h \bigl ( \gamma (-T_l),\gamma (T_l)\bigr ) + 2K\eta \\&\le u\bigl ( \gamma (T_l)\bigr ) - u\bigl ( \gamma (-T_l)\bigr ) - h \bigl ( \gamma (-T_l),\gamma (T_l)\bigr ) + 2K\eta \le 2K\eta , \end{aligned}$$

where for the last inequality we used (3.3). By the arbitrariness of $\eta$ this shows that $h(x,x)=0$, which implies that $x$ belongs to $\mathcal {A}(H)$, a contradiction. This proves the upper-semicontinuity of the Aubry set, and the conclusion follows easily from Proposition 2.18. $\square$

Thanks to Lemma 3.2, it is now sufficient to show a density result, that is, given a Tonelli Hamiltonian $H$ of class $C^2$ and $\epsilon >0$, there is $V\in C^2(M)$ with $\Vert V\Vert _{C^2(M)}<\epsilon$ such that the Aubry set of $H+V$ is minimal and hyperbolic.

3.2 The density part

Let us fix a $C^2$ Tonelli Hamiltonian $H$. First of all, up to adding a small potential (in the $C^2$ topology) we may assume that the Aubry set $\tilde{\mathcal {A}}(H)$ is minimal, i.e., all its orbits are dense in $\tilde{\mathcal {A}}(H)$ (see [21], Sect. 5.1] where we explain how to add a potential to reduce the size of the Aubry set). We can also assume that $\tilde{\mathcal {A}}(H)$ is not an equilibrium point or a periodic orbit, as otherwise we may add an arbitrarily small potential to make it hyperbolic (see [14, Theorem D]² and also [30]). Thus, the critical energy level

$$\begin{aligned} \Sigma := \Bigl \{ \theta = (x,p) \in T^*M \, \vert \, H(x,p) = \mathbf {c}[H]\Bigr \} \subset T^*M \end{aligned}$$

satisfies the assumptions of Sect. 2.4. Since we work on a surface, two cases may appear. Either the positive and negative Green bundles along $\tilde{\mathcal {A}}(H)$ satisfy

$$\begin{aligned} G^-_{\theta } \cap G_{\theta }^+ = \mathbb {R}X_H(\theta ) \quad \forall \,\theta \in \tilde{\mathcal {A}}(H), \end{aligned}$$

(3.6)

or

$$\begin{aligned} G_{\bar{\theta }}^+ = G_{\bar{\theta }}^- \quad \text{ for } \text{ some } \bar{\theta } \in \tilde{\mathcal {A}}(H). \end{aligned}$$

(3.7)

In the first case [when (3.6) holds], the hyperbolicity of $\tilde{\mathcal {A}}(H)$ follows from Proposition 2.19. Indeed, consider the projection $\Psi _t$ of the differential of the Hamiltonian flow to the bundle

$$\begin{aligned} \mathbf N_{\theta } :=\Bigl \{\xi \in T_\theta \Sigma \, \vert \, \langle D_{\theta }\pi ^* (\xi ), D_{\theta }\pi ^* (X_H(\theta ))\rangle _{\pi ^*(\theta )} = 0 \Bigr \}, \end{aligned}$$

that is $\Psi _t := \Lambda \circ D\phi ^H_t\vert _{\mathbf {N}}$ where $\Lambda :T\Sigma \rightarrow {\mathbf {N}}$ is the projection along the direction of the $X_H$:

$$\begin{aligned} \Lambda \,\xi = \xi +\beta (\xi )\, X_H(\theta )\quad \text { with } \beta (\xi )\in {\mathbb {R}} \hbox { such that } \Lambda \,\xi \in {\mathbf {N}}_\theta . \end{aligned}$$

Since the Green bundles are always transverse, the restriction of $\Psi _t$ to $\tilde{\mathcal {A}}(H)$ is quasi-hyperbolic (cf. [14, Corollary 2.3(d)]). Therefore, since we are assuming that $\tilde{\mathcal {A}} (H)$ is minimal, Proposition 2.19 implies that $\Psi _t$ is a hyperbolic action and then $\tilde{\mathcal {A}} (H)$ is a hyperbolic set.

In the second case [when (3.7) holds], the results in Sect. 2.4 show that critical solutions restricted to the Aubry sets are $C^2$ at $x=\pi ^*(\theta )$. As we will show below, this property allows us to implement the techniques developed in [21, 22] to close the orbit of $\bar{\theta }$ into a periodic orbit. However, the construction of a critical subsolution for the new Hamiltonian (which is unavoidable to close the orbit into an genuine Aubry set) becomes much more difficult than in [21, 22] because of the lack of regularity of critical solutions in a neighborhood of the orbit passing through $x$ (in [21, 22], the authors had to assume extra regularity on a critical solution to make their argument work). Still, thanks to the preparatory results on semiconcave and BV functions given in Sect. 2.7, we will be able to perform such a construction and make the whole proof work. So, the goal of the next section is to prove the following result, from which Theorem 1.1 follows.

Proposition 3.3

Let $H:T^*M \rightarrow \mathbb {R}$ be a Hamiltonian of class $C^2$, and assume that $\dim M=2$ and that $\tilde{\mathcal {A}}(H)$ is minimal. Let $\mathcal {V}$ be a neighborhood of $0$ in $C^2(M)$ and $\bar{\theta } \in \tilde{\mathcal {A}}(H)$ with $X_H(\bar{\theta })\ne 0$ be such that $G_{\bar{\theta }}^+ = G_{\bar{\theta }}^-$. Then there exists ${V} \in \mathcal {V}$ such that the Aubry set associated to the Hamiltonian $H+V$ is a hyperbolic periodic orbit (in its energy level).

3.3 Proof of Proposition 3.3

From now on, we assume that the Aubry set $\tilde{\mathcal {A}}(H)$ is a minimal set which is neither an equilibrium point nor a periodic orbit. Without loss of generality, up to adding a constant to $H$ (which does not change the dynamics), we can assume that $\mathbf {c}[H]=0$. Let $L$ denote the Lagrangian associated to $H$. Given $\epsilon >0$, our goal is to find a potential $V:M\rightarrow \mathbb {R}$ of class $C^{2}$ with $\Vert V\Vert _{C^2} <\epsilon$, together with a Lipschitz function $v_V:M\rightarrow \mathbb {R}$, and a $C^1$ curve $\gamma :[0,T'] \rightarrow M$ with $\gamma (0)=\gamma (T')$, such that the following properties are satisfied:

• (P1) $H_V \bigl (x,dv_V(x)\bigr ) \le 0 \text{ for } \text{ a.e. } x \in M$.

• (P2) $\int _0^{T'} L_V(\gamma (t),\dot{\gamma }(t)) \,dt = 0$.

Indeed, as explained in [21], sect. 5.1] (see also [23]), if we are able to do this then (P1) implies that $c[H_V]\le 0$ (see Sect. 2.1), while (P2) together with (2.2) yields $c[H_V]\ge 0$. Therefore, by (3.4) and the definition of the Peierls barrier $h$ (3.1), the closed curve $\Gamma := \gamma ([0,T'])$ is contained in the projected Aubry set of $H_V$. Now, if $W:M \rightarrow \mathbb {R}$ is any smooth function such that $W=0$ on $\Gamma$, $W>0$ outside $\Gamma$, and $\Vert W\Vert _{C^2}<\epsilon -\Vert V\Vert _{C^2}$, then the function $v$ is a critical subsolution of $H_{V-W}=H+V-W$ which is strict outside $\Gamma$, and we have $\int _0^T L_{V-W} (\gamma (t),\dot{\gamma }(t)) \,dt = 0$. By (3.4), this implies that the projected Aubry set of $H_{V-W}$ coincides with the periodic curve $t \mapsto \gamma (t)$. Moreover, as shown in [14, Theorem D], we can add a potential, small in the $C^2$ topology, which preserves the periodic orbit and makes it a hyperbolic Aubry set. Hence, we are left with finding $V$, $v_V$, and $\gamma$ such that (P1) and (P2) hold.

Fix $\epsilon >0$, and let $\bar{\theta }= (\bar{x},\bar{p}) \in \tilde{\mathcal {A}}(H)$ be as in the statement of Proposition 3.3. Let us denote by $\bar{\theta }(\cdot )=(\bar{\gamma }(\cdot ),\bar{p}(\cdot ))$ the orbit of $\bar{\theta }$ by the Hamiltonian flow, and by $\bar{\Pi }\subset M$ a local section (that is, a smooth curve) which is transverse to $\bar{\gamma }$ at $t=0$. Let $u: M \rightarrow \mathbb {R}$ be a critical solution for $H$. Recall that $u$ is differentiable on the projected Aubry set $\mathcal {A}(H)$, and that the restriction of $du$ to $\mathcal {A}(H)$ is Lipschitz (see Proposition 2.5).

The following lemma will be needed to apply Proposition 2.7.

Lemma 3.4

Let $c_0>0$ be as in Proposition 2.7. There exists $\bar{t}>0$ such that, on any time interval of the form $[t_0,t_0+\bar{t}]$ there is a time $t' \in [t_0,t_0+\bar{t}]$such that

$$\begin{aligned} \frac{d}{dt} \left\{ u\Bigl ( \phi _t^H \bigl (\bar{x}, \bar{p} \bigr ) \Bigr ) \right\} _{\vert t={t}'} = \left\langle du\bigl (\bar{\gamma } ({t}')\bigr ), \dot{\bar{\gamma }}\bigl ({t}'\bigr ) \right\rangle \ge -c_0. \end{aligned}$$

(3.8)

Proof of Lemma 3.4

If not

$$\begin{aligned} u \bigl ( \bar{\gamma }(t_0+\bar{t})\bigr ) - u \bigl ( \bar{\gamma }(t_0)\bigr ) = \int \limits _{t_0}^{t_0+\bar{t}} \bigl \langle du\bigl ( \bar{\gamma }(s)\bigr ), \dot{\bar{\gamma }}(s) \bigr \rangle \, ds \le -\int \limits _{t_0}^{t_0+\bar{t}}c_0 =-c_0 \bar{t}. \end{aligned}$$

Since $u$ is bounded, this is impossible if $\bar{t}$ is sufficiently large. $\square$

Up to replacing $H$ by $4H/\bar{t}$, we can assume that the constant $\bar{t}$ appearing in the previous lemma satisfies³

$$\begin{aligned} \bar{t}=1/4. \end{aligned}$$

(3.9)

Let us take $T>0$ to be fixed. Since $\bar{\gamma }$ can never intersect itself, there exist an open neighborhood $\mathcal {U}$ of $\bar{\gamma }([0,T])$ in $M$, and a $C^2$ diffeomorphism $\Phi : \mathcal {U} \rightarrow \mathcal {U}' := \Phi (\mathcal {U}) \subset \mathbb {R}^2$, such that, in the new system of coordinates, the curve $\Phi \left( \bar{\gamma }_{\vert [0,T]}\right)$ is a straight segment. Hence, using still $\bar{\gamma }$ instead of $\Phi (\bar{\gamma })$ to denote this curve (by a slight abuse of notation), we can assume that

• $(\pi _1)$ $\bar{\gamma } (t) = (t e_1, 0)$ for any $t\in [-1,T]$;

• $(\pi _2)$ $[-1,T]\times [-\rho ,\rho ]\subset \mathcal {U}'$.

(Here and in the sequel, $(e_1,e_2)$ denotes the canonical basis in $\mathbb {R}^2$.) Also, in this new set of coordinates, we can see $H$ as a Hamiltonian on $T^*\mathcal {U}' \subset T^*\mathbb {R}^2=\mathbb {R}^2 \times (\mathbb {R}^2)^*$, and the critical solution $u$ as a semiconcave function on $\mathbb {R}^2$. We set

$$\begin{aligned} \Pi ^t_r:= \Bigl \{ (t e_1,y) \, \vert \, y \in [-r,r]\Bigr \} \quad \forall \,t,r\in \mathbb {R}. \end{aligned}$$

The intersection of the Aubry set (resp. projected Aubry set) with $T^*\mathcal {U}$ (resp. with $\mathcal {U}$) is transported by $\Phi$. Let us denote by $\tilde{\mathcal {A}}$ and $\mathcal {A}$ their respective images in $T^*\mathcal {U}'$ and $\mathcal {U}'$. The Green bundles $G_{\phi _t^H(\bar{\theta })}^+=G_{\phi _t^H(\bar{\theta })}^-$ for $t\in [-1,T]$, and $G^+_{\theta }$ with $\theta \in T^*\mathcal {U}$, are also transported by $\Phi$. We denote them respectively by $G_t$ and $G^+_{\theta }$ in $T(T^*\mathcal {U}')$. We now apply (3.7) and Corollary 2.15 to deduce that, up to reduce the size of $\rho$ and $\mathcal U'$, there is a function $f:\mathcal {U}'\rightarrow \mathbb {R}$ of class $C^{1,1}$ such that the $1$-form $\Psi :=df$ on $\mathcal {U}'$ satisfies the following properties:

• $(\pi _3)$ $\Psi$ is $C^1$ along $\bar{\gamma }([-1,T])$;

• $(\pi _4)$ $\tilde{\mathcal {A}} \cap T^*\mathcal {U'} \subset \text{ Graph } (\Psi )$;

• $(\pi _5)$ for every $t\in [-1,T], G_t = \text{ Graph } \bigl (L_t:=d_{\bar{\gamma }(t)}\Psi \bigr ) \subset \mathbb {R}^2 \times (\mathbb {R}^2)^*$.

3.3.1 Some preliminary regularity estimates on $u$

Let us recall that $u$ is semiconcave (see Proposition 2.3), so the discussion in Sect. 2.7 (see in particular Sect. 2.7.2) applies. Also, since $\bar{\gamma }([0,T])=\{te_1\}_{t \in [0,T]}$ (see ($\pi _1$)) and $te_1 \in \mathcal A$ (hence $u$ is differentiable at $te_1$, see Proposition 2.5), by upper-semicontinuity of the limiting differential of semiconcave functions there is a modulus of continuity $\omega : \mathbb {R}^+ \rightarrow \mathbb {R}^+$ (that is, $\omega$ is nondecreasing with $\lim _{r\downarrow 0} \omega (r)=0$), possibly depending on $T$, such that

$$\begin{aligned}&\Bigl |(x,p) - (te_1,\nabla u(te_1)) \Bigr | \le \omega (r) \quad \forall \,x \in \Pi _r^t,\nonumber \\&\quad p \in D_x^*u(x), \,t \in [0,T], \,r \in (0,\rho ) \end{aligned}$$

(3.10)

and $\left( \hbox {since} \frac{\partial H}{\partial p}\bigl (te_1,\nabla u(te_1)\bigr )=e_1, \hbox {see} (\pi _1)\right)$

$$\begin{aligned} \biggl |\frac{\partial H}{\partial p}(x,p) - e_1 \biggr | \le \omega (r) \quad \forall \,x \in \Pi _r^t,\,p \in D_x^*u(x),\, t \in [0,T], \, r \in (0,\rho ). \end{aligned}$$

(3.11)

As in (2.20), for a.e. $t\in [0,T]$ we define the function $w_{t}: [-\rho ,\rho ] \rightarrow \mathbb {R}^n$ by

$$\begin{aligned} w_t(s) :=\nabla u(t,s) \quad \text{ for } \text{ a.e. } s \in [-\rho ,\rho ], \end{aligned}$$

and we recall the following decomposition for $Dw_t$ (see Sect. 2.7):

$$\begin{aligned} Dw_{t}= \nabla w_{t}\,ds+ D_{\mathcal S} w_{t}, \end{aligned}$$

where $\nabla w_{t}\,ds$ is absolutely continuous and $D_{\mathcal S}w_t$ is singular with respect to $ds$.

We notice that (3.11) implies that (2.29) holds with $\Xi (x)\equiv e_1$, hence it follows from (2.31) that

$$\begin{aligned} |D_{\mathcal S}w_{t}|\le 2| D_{\mathcal S}w_{t}\cdot e_2| \quad \text {inside } \mathcal U', \hbox { for a.e. } t \in [0,T]. \end{aligned}$$

(3.12)

Also, Proposition 2.17 combined with the upper semicontinuity of the positive Green bundle provides an upper bound on $D^2u$ in a neighborhood of a the curve $\bar{\gamma }([0,T])$. More precisely, we recall that $\langle D_{\mathcal {S}}^2 v\cdot e,e\rangle$ is a nonpositive measure for any vector $e \in \mathbb {R}^n$ (see Sect. 2.7). Also, by ($\pi _5$) and Propositions 2.10 and 2.17 we deduce that there exists a modulus of continuity $\omega ':\mathbb {R}^+\rightarrow \mathbb {R}^+$, possibly depending on $T$, such that, for a.e. $t \in [0,T]$,

$$\begin{aligned} \nabla ^2u(x) \le L_t + \omega '(r)\mathrm{Id} \quad \text {for a.e. } x \in \Pi _r^t. \end{aligned}$$

(3.13)

(Recall that $\nabla ^2u$ denotes the pointwise Hessian of $u$, which exists almost everywhere.) We denote by $\fancyscript{O}$ the orbit of $\bar{\gamma }$ in $\mathcal {U}'$, that is $\fancyscript{O}:=\bar{\gamma }(\mathbb {R})\cap \mathcal {U}'$.

In the next lemma we use (3.7) to show that, for a.e. $t$, $Dw_{t}$ is close in total variation to a constant matrix. From now on, we always denote a modulus of continuity by $\omega$ and a positive constant by $C$, their values might change from line to line but otherwise they depend only on $T$ and the data (i.e., $H$, $u$, etc.).

Lemma 3.5

Let $\Psi$ be as in ($\pi _3$)-($\pi _5$). There exist a modulus of continuity $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$ and a constant $C >0$ such that the following properties hold for any $r \in (0,\rho ]$:

• (i) For a.e. $t\in [0,T]$, for every $y_1=(t,\ell _1),y_2=(t,\ell _2) \in \fancyscript{O} \cap \Pi _r^t$ with $\ell _2>\ell _1$,

  $$\begin{aligned} \int \limits _{\ell _1}^{\ell _2} \left| \nabla w_{t}(s)-L_t\cdot e_2\right| \,ds +\int \limits _{\ell _1}^{\ell _2} d|D_{\mathcal {S}}w_t|(s)\le \omega (r)\,|\ell _2-\ell _1|. \end{aligned}$$

• (ii) For every $y_1,y_2 \in \fancyscript{O} \cap \Pi _\rho ^T$ there exists a family of matrices $\{M_{-t}\}_{t \in [0,T]}$, with

  $$\begin{aligned} \bigl |M_{-t} \bigr |+ \bigl |(M_{-t})^{-1} \bigr |\le C, \end{aligned}$$

  such that the following holds for any constant $N \ge 1$: for every $z, z' \in \Pi _\rho ^T\cap [y_1,y_2]$ such that $u$ is differentiable at $z,z'$ and $|z'-z|\ge \frac{|y_1-y_2|}{N}$, we have

  $$\begin{aligned}&\Bigl |\pi ^*\bigl ( \phi _{-t}^H(z,\nabla u(z))\bigr )- \pi ^*\bigl (\phi _{-t}^H(z',\nabla u(z'))\bigr ) - M_{-t}(z-z')\Bigr |\\&\quad \le N\,\omega (r)\, |z-z'|. \end{aligned}$$

Proof of Lemma 3.5

We begin by observing that $|y_2-y_1|=|\ell _2-\ell _1|$. Since the graph of $\nabla u$ restricted to $\bar{\gamma }([0,T])=\{te_1\}_{t \in [0,T]}$ is contained inside the graph of $\Psi$ and the latter is $C^1$ there [see ($\pi _1$) and ($\pi _3$)], for a.e. $t\in [0,T]$ we get

$$\begin{aligned}&\Bigl | w_t(\ell _2)-w_t(\ell _1)-L_t\cdot (y_2-y_1)\Bigr |\nonumber \\&\quad = \Bigl | \Psi (y_2)- \Psi (y_1) - d_{\bar{\gamma }(t)}\Psi \cdot (y_2-y_1)\Bigr |\nonumber \\&\quad =\left| \int \limits _0^1 d_{y_1+s(y_2-y_1)}\Psi \cdot (y_2-y_1) \, ds - d_{\bar{\gamma }(t)}\Psi \cdot (y_2-y_1)\right| \nonumber \\&\quad \le \int \limits _0^1 \left| d_{y_1+s(y_2-y_1)}\Psi - d_{\bar{\gamma }(t)}\Psi \right| \left| y_2-y_1\right| \, ds \nonumber \\&\quad \le \omega (r)\,|\ell _2-\ell _1|, \end{aligned}$$

(3.14)

for some modulus of continuity $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$. So, rewriting the above expression using the fundamental theorem of calculus [see (2.21)], for a.e. $t\in [0,T]$ we have (observe that $\frac{y_2-y_1}{|y_2-y_1|}=e_2$)

$$\begin{aligned} \left| \, \int \limits _{\ell _1}^{\ell _2} \bigl [\nabla w_{t} (s)-L_t \cdot e_2\bigr ] \,ds+\int \limits _{\ell _1}^{\ell _2} d[D_{\mathcal {S}}w_t ](s) \right| \le \omega (r)\,|\ell _2-\ell _1|, \end{aligned}$$

which implies in particular that

$$\begin{aligned} \,\,\left| \,\int \limits _{\ell _1}^{\ell _2} \bigl [\nabla w_{t} (s)\cdot e_2 -\langle L_t \cdot e_2,e_2\rangle \bigr ]\,ds+\int \limits _{\ell _1}^{\ell _2} d[D_{\mathcal {S}}w_t\cdot e_2](s) \right| \le \omega (r)\,|\ell _2-\ell _1|. \end{aligned}$$

This estimate combined with (2.23), (3.13), and (2.25), gives

$$\begin{aligned}&\int \limits _{\ell _1}^{\ell _2}\left| \nabla w_{t} (s)\cdot e_2-\langle L_t \cdot e_2,e_2\rangle \right| \,ds +\int \limits _{\ell _1}^{\ell _2} d|D_{\mathcal {S}}w_t\cdot e_2|(s) \le \omega (r)\,|\ell _2-\ell _1|\\&\quad \text {for a.e. } t\in [0,T], \end{aligned}$$

which shows that $Dw_t\cdot e_2$ is $L^1$-close to $\langle L_t\cdot e_2,e_2\rangle$.

We now need to control $Dw_t\cdot e_1$. For this, we first apply (3.12) to obtain that the singular part of $Dw_t$ is controlled by $D_{\mathcal S}w_t\cdot e_2$: indeed (3.12) and the bound above imply

$$\begin{aligned} \int \limits _{\ell _1}^{\ell _2} d |D_{\mathcal {S}}w_t|(s) \le 2 \int \limits _{\ell _1}^{\ell _2} d |D_{\mathcal {S}}w_t\cdot e_2| (s) \le 2\,\omega (r)\,|\ell _2-\ell _1|\quad \text {for a.e. } t\in [0,T]. \end{aligned}$$

Hence it suffices to control only the absolutely continuous part of $Dw_t$.

Recall that, thanks to (2.23), for a.e. $t \in [0,T]$ we have

$$\begin{aligned} \nabla w_t(s)=\nabla ^2u(t,s)\cdot e_2\quad \text {for a.e. } s\in [-r,r], \end{aligned}$$

where $\nabla ^2u$ is the Hessian of $u$, which exists at almost every point. Hence it suffices to prove the closeness of $\nabla w_t$ to $L_t\cdot e_2$ only at points where $u$ is twice differentiable.

For every $x_\ell :=(t,\ell ) \in \Pi _r^t$ where $u$ is twice differentiable, consider the curve

$$\begin{aligned} \bigl (x_\ell (\tau ),p_\ell (\tau )\bigr ):=\phi _\tau ^H(x_\ell ,\nabla u(x_\ell )) . \end{aligned}$$

It follows from (3.10) and ($\pi _1$) that

$$\begin{aligned} \bigl |x_\ell (-\tau ) - (t-\tau )e_1\bigr | \le \omega (r) \quad \forall \,t \in [0,T],\, \tau \in [0,1]. \end{aligned}$$

Also, since the trajectories do not cross backward in time, $u$ is differentiable along them, and $p_\ell (-\tau )=\nabla u(x_\ell (-\tau ))$ (see Proposition 2.4), we have [here we use $\dot{x}_s(\tau )$ to denote the derivative with respect to $\tau$]

$$\begin{aligned} \frac{d}{d\tau }\bigl [ p_\ell (-\tau )\bigr ]&= \frac{d}{d\tau }\bigl [ \nabla u(x_\ell (-\tau ))\bigr ]\nonumber \\&= -\nabla ^2 u(x_\ell (-\tau ))\cdot \dot{x}_\ell (-\tau ) \quad \forall \,\tau \in [0,1], \end{aligned}$$

(3.15)

Since $p_\ell$ is uniformly bounded and solves the Hamiltonian system, also $\frac{d}{d\tau }\bigl [ p_\ell (-\tau )\bigr ]$ is uniformly bounded, hence we have

$$\begin{aligned}&\Bigl | \left\langle \nabla ^2 u(x_\ell (-\tau )) \cdot e, \dot{x}_{\ell }(-\tau )\right\rangle - \left\langle \nabla ^2 u(x_\ell (0))\cdot e, \dot{x}_{\ell }(0) \right\rangle \Bigr | \nonumber \\&\quad \le C\tau \, |e|\quad \forall \,e \in \mathbb {R}^2, \, \forall \,\tau \in [0,1]. \end{aligned}$$

(3.16)

To simplify the notation, set $x_s:=x_{\ell _1+s(\ell _2-\ell _1)}$. Then, it follows from (3.15) to (3.16) and the smoothness in $\tau$ of the curves $\tau \mapsto x_s(\tau )$ that, for every $\tau \in [0,1]$,

$$\begin{aligned}&\int \limits _0^1\left| \nabla w_t(\ell _1+s(\ell _2-\ell _1)) \cdot \dot{x}_s(0) - \langle L_t\cdot e_2,\dot{x}_s(0)\rangle \right| \,ds \nonumber \\&\quad = \int \limits _0^1\left| \langle \nabla ^2 u(x_s(0))\cdot e_2, \dot{x}_s(0)\rangle - \langle L_t\cdot e_2 ,\dot{x}_s(0)\rangle \right| \,ds\nonumber \\&\quad \le \int \limits _0^1\biggl |\frac{1}{\tau }\int \limits _{-\tau }^0 \langle \nabla ^2 u(x_s(\sigma ))\cdot \dot{x}_s(\sigma ), e_2\rangle \,d\sigma - \langle L_t\cdot e_2,\dot{x}_s(0)\rangle \biggr |\,ds \nonumber \\&\quad \quad + \int \limits _0^1\frac{1}{\tau }\int \limits _{-\tau }^0 \Bigl | \langle \nabla ^2 u(x_s(\sigma ))\cdot e_2,\dot{x}_s(\sigma )\rangle -\langle \nabla ^2 u(x_s(0))\cdot e_2, \dot{x}_s(0)\rangle \Bigr | \,d\sigma \,ds \nonumber \\&\quad \le \int \limits _0^1\biggl | \left\langle \frac{\nabla u(x_s(0)) - \nabla u(x_s(-\tau ))}{\tau }, e_2\right\rangle - \langle L_t\cdot e_2,\dot{x}_s(0)\rangle \biggr |\,ds + C\tau . \end{aligned}$$

(3.17)

By ($\pi _1$) and ($\pi _5$) (note that $\nabla u$ varies smoothly along $\bar{\gamma }([0,T])$, since it solves the Hamiltonian system) we have

$$\begin{aligned} \biggl |\frac{\nabla u(te_1)\cdot e_2 - \nabla u((t-\tau )e_1)\cdot e_2}{\tau }- \langle L_t\cdot e_1,e_2\rangle \biggr | \le C \tau \quad \forall \,\tau \in [0,1]. \end{aligned}$$

Hence, by (3.10), (3.11), and (3.17), for every $\tau \in [0,1]$ we get

$$\begin{aligned} \int \limits _0^1\bigl |\nabla w_t(\ell _1+s(\ell _2-\ell _1)) \cdot \dot{x}_s(0) - \langle L_t\cdot e_2,\dot{x}_s(0)\rangle \bigr |\,ds \le C\tau + \frac{\omega (r)}{\tau }. \end{aligned}$$

Thus, choosing $\tau :=\sqrt{\omega (r)}$ and using that $|\dot{x}_s(0)-e_1|\le \omega (r)$ and that $L_t$ is bounded (since $u$ is universally $C^{1,1}$ on the Aubry set), we get

$$\begin{aligned} \frac{1}{|\ell _2-\ell _1|}\int \limits _{\ell _1}^{\ell _2}|\nabla w_t(s)\cdot e_1 - \langle L_t\cdot e_2,e_1\rangle \bigr |\,ds&= \int \limits _0^1\bigl |\nabla w_t(\ell _1+s(\ell _2-\ell _1)) \cdot e_1 \\&\quad - \langle L_t\cdot e_2,e_1\rangle \bigr |\,ds\\&\le C\, \sqrt{\omega (r)}, \end{aligned}$$

concluding the proof of (i).

Let us now prove the second assertion. To simplify the notation, for a.e. $\bar{t} \in [T-1,T]$ we define the functions⁴

$$\begin{aligned} \psi _{-t}^{\bar{t}}(s)&:= \pi ^*\bigl (\phi _{-t}^H(z,\nabla u(z))\bigr )\\&= \pi ^*\bigl (\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr )\bigr ),\quad \text {for a.e. }z=(\bar{t},s) \in \Pi _r^{\bar{t}},\; \forall \,t \in [0,T]. \end{aligned}$$

By the chain-rule formula for BV functions [2, Theorem 3.96], the following hold: if we decompose the distributional derivative $D\psi _{-t}^{\bar{t}}$ into its absolutely continuous part $\nabla \psi _{-t}^{\bar{t}}$ and its singular part $D_{\mathcal S}\psi _{-t}^{\bar{t}}$, we have

$$\begin{aligned} \nabla \psi _{-t}^{\bar{t}}(s)&= d\pi ^*\bigl ( \phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr ) \bigr )\cdot \Bigl (\partial _{x_2}\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr )\\&\quad +\,\partial _p\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr )\cdot \nabla w_{\bar{t}}(s)\Bigr ) \end{aligned}$$

for a.e. $s \in [-r,r]$, and

$$\begin{aligned} \bigl |D_{\mathcal S} \psi _{-t}^{\bar{t}}\bigr | \le C\bigl |D_{\mathcal S} w_{\bar{t}}\bigr |. \end{aligned}$$

Given $z,z' \in \Pi _r^{\bar{t}}$, let us denote by $\int _{z}^{z'}d\mu$ the integral of a measure $\mu$ over the segment joining $z$ to $z'$. Then, by (i) and (3.10), for every $\tau \in [0,T]$ we have

$$\begin{aligned} \int \limits _{y_1}^{y_2} d|D\psi _{-t}^{\bar{t}}- M_{-t}^{\bar{t}}\cdot e_2|&= \int \limits _{y_1}^{y_2} \bigl |\nabla \psi _{-t}^{\bar{t}}(s)-M_{-t}^{\bar{t}}\cdot e_2 \bigr | \,ds+\int \limits _{y_1}^{y_2}d|D_{\mathcal S}\psi _{-t}^{\bar{t}}|(s)\nonumber \\&\le \omega (r)|y_2-y_1|, \end{aligned}$$

(3.18)

where

$$\begin{aligned} M_{-t}^{\bar{t}}:=d\pi ^*\bigl ( \phi _{-t}^H(\bar{t}e_1,e_1)\bigr )\cdot \Bigl (\partial _x\phi _{-t}^H(\bar{t}e_1,e_1)+\partial _p\phi _{-t}^H(\bar{t}e_1,e_1)L_{\bar{t}}\Bigr ), \end{aligned}$$

and we used that $|L_{\bar{t}}|$ is universally bounded (because $u$ is universally $C^{1,1}$ on the Aubry set) to estimate

$$\begin{aligned}&\Bigl |\partial _p\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr )\cdot \nabla w_{\bar{t}}(s) - \partial _p\phi _{-t}^H(\bar{t}e_1,e_1)L_{\bar{t}}\cdot e_2\Bigr |\\&\quad \le \Bigl |\partial _p\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr )\cdot \bigl (\nabla w_{\bar{t}}(s) -L_{\bar{t}}\cdot e_2\bigr )\Bigr |\\&\quad \quad +\Bigl |\Bigl (\partial _p\phi _{-t}^H\bigl ((\bar{t},s),w_{\bar{t}}(s)\bigr ) - \partial _p\phi _{-t}^H(\bar{t}e_1,e_1)\Bigr ) L_{\bar{t}}\cdot e_2\Bigr | \end{aligned}$$

The boundedness of $|L_{\bar{t}}|$ implies also that the norm $M_{-t}^{\bar{t}}$ is bounded on $[0,T]$ by a constant depending only on $T$. Also, since $u$ is semiconcave, a simple Gronwall argument shows that the backward flow $t \mapsto \psi _{-t}(z)$ is not “too much contractive”: there exists a universal constant $C>0$ such that

$$\begin{aligned} \left| \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right| \ge e^{-Ct}|z-z'| \quad \forall \,t \in [0,T], \, \forall z,z' \in \Pi _r^{\bar{t}}. \end{aligned}$$

(3.19)

Before proving the validity of the above estimate, we first show how we use it to conclude the proof.

From (3.19) we deduce that $|(M_{-t}^{\bar{t}})^{-1}| \le e^{CT}$ and that the trajectories cannot cross backward in time. Also, from (3.18) and the assumption $|z'-z|\ge \frac{|y_2-y_1|}{N}$ we deduce that

$$\begin{aligned} \bigl |\psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z') - M_{-t}^{\bar{t}}(z-z')\bigr |&\le \int \limits _{z'}^z d\left| D\psi _{-t}^{\bar{t}} - M_{-t}^{\bar{t}}\cdot e_2\right| \\&\le \int \limits _{y_1}^{y_2} d\left| D\psi _{-t}^{\bar{t}}- M_{-t}^{\bar{t}}\cdot e_2\right| \\&\le \omega (r)\,|y_1-y_2| \le N\omega (r)\,|z'-z| \end{aligned}$$

for a.e. $t \in [0,T]$ and a.e. $z,z' \in \Pi _r^t$. By a simple approximation argument, the above estimate extends to $\bar{t}=T$ and every $z,z' \in \Pi _r^T$ such that $u$ is differentiable at $z,z'$, which proves Lemma 3.5 with $M_{-t}:=M_{-t}^T$.

To finish the proof, we need to show the validity of (3.19), and we notice that (by triangle inequality in $z$ and because $t\mapsto \psi _{-t}^{\bar{t}}$ enjoys the semigroup property) it is sufficient to prove the result for $z, z'$ close to each other and for small times. By semiconcavity and compactness, there is a constant $K>0$ such that for every $(\bar{x},\bar{p})\in T^*\mathcal {U}'$ with $H(\bar{x},\bar{p})\le 0$ and every $x,x'\in \mathcal {U}'$ and $p \in \text{ conv } \left( D_x^*u\right) , p' \in \text{ conv } \left( D^*_{x'}u\right)$, we have (see [10])

$$\begin{aligned} \left\langle \frac{\partial ^2 H}{\partial p^2} \bigl (\bar{x},\bar{p}\bigr ) (p-p'), x-x'\right\rangle \le K \bigl | x-x'\bigr |^2. \end{aligned}$$

(3.20)

In particular, the above inequality holds for any $p=\nabla u(x), p'=\nabla u(x')$ with $u$ differentiable at $x, x'$. For any $z, z'\in \Pi _r^{\bar{t}}$ close enough and $t>0$ small, there is $(\bar{x},\bar{p}) \in T^*\mathcal {U}'$ with $H(\bar{x},\bar{p})\le 0$ such that

$$\begin{aligned}&\left\langle \frac{d}{dt} \left( \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right) , \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right\rangle \\&\quad = \left\langle -\frac{\partial H}{\partial p} \left( \phi _{-t}^H(z,\nabla u(z))\right) + \frac{\partial H}{\partial p} \left( \phi _{-t}^H(z',\nabla u(z'))\right) , \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right\rangle \\&\quad = \left\langle -\frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z)\bigr ) \right) + \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z'), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) , \right. \\&\qquad \qquad \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\Bigg \rangle \\&\quad = \left\langle -\frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z)\bigr ) \right) + \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) ,\right. \\&\quad \quad \qquad \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\Bigg \rangle \\&\quad \quad + \left\langle \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z'), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) - \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) ,\right. \\&\quad \quad \qquad \left. \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right\rangle \\&\quad = \left\langle \frac{\partial ^2 H}{\partial p^2} \bigl (\bar{x},\bar{p}\bigr ) \left( \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) - \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z)\bigr ) \right) , \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right\rangle \\&\quad \quad + \left\langle \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z'), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) - \frac{\partial H}{\partial p} \left( \psi _{-t}^{\bar{t}}(z), \nabla u \bigl ( \psi _{-t}^{\bar{t}}(z')\bigr ) \right) , \right. \\&\quad \quad \qquad \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\Bigg \rangle . \end{aligned}$$

By (3.20) and $C^2$ regularity of $H$, we infer that there is some universal constant $K'>0$ such that

$$\begin{aligned} \left\langle \frac{d}{dt} \left( \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right) , \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right\rangle \ge - K' \left| \psi _{-t}^{\bar{t}}(z)-\psi _{-t}^{\bar{t}}(z')\right| ^2. \end{aligned}$$

We conclude easily by Gronwall’s lemma. $\square$

The following bound will be crucial to estimate the action.

Lemma 3.6

There exist a modulus of continuity $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$ and a constant $K'>0$ such that the following holds: Let $y_1,y_2 \in \fancyscript{O}\cap \Pi ^t_\rho$ for some $t \in [0,T]$. Then, for every $r \in (0,\rho ]$, and for every $z_1,z_2 \in \Pi _r^t\cap [y_1,y_2]$ such that $u$ is differentiable at $z_1,z_2$ and $\frac{|y_1-y_2|}{10N} \le |z_1-z_2|\le \frac{|y_1-y_2|}{N}$,

• (i)
  $$\begin{aligned} \bigl | u(z_2)-u(z_1)-\langle \nabla u(z_1),z_2-z_1\rangle \bigr | \le K' \frac{|y_1 - y_2|^2}{N}; \end{aligned}$$
• (ii)
  $$\begin{aligned} \bigl |\nabla u(z_2)-\nabla u(z_1)\bigr | \le K'\Bigl (\omega (r )+\frac{1}{N}\Bigr ) |y_1-y_2| . \end{aligned}$$

Proof of Lemma 3.6

Since $u$ is semiconcave, there exists a universal constant $C$ such that $v:=u-C|x|^2$ is concave. Since

$$\begin{aligned}&\Bigl | \bigl (u(z_2)-u(z_1)-\langle \nabla u(z_1),z_2-z_1\rangle \bigr ) - \bigl (v(z_2)-v(z_1) -\langle \nabla v(z_1),z_2-z_1\rangle \bigr ) \Bigr | \\&\quad \le C|z_1-z_2|^2 \le C \frac{|y_1 - y_2|^2}{N^2}, \end{aligned}$$

it suffices to prove the result (i) with $v$ in place of $u$.

By concavity of $v$, since $z_2-z_1$ is parallel to $y_2-y_1$, and $|z_1-z_2|\ge \frac{|y_1-y_2|}{10N}$, we get

$$\begin{aligned} 0&\ge v(z_2)-v(z_1)-\langle \nabla v(z_1),z_2-z_1\rangle \\&\ge \langle \nabla v(z_2) - \nabla v(z_1),z_2-z_1\rangle \\&\ge \langle \nabla v(y_2) - \nabla v(y_1),z_2-z_1\rangle \\&\ge \frac{1}{10N} \langle \nabla v(y_2) - \nabla v(y_1),y_2-y_1\rangle \\&\ge -\, C \frac{|y_1 - y_2|^2}{N}, \end{aligned}$$

where for the last estimate we used that $u$ (and hence $v$) is $C^{1,1}$ with a universal bound on the Aubry set. This proves (i).

For (ii), we recall that $\int _{z}^{z'}d\mu$ denotes the integral of a measure $\mu$ over the segment joining $z$ to $z'$. Hence, using the same notation as before, we apply Lemma 3.5(i) and use that $|L_t|$ is universally bounded (because of the $C^{1,1}$ regularity of $u$ on the Aubry set) to get, for a.e. $t \in [0,T]$,

$$\begin{aligned} \bigl |\nabla u(z_2)-\nabla u(z_1)\bigr |&\le \int \limits _{z_1}^{z_2} d|Dw_t |\\&\le \int \limits _{z_1}^{z_2} d|Dw_t -L_t\cdot e_2|+ |L_t|\,|z_1-z_2|\\&\le \int \limits _{y_1}^{y_2} d|Dw_t -L_t\cdot e_2|+ C\,|z_1-z_2|\\&\le \omega ( r )\,|y_1-y_2|+ C\,\frac{|y_1-y_2|}{N}. \end{aligned}$$

By approximation, this estimate extends to every $t \in [0,T]$. $\square$

3.3.2 The connection

Given $y_1, y_2 \in \mathbb {R}$, we set

$$\begin{aligned} I^{1/3} \bigl (y_1;y_2\bigr ) := \Bigl \{ y \in \mathbb {R}\, \vert \, \text{ dist } \bigl ( y,[y_1,y_2]\bigr ) < \bigl | y_1-y_2\bigr | /3\Bigr \}. \end{aligned}$$

Lemma 5.2 in [22] (see also [3], Remarque 6.3.3]) applied with $n=1$ yields the following result:

Lemma 3.7

Let $\hat{r}>0$ and $Y$ be a finite set in $\mathbb {R}$ such that $B(0,\hat{r}/12)\cap Y$ contains at least two points. Then there are $y_1\ne y_2 \in Y$ such that the interval $I^{1/3} (y_1;y_2)$ is included in $B(0,\hat{r})$ and does not intersect $Y\setminus \{y_1,y_2\}$.

Given $\hat{r} \in (0,\rho )$ small enough ($\hat{r}$ much smaller than $\rho$ and $\epsilon$), let $T_{\hat{r}}\gg T$ be the first time such that $\bar{\gamma }(T_{\hat{r}}) \in \Pi ^T_{\hat{r}/12}$, and define the set

$$\begin{aligned} W := \Bigl \{ w_0:= \bar{x},w_1:= \bar{\gamma }(t_1), \ldots , w_J := \bar{\gamma }(T_{\hat{r}})\Bigr \}\subset \mathcal {A} \end{aligned}$$

(3.21)

obtained by intersecting the curve

$$\begin{aligned}{}[T,T_{\hat{r}}]\ni t \mapsto \bar{\gamma }(t) \end{aligned}$$

with $\Pi ^T_{\hat{r}}$. We apply Lemma 3.7 with $Y=W\subset \Pi ^T_{\hat{r}}$ to find two points

$$\begin{aligned} \hat{y}_1 = w_j \quad \text{ and } \quad \hat{y}_2 = w_l \quad \text{ with } j>l \end{aligned}$$

(3.22)

which satisfy the properties described in the statement of the lemma. Set

$$\begin{aligned} N := \left\lfloor \frac{1}{\epsilon }\right\rfloor +1, \quad \eta := 2N+1, \end{aligned}$$

and consider a sequence of points $\hat{z}_1,\ldots , \hat{z}_{\eta }$ in the segment $[\hat{y}_1,\hat{y}_2] \subset \Pi _\rho ^T$ which satisfy⁵

$$\begin{aligned}&\hat{z}_1:=\hat{y}_1, \quad \hat{z}_\eta :=\hat{y}_2,\nonumber \\&u \text { is differentiable at }\hat{z}_i\quad \forall \, i=1, \ldots , \eta , \end{aligned}$$

(3.23)

and

$$\begin{aligned}&\hat{z}_i \in \Bigl [ \hat{y}_1 + \frac{i-4/3}{2N} \bigl ( \hat{y}_2-\hat{y}_1\bigr ), \hat{y}_1 + \frac{i-2/3}{2N} \bigl (\hat{y}_2-\hat{y}_1\bigr )\Bigr ]\nonumber \\&\quad \forall \, i=2, \ldots , \eta -1. \end{aligned}$$

(3.24)

Notice that

• $(\pi _6)$ $\frac{\left| \hat{y}_2 -\hat{y}_1\right| }{6N}\le \bigl | \hat{z}_{i+1}-\hat{z}_i \bigr | \le \frac{5\left| \hat{y}_2 -\hat{y}_1\right| }{6N}\quad$ $\forall \, i=1,\ldots ,\eta -1$.

We now fix $T$ to be an arbitrary time in $[\eta +1,\eta +2]$ chosen so that (3.23) and (3.24) hold. Applying Lemma 3.4 [recall also (3.9)], for any $i=1,\ldots ,\eta -1$ we can find a time $t_i \in [i-1-1/8,i-1+1/8]$ such that

$$\begin{aligned} \bar{\gamma }(t_i) \text { satisfies (3.8) }\quad \forall \,i=1, \ldots , \eta -1. \end{aligned}$$

(3.25)

Given $z \in [\hat{y}_1,\hat{y}_2]$ such that $u$ is differentiable at $z$, we can consider the calibrated curve $\gamma _z:(-\infty ,0] \rightarrow M$ as in (2.6). Notice that, since $\hat{y}_1$ and $\hat{y}_2$ belong to the projected Aubry set, $u$ is differentiable at $\hat{y}_1$ and $\hat{y}_2$ and those curves are transverse to $\Pi _\rho ^T$ [remember (3.10)–(3.11)], the curves $\gamma _{\hat{y}_1}$ and $\gamma _{\hat{y}_2}$ are unique and disjoint and moreover the curves $\gamma _z$ cannot intersect $\gamma _{\hat{y}_1}$ and $\gamma _{\hat{y}_2}$ (see Propositions 2.4 and 2.5). Hence, provided $\hat{r}$ is sufficiently small, for any $z \in [\hat{y}_1,\hat{y}_2]$ where $u$ is differentiable there exist $T_z \in [T-1,T+1]$ such that

$$\begin{aligned} \gamma _{z}(-T_z) \in \Pi _{\rho }^0, \quad \gamma _{z}([-T_z,0]) \subset [-1,T]\times [-\rho ,\rho ]. \end{aligned}$$

Recalling (3.23), we now define the following points for all $i=1, \ldots , \eta -1$ (see Fig. 1), where $\bar{\tau }\in (0,1/10)$ is the same as in Proposition 2.7:

$$\begin{aligned} z_{i,-}:=\gamma _{\hat{z}_i}([-T_{\hat{z}_i},0])\cap \Pi _\rho ^{t_i},\quad z_{i,+}:=\gamma _{\hat{z}_i}([-T_{\hat{z}_i},0])\cap \Pi _\rho ^{t_i+\bar{\tau }}, \end{aligned}$$

$$\begin{aligned} z_{i,-}':=\gamma _{\hat{z}_{i+1}}([-T_{\hat{z}_{i+1}},0])\cap \Pi _\rho ^{t_i},\quad z_{i,+}':=\gamma _{\hat{z}_{i+1}}([-T_{\hat{z}_{i+1}},0])\cap \Pi _\rho ^{t_i+\bar{\tau }}. \end{aligned}$$

Fig. 1

The points $z_{i,-}$ and $z_{i,+}$ (resp. $z_{i,-}'$ and $z_{i,+}'$) are obtained from $z_i$ (resp. $z_{i+1}$) by intersecting the trajectory passing through $\hat{z}_i$ (resp. $\hat{z}_{i+1}$) with the segments $\Pi _\rho ^{t_i}$ and $\Pi _\rho ^{t_i+\bar{\tau }}$. Analogously, the points $y_{1}^t$ (resp. $y_2^t$) defined in (3.26) are obtained by intersecting the trajectory passing through $\hat{y}_1$ (resp. $\hat{y}_2$) with the segment $\Pi _\rho ^{t}$. Our goal is to connect $z_{i,-}$ to $z_{i,+}'$ with a control on the action, in order to obtain a closed curve which satisfies (P2)

Also, we set

$$\begin{aligned} y_{1}^t:=\gamma _{\hat{y}_1}([-T_{y_1},0])\cap \Pi _\rho ^{t},\quad y_{2}^t:=\gamma _{\hat{y}_2}([-T_{y_2},0])\cap \Pi _\rho ^{t}. \end{aligned}$$

(3.26)

By Lemma 3.5(ii), provided $\hat{r}$ is sufficiently small, ($\pi _6$) yields

• $(\pi _7)$ $\frac{\left| y_{1}^{t_i+\bar{\tau }}-y_{2}^{t_i+\bar{\tau }}\right| }{7N}\le |z_{i,+}-z_{i,+}'| \le \frac{\left| y_{1}^{t_i+\bar{\tau }}-y_{2}^{t_i+\bar{\tau }}\right| }{N}\le \frac{C}{N}\hat{r}\quad$ $\forall \,i=1,\ldots ,\eta -1$.

Also, using again Lemma 3.5(ii), it follows from the construction of $\hat{y}_1$ and $\hat{y}_2$ (see Lemma 3.7) that, for any $t \in [0,T]$, all points of $\bar{\gamma }([0,T_{\hat{r}}])$ on $\Pi _\rho ^t$ are at distance at least $\frac{|y_{1}^t-y_{2}^t|}{4}$ from $\{y_{1}^t,y_{2}^t\}$ (besides the points $\{y_{1}^t,y_{2}^t\}$ themselves), that is

• $(\pi _8)$ $\mathrm{dist}\Bigl (\Bigl (\bigl (\bar{\gamma }([0,T_{\hat{r}}]) \cap \Pi _\rho ^t\bigr )\setminus \{y_{1}^t,y_{2}^t\}\Bigr ) ,\{y_{1}^t,y_{2}^t\}\Bigr ) \ge \frac{|y_{1}^t-y_{2}^t|}{4}\quad$ for all $t \in [0,T]$.

Since $u$ is differentiable at $z_{i,-},z_{i,-}',z_{i,+},z_{i,+}'$ [see (3.23) and Proposition 2.4], by Lemma 3.6(ii) and ($\pi _7$) it follows that (provided $\hat{r}$ is small enough)

$$\begin{aligned} \bigl |\nabla u(z_{i,+})-\nabla u(z_{i,+}')\bigr | \le \frac{2K'}{N} \, \left| y_{1}^{t_i+\bar{\tau }}- y_{2}^{t_i+\bar{\tau }} \right| . \end{aligned}$$

Hence, since $\mathcal {P}_{0,\bar{\tau }}^*\bigl (z_{i,-},\nabla u(z_{i,-})\bigr ) =\bigl (z_{i,+},\nabla u(z_{i,+})\bigr )$ and thanks to (3.25), if $\hat{r}$ is sufficiently small (so that $z_{i,-}$ is close to $\bar{\gamma }(t_i)$) we can apply Proposition 2.7 with $x^0=z_{i,-}$, $x^f=z_{i,+}'$, $p^0=\nabla u(z_{i,-})$, $p^f=\nabla u(z_{i,+}')$, $r=|y_{1}^{t_i+\bar{\tau }}- y_{2}^{t_i+\bar{\tau }}|$, and $\hat{\epsilon }=2(1+K')/N$ (with $K'$ as in Lemma 3.6), to find a potential $V_i$ which permits to connect $z_{i,-}$ to $z_{i,+}'$ on a time interval $[\hat{t}_i,\hat{t}_i+T_i^f]$. Notice that the constant $\sigma =\sigma _i$ appearing in the value of the action is an arbitrary number less than $\hat{\epsilon }r^2=2 (1+K') \frac{|y_{1}^{t_i+\bar{\tau }}- y_{2}^{t_i+\bar{\tau }}|^2}{N}$.

We now construct a curve $\gamma :[0,T']\rightarrow M$ by concatenating $\gamma _1:[0,T_1]\rightarrow M$ with $\gamma _2:[T_1,T']\rightarrow M$, where

$$\begin{aligned} \gamma _2(t):=\pi ^* \left( \phi _{t-T_1}^{H}\left( \hat{y}_2, \nabla u(\hat{y}_2 )\right) \right) \quad \text {connects}\; z_{\eta -1,+}' \hbox { to } z_{1,-}, \end{aligned}$$

while $\gamma _1$ is obtained as a concatenation of $2\eta -1$ pieces: defining by $V:=\sum _i V_i$ (notice that the support of the $V_i$’s are all disjoint, so the $C^2$ norm of $V$ is bounded by $\max _i \Vert V_i\Vert _{C^2}$), for every $i=1,\ldots ,\eta -1$ we use the flow $(t,z) \mapsto \pi ^* \left( \phi _t^{H+V}(z,\nabla u(z)) \right)$ to connect $z_{i,-}$ to $z_{i,+}'$ on a time interval $[\hat{t}_i,\hat{t}_i+T_i^f]$, while on $[\hat{t}_i+T_i^f,\hat{t}_{i+1}]$ ($i=1,\ldots ,\eta -1$) we just use the original flow $(t,z) \mapsto \pi ^* \left( \phi _t^{H}(z,\nabla u(z)) \right)$ to send $z_{i,+}'$ onto $z_{i+1,-}$. (See [21], Subsection 5.3] for more detail.)

In this way we obtain a closed curve $\gamma :[0,T']\rightarrow M$ (see Fig. 1) whose action is given by the following formula (see [21, Section 5.4]):

$$\begin{aligned}&\int \limits _0^{T'} L_V(\gamma (t),\dot{\gamma }(t)) \,dt \\&\quad = \sum _{i=1}^{\eta -1}\left[ \bigl \langle \nabla {u} (z_{i,+}), {z}_{i,+}' - z_{i,+}\bigr \rangle - \Bigl ( {u}({z}_{i,+}') - {u}(z_{i,+})\Bigr ) \right] + \sigma _i. \end{aligned}$$

Thanks to ($\pi _7$) and Lemma 3.6 we deduce that

$$\begin{aligned} \left| \bigl \langle \nabla {u} (z_{i,+}), {z}_{i,+}' - z_{i,+}\bigr \rangle - \Bigl ( {u}({z}_{i,+}') - {u}(z_{i,+})\Bigr ) \right| \le K' \frac{|y_{1}^{t_i+\bar{\tau }}- y_{2}^{t_i+\bar{\tau }}|^2}{N}. \end{aligned}$$

Hence, since $\sigma _i$ can be any arbitrary number less than $2(1+K') \frac{|y_{1}^{t_i+\bar{\tau }}- y_{2}^{t_i+\bar{\tau }}|^2}{N}$, we can choose

$$\begin{aligned} \sigma _i:= \Bigl ( {u}({z}_{i,+}') - {u}(z_{i,+})\Bigr ) -\bigl \langle \nabla {u} (z_{i,+}), {z}_{i,+}' - z_{i,+}\bigr \rangle \end{aligned}$$

to enforce

$$\begin{aligned} \int \limits _0^{T'} L_V(\gamma (t),\dot{\gamma }(t)) \,dt = 0, \end{aligned}$$

as desired. This concludes the proof of (P2).

3.3.3 A “good” critical subsolution for $H$

To prove (P1), we first need to construct a $C^{1,1}$ critical subsolution $v$ which is “$C^2$ in average”. Recall that, for every $t>0$, the function $h_t:M \times M \rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} h_t(x,y) := \inf \int \limits _0^t L\bigl ( \gamma (s),\dot{\gamma }(s)\bigr ) \, ds, \end{aligned}$$

where the infimum is taken over all Lipschitz curves $\gamma : [0,t] \rightarrow M$ such that $\gamma (0)=x$ and $\gamma (t)=y$.

Lemma 3.8

Let $\Psi$ be as in ($\pi _3$)–($\pi _5$), and let $y_1^t=(t,\ell _1^t),y_2^t=(t,\ell _2^t) \in \fancyscript{O} \cap \Pi _\rho ^t$ be as in (3.26). There exists $s_0>0$ small but universal such that the critical subsolution $v:M\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} v(x):=\mathcal {T}_{s_0}^+u(x)=\sup _{y \in M} \Bigl \{u(y) - h_{s_0}(x,y)\Bigr \} \quad \forall \,x \in M, \end{aligned}$$

is universally $C^{1,1}$ and, for a.e. $t\in [0,T]$, it satisfies ⁶

$$\begin{aligned} \int \limits _{0}^{1} \left| \left\langle \nabla ^2v\bigl (y_1^t+s(y_2^t-y_1^t)\bigr )\cdot e_2,e_2\right\rangle - \langle L_t\cdot e_2,e_2\rangle \right| \,ds \le \omega \bigl (|\ell _1^t|+|\ell _2^t|\bigr ) \end{aligned}$$

for some universal modulus of continuity $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$.

Proof of Lemma 3.8

The fact that $v$ is a critical subsolution is standard, see for instance [8].

By semiconcavity, there is a bounded family of $C^2(\mathcal {U}',\mathbb {R})$ such that

$$\begin{aligned} u= \inf _{f\in \mathcal F} \bigl \{f\bigr \}. \end{aligned}$$

Moreover, thanks to the estimate (3.13) on $D^2u$ provided by the Green bundles, we may assume that

$$\begin{aligned} \nabla ^2f (x) \le L_t+ \omega '(r)\mathrm{Id} \quad \forall \,x \in \Pi _r^t, \,r \in (0,\rho ], \,t \in [0,T], \,f\in \mathcal F. \end{aligned}$$

(3.27)

Then, for every $s_0>0$ we have

$$\begin{aligned} v= \inf _{f\in \mathcal F} \bigl \{T_{s_0}^+f\bigr \}. \end{aligned}$$

By [8] it is known that, for $s_0>0$ small enough, $v \in C^{1,1}$, $v=u$, and $\nabla v=\nabla u$ on the projected Aubry set. Since $\nabla v=\nabla u$ on $\fancyscript{O}$, by (3.14) we get

$$\begin{aligned} \nabla v\left( y_2^t\right)&= \nabla v\left( y_1^t\right) +L_t\cdot \left( y_2^t-y_1^t\right) \\&\quad +\,\omega \bigl (|\ell _1^t|+|\ell _2^t|\bigr )\,\left| y_1^t-y_2^t\right| \quad \forall \,t \in [0,T]. \end{aligned}$$

Since $y_2^t-y_1^t$ is parallel to $e_2$, rewriting the above expression using the fundamental theorem of calculus (recall that $v \in C^{1,1}$) we have

$$\begin{aligned}&\left| \int \limits _{0}^{1}\langle \nabla ^2v\bigl (y_1^t+s(y_2^t-y_1^t)\bigr )\cdot e_2,e_2\rangle \,ds-\langle L_t\cdot e_2,e_2\rangle \right| \nonumber \\&\quad \le \omega \bigl (|\ell _1^t|+|\ell _2^t|\bigr )\quad \text {for a.e. } t \in [0,T], \end{aligned}$$

(3.28)

where $\nabla ^2 v$ is the pointwise Hessian of $v$.

By [8, Lemma 3] there is $s_0>0$ such that, for every $s\in [0,s_0]$, $\mathcal {T}^+_s (\mathcal F)$ is a bounded set in $C^2(M,\mathbb {R})$. Since the Hessian of a $C^2$ function $f$ is transported by the linearized Hamiltonian flow along the calibrating trajectories (see for instance the discussion in [21] after Lemma 5.3), and since all the trajectories are close (as a function of $r$) to the trajectory passing through $te_1$ [see (3.11)], we deduce from (3.27) that, for all $r \in (0,\rho ]$,

$$\begin{aligned} \nabla ^2v \le L_t+\omega '(r)\mathrm{Id} \quad \text {a.e. on } \Pi _r^t, \hbox { for a.e. } t \in [0,T]. \end{aligned}$$

Hence, combining this bound with (3.28) we easily get

$$\begin{aligned}&\int \limits _{0}^{1}\left| \left\langle \left[ \nabla ^2v\left( y_1^t+s\left( y_2^t-y_1^t\right) \right) -L_t\right] \cdot e_2,e_2\right\rangle \right| \,ds\\&\quad \le \omega \left( \left| \ell _1^t\right| +\left| \ell _2^t\right| \right) +\omega '\left( \left| \ell _1^t\right| +\left| \ell _2^t\right| \right) , \end{aligned}$$

as desired. $\square$

Combining Lemmas 3.8 and 2.20, we can also prove that there are many points where $v$ is “$C^{2}$ in average”.

Lemma 3.9

With the same notation as in Lemma 3.8, let $y_1^t,y_2^t \in \fancyscript{O}\cap \Pi ^t_r$, $r:=|\ell _1^t|+|\ell _2^t| \in (0,\rho ]$. Then, for a.e. $t \in [0,T]$ there exists a set $A_t\subset [y_1^t,y_2^t]$ such that

$$\begin{aligned} |A_t|\ge \bigl (1-\sqrt{\omega (r)}\bigr )|y_1^t-y_2^t| \end{aligned}$$

and

$$\begin{aligned} \frac{1}{R}\int \limits _{0}^R \left| \langle \nabla ^2v(z+se_2)\cdot e_2,e_2\rangle -\langle L_t\cdot e_2,e_2\rangle \right| \,ds \le C_1\sqrt{\omega (r)} \end{aligned}$$

for all $z \in A_t$ and $R \in \bigl [-|y_1^t-z|,|y_2^t-z|\bigr ]$.

Proof of Lemma 3.9

We simply apply Lemma 2.20 to the one dimensional function

$$\begin{aligned} f(s):=\left| \langle \nabla ^2v(z+se_2)\cdot e_2,e_2\rangle -\langle L_t\cdot e_2,e_2\rangle \right| \,\chi _{[-|y_1-z|,|y_2-z|]}(s) \end{aligned}$$

with $\delta =C_1\sqrt{\omega (r)}$, and use Lemma 3.8. $\square$

From the results above we see that, provided $r$ is sufficiently small, we can shift the system of coordinates in the variable $t$ by an arbitrary small amount (sat, $t\mapsto t+\tau _0$ with $|\tau _0|$ arbitrarily small) to ensure that the above lemmas all apply with $t=t_i$ for all $i=1,\ldots ,\eta -1$, and then the points $\hat{z}_1,\ldots ,\hat{z}_{\eta }$ can be chosen so such that

• $(\pi _9)$ $z_{i,-},z_{i,-}' \in A_{t_i}$    for all $i=1,\ldots ,\eta -1$.

3.3.4 Construction of a global critical subsolution

As before, we will denote by $\omega :\mathbb {R}^+\rightarrow \mathbb {R}^+$ a modulus of continuity which may change from line to line.

Our goal is to construct a critical subsolution $v_V:M\rightarrow \mathbb {R}$ satisfying (P1). We proceed as follows: first, for any $i=1,\ldots ,t_i$ we define $u_0^i$ and $u_V^i$ as the $C^{1,1}$ solutions of the Dirichlet problems

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} {H} \bigl ( z, \nabla {u}_{0}^i (z)\bigr ) =0 &{}\text {in } [t_i,t_i+3\bar{\tau }]\times \Pi _\rho ^{t_i},\\ {u}_{0}^i = v &{}\text {on } \Pi _\rho ^{t_i}, \end{array} \right. \\&\left\{ \begin{array}{l@{\quad }l} {H}_V \bigl ( z, \nabla {u}_{V}^i (z)\bigr ) =0 &{}\text {in } [t_i,t_i+3\bar{\tau }]\times \Pi _\rho ^{t_i},\\ {u}_{V}^i = v &{}\text {on } \Pi _\rho ^{t_i}, \end{array} \right. \end{aligned}$$

where $v$ is as in Lemma 3.8 [see Lemma 2.6(iv)]. Let $\gamma :[0,T']\rightarrow M$ be the closed trajectory constructed in Sect. 3.3.2, and define $\Gamma _i:=\gamma ([\hat{t}_i+T_i^f,\hat{t}_{i+1}])$ to be the piece of curve which connects $z_{i,+}'$ to $z_{i+1,-}$ (see Fig. 2). In complete analogy with [21, Section 5.5, Property ($\pi 3$)], we have

$$\begin{aligned} u_0^i=u_V^i,\quad \nabla u_0^i=\nabla u_V^i\quad \text {on } \Gamma _i\cap \mathcal C_i, \hbox { where} \quad \mathcal C_i:=\bigcup _{t\in [t_i,t_i+3\bar{\tau }]} [y_{1}^{t},y_{2}^{t}]. \end{aligned}$$

(3.29)

Also, because $y_1^t$ and $y_2^t$ are in the projected Aubry set and $V$ is supported in the interior of $\mathcal C_i$, we get that the values of $u_0^i,u_V^i,v$ are all transported along the curves, and the same happens to their gradients (see [21, Lemma 5.3] and the discussion immediately after it), so in analogy with [21, Section 5.5, Property ($\pi 7$)] we get

$$\begin{aligned}&u_0^i=u_V^i=v,\quad \nabla u_0^i=\nabla u_V^i=\nabla v\quad \text {on } \partial _{lat}\mathcal C_i, \nonumber \\&\quad \hbox { where} \quad \partial _{lat}\mathcal C_i :=\bigcup _{t\in [t_i,t_i+3\bar{\tau }]}\bigl \{y_{1}^{t},y_{2}^{t}\bigr \}. \end{aligned}$$

(3.30)

We claim that

$$\begin{aligned} \left| u_0^i(x)-u_V^i(x)\right| \le \omega (\hat{r}+\epsilon )\,\mathrm{dist}(x,\Gamma _i)^2 \quad \forall \, x \in \mathcal C_i. \end{aligned}$$

(3.31)

Indeed, since by ($\pi _1$), ($\pi _9$), and Lemmas 3.8 and 3.9

$$\begin{aligned} \frac{1}{R}\int \limits _{0}^R \left| \langle \nabla ^2v(z+se_2)\cdot e_2,e_2\rangle -\langle L_{t_i}\cdot e_2,e_2\rangle \right| \,ds \le C_1\sqrt{\omega \left( \left| \ell _1^{t_i}\right| +\left| \ell _2^{t_i}\right| \right) } \end{aligned}$$

for $z=z_{i,-},z_{i,-}'$ and $R \in \bigl [-|y_1^{t_i}-z|,|y_2^{t_i}-z|\bigr ]$, and the flow of the vector field $\frac{\partial H}{\partial p}(x,\nabla v(x))$ is bi-Lipschitz (since $v$ is $C^{1,1}$), we deduce that (recall that the Hessian of a solution is propagated along the linearized flow)

$$\begin{aligned} \frac{1}{R}\int \limits _{0}^R \left| \left\langle \nabla ^2u_0^i\left( z_i^t+se_2\right) \cdot e_2,e_2\right\rangle -\langle L_t\cdot e_2,e_2\rangle \right| \,ds \le C\sqrt{\omega \left( \left| \ell _1^{t_i}\right| +\left| \ell _2^{t_i}\right| \right) } \end{aligned}$$

for all $R \in \bigl [-|y_1^t-z|,|y_2^t-z|\bigr ]$ and $t\in [t_i+2\bar{\tau },t_i+3\bar{\tau }]$, where

$$\begin{aligned} z_i^t:=\gamma _{\hat{z}_{i+1}}([-T_{\hat{z}_{i+1}},0])\cap \Pi _\rho ^{t}. \end{aligned}$$

Also, because $\Vert V\Vert _{C^2}\le \epsilon$, the linearized flows of $H$ and $H_V$ are close in terms of $\epsilon$ (see [21], Proof of Lemma 5.5]), hence

$$\begin{aligned}&\frac{1}{R}\int \limits _{0}^R \left| \left\langle \nabla ^2u_V^i\left( z_i^t+se_2\right) \cdot e_2,e_2\right\rangle -\langle L_t\cdot e_2,e_2\rangle \right| \,ds\\&\quad \le C\sqrt{\omega \left( \left| \ell _1^{t_i}\right| +\left| \ell _2^{t_i}\right| \right) }+\omega (\epsilon ) \end{aligned}$$

for all $R \in \bigl [-|y_1^t-z|,|y_2^t-z|\bigr ]$ and $t\in [t_i+2\bar{\tau },t_i+3\bar{\tau }]$. Since $|\ell _1^{t_i}|+|\ell _2^{t_i}| \le C\hat{r}$ (because $\hat{y}_1,\hat{y}_2\in \Pi _{\hat{r}}^T$ and the flow is Lipschitz on the Aubry set), this estimate combined with (3.29) and a simple Taylor expansion proves (3.31).

We consider now for every $i$ a smooth nonincreasing function $\Theta _i : \mathbb {R}\rightarrow [0,1]$ such that

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \Theta _i(\lambda ) = 1 &{}\quad \text{ if } \lambda \in [t_i,t_i+3\bar{\tau }/2],\\ \Theta _i(\lambda )= 0 &{}\quad \text{ if } \lambda \in [t_i+5\bar{\tau }/2,t_i+3\bar{\tau }], \end{array} \right. \end{aligned}$$

and we set

$$\begin{aligned} \Theta _i (z) :=\Theta _i(z_1) \quad \forall z=(z_1,z_2)\in \mathbb {R}^2. \end{aligned}$$

Then we define $\tilde{u}_i$ as

$$\begin{aligned} \tilde{u}_i (z) := \left\{ \begin{array}{l@{\quad }l} \Theta _i(z) {u}_V^i (z) + \bigl (1-\Theta _i (z)\bigr ) v (z) &{} \text {for } z \in \mathcal {C}_i,\\ v(z) &{} \text {for }z \in \mathcal C_i'\setminus \mathcal {C}_i, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \mathcal C_i':=\bigcup _{t\in [t_i,t_i+3\bar{\tau }]}\left[ y_{1}^{t,'},y_{2}^{t,'}\right] , \quad y_{1}^{t,'}:=y_{1}^t-\frac{y_2^t-y_1^t}{4},\quad y_{2}^{t,'}:=y_{2}^t+\frac{y_2^t-y_1^t}{4} \end{aligned}$$

(see Fig. 2).

Fig. 2

The curve $\Gamma _i$ corresponds to the horizontal segment starting from the point $z_{i,+}'$ and going to the right. The function $\tilde{u}_i$ is obtained by interpolating (using a cut-off function) between $u_V^i$ (the viscosity solution for $H_V$) and $v$ (the critical subsolution for $H$ constructed in Sect. 3.3.3) inside the “cylinder” $\mathcal {C}_i'$. Then, by adding a new potential $V_i'$, small in $C^2$ topology and supported inside $\mathcal {C}_i' \cap \left\{ z=\left( z_1,\hat{z}\right) \, \vert \, z_1 \in \left[ \bar{\tau },3\bar{\tau }\right] \right\}$, we can ensure that ${H}_{V_i+V_i'} \left( z, \nabla \tilde{u}_i(z)\right) \le 0$. Since the cylinders $\mathcal {C}_i'$ are disjoint, we can repeat this construction for $i=1,\ldots ,\eta -1$ to find a global critical subsolution $\tilde{u}$ and a potential $\tilde{V}$ so that (P1) and (P2) hold

Thanks to (3.30), $\tilde{u}_i$ is of class $C^{1,1}$ inside $\mathcal C_i'$. Moreover, for every $z\in \mathcal {C}_i$ we have

$$\begin{aligned} \nabla \tilde{u}_i(z) = \bigl ( {u}_V^i (z)- {v}(z)\bigr ) \nabla \Theta _i (z)+ \Theta _i (z) \nabla {u}_V^i(z) + \bigl ( 1- \Theta _i (z)\bigr ) \nabla {v}(z). \end{aligned}$$

Set

$$\begin{aligned} P_i(z) := \Theta _i(z) \nabla {u}_V^i(z) + \bigl ( 1- \Theta _i (z)\bigr ) \nabla {v}(z) \quad \forall \,z \in \mathcal {C}_i. \end{aligned}$$

By convexity of ${H}$ in the $p$ variable we get

$$\begin{aligned} {H}_V \left( z, P_i(z) \right) \le 0 \quad \forall \,z \in \mathcal {C}_i. \end{aligned}$$

Moreover, since $v$ is a subsolution for $H$,⁷

$$\begin{aligned} {H}_V \left( z, \nabla \tilde{u}_i (z) \right) \le 0 \quad \forall \,z \in \mathcal {C}_i'\setminus \mathcal {C}_i,\quad {u}_0^i (z)- v(z) \ge 0 \quad \forall \, z\, \in \mathcal {C}_i. \end{aligned}$$

Also, since $\nabla \Theta _i$ points in the direction of $-e_1$, by (3.11) we get

$$\begin{aligned} \left\langle \frac{\partial H}{\partial p} \left( z, P_i(z) \right) , \nabla \Theta _i (z)\right\rangle \le -\frac{1}{2} |\nabla \Theta _i (z)| \quad \forall \,z \in \mathcal {C}_i. \end{aligned}$$

Then, using that $u_0^i \ge v$ and Taylor’s formula, we obtain

$$\begin{aligned} {H}_V \left( z, \nabla \tilde{u}_i(z)\right)&\le {H}_V \left( z, P_i(z) \right) + \left( {u}_V^i (z)- {v}(z) \right) \left\langle \frac{\partial H}{\partial p} \left( z, P_i(z) \right) , \nabla \Theta _i (z)\right\rangle \\&\quad + K \,|\nabla \Theta _i (z)|^2 \left| u_V^i(z)-v(z)\right| ^2 \\&\le \left( {u}_V^i (z)- u_0^i(z) \right) \left\langle \frac{\partial H}{\partial p} \left( z, P_i(z) \right) , \nabla \Theta _i (z)\right\rangle \nonumber \\&\quad -\frac{1}{2} \left| {u}_0^i (z)- v(z) \right| \, |\nabla \Theta _i (z)|\\&\quad +2K\,|\nabla \Theta _i (z)|^2 \, \left| u_V^i(z)-u_0^i(z)\right| ^2\\&\quad +2K \,|\nabla \Theta _i (z)|^2 \, \left| u_0^i(z)- v(z)\right| ^2\\&\le C \,|\nabla \Theta _i (z)|\,\left| {u}_V^i (z)- u_0^i(z) \right| , \end{aligned}$$

where we used that $|u_V^i-u_0^i|$ and $|u_0^i- v|$ are small to absorb the quadratic terms into the linear ones. Since the last term in the above equation is of order $\omega (\hat{r}+\epsilon )\,\mathrm{dist}(z,\Gamma _i)^2$ [by (3.31)] and vanishes both outside $\mathcal C_i$ and outside the support of $\Theta$, we deduce that

$$\begin{aligned} {H}_V \left( z, \nabla \tilde{u}_i(z)\right) \le 0 \end{aligned}$$

inside $\mathcal C_i' \cap \bigl ([t_i,t_i+3\bar{\tau }/2]\cup [t_i+5\bar{\tau }/2,t_i+3\bar{\tau }]\bigr )$, and [using ($\pi _8$)]

$$\begin{aligned} {H}_V \left( z, \nabla \tilde{u}_i(z)\right) \le \omega (\hat{r}+\epsilon )\,\min \bigl \{\mathrm{dist}(z,\Gamma _i)^2,\mathrm{dist}(z,\partial _{lat}\mathcal C_i')^2 \bigr \} \end{aligned}$$

inside $\mathcal C_i' \cap [t_i+3\bar{\tau }/2,t_i+5\bar{\tau }/2]$, where

$$\begin{aligned} \partial _{lat}\mathcal C_i':=\bigcup _{t\in [t_i+2\bar{\tau },t_i+3\bar{\tau }]}\bigl \{y_{1}^{t,'},y_{2}^{t,'}\bigr \}. \end{aligned}$$

By choosing $\hat{r}$ and $\epsilon$ sufficiently small, it is easy to see that we can add a potential $V_i' \le 0$, small in $C^2$ topology, which vanishes on $\Gamma$ and supported inside $\mathcal C_i'$, so that

$$\begin{aligned} {H}_{V+V_i'} \left( z, \nabla \tilde{u}_i(z)\right) \le 0 \quad \text {in }\mathcal C_i' \end{aligned}$$

and property (P2) is preserved. Then the function $\tilde{u}$ obtained by gluing together the functions $\tilde{u}_i$ with $v$ is a global critical subsolution for $H_{\tilde{V}}$ with $\tilde{V}:=V+\sum _{i}V_i'$ (notice that the support of the $V_i'$’s are all disjoint, so the $C^2$ norm of $\sum _{i}V_i'$ is bounded by $\max _i \Vert V_i'\Vert _{C^2}$), yielding (P1) and concluding the proof of Proposition 3.3.

4 Examples

Recall that a minimal set of a Lipschitz vector field on a surface is called exceptional if it is neither a fixed point, nor a closed trajectory, nor the whole surface (see [33]). By the Poincaré-Bendixon Theorem, exceptional minimal sets do not exist on the two-dimensional sphere. The purpose of this section is to construct Tonelli Hamiltonians with exceptional minimal Aubry sets on orientable surfaces with positive genus. Such a counter-example in the setting of twist maps was given by Goroff [24].

4.1 Preliminaries on the Mather functions

Let $M$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 2$ and $H:T^*M \rightarrow \mathbb {R}$ a Tonelli Hamiltonian of class $C^2$. Denote by $L:TM \rightarrow \mathbb {R}$ the Tonelli Lagrangian of class $C^2$ associated with $H$ by Legendre–Fenchel duality (see Sect. 2.1). The flow $\phi ^L_t$ of $L$ is conjugated with the Hamiltonian flow through the Legendre transform ${\mathcal L}:T^*M\rightarrow TM$ defined by ${\mathcal L}(x,p) := \left( x, \frac{\partial H}{\partial p}(x,p)\right)$, that is

$$\begin{aligned} \phi ^L_t = {\mathcal L}\circ \phi ^H_t \circ {\mathcal L}^{-1}. \end{aligned}$$

Denote by $\mathcal {M}(L)$ the set of probability measures on $TM$ which are invariant under the Lagrangian flow. Recall that the homology $\rho (\mu )\in H_1(M,\mathbb {R})=H^1(M,\mathbb {R})^*$ of a measure $\mu \in \mathcal {M}(L)$ is determined by

$$\begin{aligned} \langle \rho (\mu ), [\omega ]\rangle = \int \limits _{TM} \omega _x ( v) \,d\mu (x,v), \end{aligned}$$

where $\omega$ is any closed 1-form on $M$ and $[\omega ]\in H^1(M,\mathbb {R})$ is its cohomology class. The action of $\mu$ with respect to $L$ is defined as

$$\begin{aligned} \mathbb {A}_L(\mu ) := \int \limits _{TM} L \; d\mu . \end{aligned}$$

The Mañé critical value of $L$ and $H$ can be recovered as

$$\begin{aligned} \mathbf {c}[H]=\mathbf {c}[L] := - \min \Bigl \{ \mathbb {A}_L (\mu ) \, \vert \, \mu \in \mathcal {M} (L)\Bigr \}. \end{aligned}$$

The Mather $\alpha$ and $\beta$ functions associated with $L$ (or equivalently with $H$),

$$\begin{aligned} \alpha _L \, : \, H^1(M,\mathbb {R}) \longrightarrow \mathbb {R}\quad \beta _L\, :\,H_1(M,\mathbb {R})\longrightarrow \mathbb {R}, \end{aligned}$$

are defined as

$$\begin{aligned} \beta _L(h) := \min \Big \{\mathbb {A}_L(\mu ) \, \vert \, \mu \in {\mathcal M}(L), \, \rho (\mu ) = h \Big \} \quad \forall \,h \in H_1(M,\mathbb {R}) \end{aligned}$$

and

$$\begin{aligned} \alpha _L \bigl ([\omega ]\bigr ) := \mathbf {c}[ L - \omega ] \quad \forall \, [\omega ] \in H^1(M,\mathbb {R}). \end{aligned}$$

They are convex functions with superlinear growth which are conjugate (see [31, Theorem 1]), that is

$$\begin{aligned} \alpha _L(c) = \max \Bigl \{ \langle h,c\rangle - \beta _L(h) \, \vert \, h \in H_1(M,\mathbb {R})\Bigr \} \quad \forall \,c \in H^1(M,\mathbb {R}) \end{aligned}$$

and

$$\begin{aligned} \beta _L(h) = \max \Bigl \{ \langle h,c\rangle - \alpha _L(c) \, \vert \, c \in H^1(M,\mathbb {R})\Bigr \} \quad \forall \,h \in H_1(M,\mathbb {R}). \end{aligned}$$

Let us now introduce some definitions and notation. We call flat of $\beta _L$ any non-trivial maximal convex domain in $H_1(M,\mathbb {R})$ on which $\beta _L$ is an affine function. Moreover we say that a flat is radial if it is contained in a set of the form $\langle h \rangle =\{t h \; | \; t\in \mathbb {R}\}$ with $h\in H_1(M,\mathbb {R})$. By conjugation, any flat $F$ of $\beta _L$ is associated with a non-differentiability point of $\alpha _L$: more precisely, if $c\in H^1(M,\mathbb {R})$ satisfies

$$\begin{aligned} \alpha _L(c) = \langle h,c \rangle - \beta _L(h) \quad \forall \,h\in F, \end{aligned}$$

then all affine functions $c' \mapsto \langle h,c' \rangle - \beta _L(h)$ with slope $h\in F$ are supporting functions for $\alpha$ at $c$. Given $h\in H_1(M,\mathbb {R})$ and $c=[\omega ]\in H^1(M,\mathbb {R})$, let

$$\begin{aligned} {\mathcal M}_h(L)&:= \arg \min \Big \{\, \mathbb {A}_L(\mu )\;\big |\; \mu \in {\mathcal M}(L),\, \rho (\mu ) = h \,\Big \},\\ {\mathcal M}^{c}(L)={\mathcal M}^\omega (L)&:= \arg \min \Big \{ \; \mathbb {A}_{L-\omega }(\mu )\;\big |\; \mu \in {\mathcal M}(L)\;\Big \}. \end{aligned}$$

Note that, by the above properties, for every $c\in H^1(M,\mathbb {R})$ we have

$$\begin{aligned} \rho \bigl ( {\mathcal M}^{c}(L)\bigr ) = \Bigl \{ h \in H_1(M,R) \, \vert \, \alpha _L(c) + \beta _L(h) = \langle h,c\rangle \Bigr \}. \end{aligned}$$

(4.1)

Finally, we recall that a homology class is rational if there is $t\in \mathbb {R}$ such that $th\in H_1(M,\mathbb {Z})$.

4.2 Exceptional minimal hyperbolic Aubry sets on the $2$-torus

Let $M$ be a torus of dimension $2$ and fix $P$ a point in $M$. The open manifold $M \setminus \{P\}$ can be equipped with a hyperbolic metric of curvature $-1$ (we refer the reader to [34] and references therein for further details in hyperbolic geometry). Let us fix a simple close curve $\chi$ with length $\ell >0$ which bounds a small open disc $D$ containing $P$, and another simple closed curve $\chi '$ with length $\ell '\in (0,\ell )$ which is contained in $D$ and which bounds a small open disc $D'$ containing $P$. We can choose $\chi '$ so small that $d(\chi ,\chi ') > \ell$, where $d$ denotes the distance with respect to the hyperbolic metric. We now change the hyperbolic metric on $D'\setminus \{P\}$ into a smooth metric on $D'$ which coincides with the former metric on the boundary of $D'$. In this way we obtain a smooth metric $g$ on $M$. We will be concerned with the geodesic Lagrangian $L : TM \rightarrow \mathbb {R}$ defined as

$$\begin{aligned} L (x,v) := \tfrac{1}{2} \left\| v\right\| _x^2 \quad \forall \,(x,v) \in TM, \end{aligned}$$

(4.2)

and we denote by $H$ the associated Hamiltonian. We notice that, for every $c=[\omega ] \in H^{1}(M,\mathbb {R})$, $\mathbf {c}[L-\omega ] \ge 0$ and $\mathbf {c}[L- \omega ] = 0 \Leftrightarrow [\omega ]=0$. For every $c=[\omega ] \in H^{1}(M,\mathbb {R})$ we denote respectively by $\tilde{\mathcal {A}}(c)$ and $\mathcal {A}(c)$ the Aubry set and projected Aubry set of the Hamiltonian associated with the Lagrangian $L-\omega$, that is, of the Hamiltonian given by $H(x,p)=\frac{1}{2} \Vert p+\omega _x\Vert ^2$, where $\Vert \cdot \Vert$ denotes the cometric on $T^*M$. In the sequel, by abuse of notation, we will look at the Aubry set as a subset of $TM$ via the identification between $TM$ and $T^*M$ given by the Legendre transform.

Lemma 4.1

For every closed form $\omega$ with $c=[\omega ]\ne 0$, we have

$$\begin{aligned} \mathcal {A} (c) \cap D' = \emptyset . \end{aligned}$$

Proof of Lemma 4.1

Let us first show that $\mathcal {A} (c)$ cannot be included in $D$. Argue by contradiction and pick a positively recurrent point $\theta =(x,p)$ of the Aubry set $\tilde{\mathcal {A}}(c)$. Let $\gamma (t) = \pi ^*\left( \phi ^H_t(x,p)\right)$ for $t\in \mathbb {R}$. Then there exists a sequence $t_k\rightarrow +\infty$ such that $\theta = \lim _{k\rightarrow \infty } \phi ^H_{t_k}(\theta )$. Since $\theta$ belongs to the Aubry set, the curve $\gamma$ is calibrated, that is, for every critical solution $u:M\rightarrow \mathbb {R}$ we have

$$\begin{aligned} 0&= \lim _{k \rightarrow \infty } u\bigl (\gamma (t_k)\bigr ) - u(x)\\&= \lim _{k \rightarrow \infty } \int \limits _0^{t_k}\bigl [ L\bigl ( \gamma (t),\dot{\gamma }(t)\bigr ) - \omega _{\gamma (t)} \bigl ( \dot{\gamma }(t)\bigr ) \bigr ] \, dt + \mathbf {c}[L-\omega ] \, t_k. \end{aligned}$$

Let $f :D \rightarrow \mathbb {R}$ be a smooth function such that $df=\omega$ on $D$. Since $\gamma ([0,\infty )) \subset \mathcal {A}(c) \subset D$ by assumption, we have

$$\begin{aligned} \int \limits _0^{t_k}\omega _{\gamma (t)} \bigl ( \dot{\gamma }(t)\bigr ) \, dt = f\bigl ( \gamma (t_k)) - f\bigl ( \gamma (0)). \end{aligned}$$

Hence, since $L \ge 0$, combining the two estimates above and letting $k \rightarrow \infty$ we obtain $\mathbf {c}[L-\omega ]\le 0$. Recalling that $\mathbf {c}[L-\omega ] \ge 0$ and $\mathbf {c}[L- \omega ] = 0 \Leftrightarrow [\omega ]=0$, we infer that $\mathbf {c}[L-\omega ]=0$ which means that $\omega$ is exact, a contradiction.

Assume now that $\mathcal {A}(c)$ intersects both $D'$ and $M\setminus D$. Then there are a calibrated curve $\gamma : \mathbb {R}\rightarrow \mathcal {A}(c)$ and $T_1$ such that $\gamma (0)\in \partial D'$ and $\gamma (T_1)\in \partial D$. The $\alpha$-limit and $\omega$-limit sets of $\gamma$ contain positively recurrent points, so (by the previous argument) they cannot be contained in $D$. Therefore we may assume that $T_1<0$ and that there is $T_2>0$ such that $\gamma (T_2) \in \partial D$ and $\gamma ( (T_1,T_2)) \subset D$. Let $\bar{\chi }: [T_1,T_2] \rightarrow \partial D$ be a smooth constant-speed curve corresponding to piece of the curve $\chi$ joining $\gamma (T_1)$ to $\gamma (T_2)$ with constant speed. Since $d(\chi ,\chi ')>\ell$ and while the length of $\chi$ is $\ell$

$$\begin{aligned} \int \limits _{T_1}^{T_2} \bigl \Vert \dot{\gamma }(t)\bigr \Vert ^2_{\gamma (t)} \, dt > \frac{\ell ^2}{T_2-T_1} \ge \int \limits _{T_1}^{T_2} \bigl \Vert \dot{\bar{\chi }}(t)\bigr \Vert ^2_{\bar{\chi }(t)} \, dt, \end{aligned}$$

which shows that (since both curves are contained in $\bar{D}$ the integral of $\omega$ along them just depends on their end-points)

$$\begin{aligned}&\int \limits _{T_1}^{T_2}\Bigl [ \bigl \Vert \dot{\gamma }(t)\bigr \Vert ^2_{\gamma (t)} - \omega _{\gamma (t)} \bigl ( \dot{\gamma }(t)\bigr ) \Bigr ] dt\\&\quad > \frac{\ell ^2}{T_2-T_1} \ge \int \limits _{T_1}^{T_2} \Bigl [\bigl \Vert \dot{\bar{\chi }}(t)\bigr \Vert ^2_{\bar{\chi }(t)} - \omega _{\bar{\chi }(t)} \bigl ( \dot{\bar{\chi }}(t)\bigr ) \Bigr ]\, dt. \end{aligned}$$

This contradicts the minimality of $\gamma$ [see (2.2) and (2.9)], proving the result. $\square$

Lemma 4.2

The function $\beta _L$ has no flat.

Proof of Lemma 4.2

By homogeneity of $L$, the function $\beta _L$ is quadratic in the radial direction, that is $\beta (th)=t^2\beta (h)$ for any $h\in h_1(M,\mathbb {R})$ and $t\ge 0$. Thus it suffices to show that any flat of $\beta _L$ has to be radial. Argue by contradiction and suppose that there is a flat $F\subset H_1(M,\mathbb {R})$ which is not radial. Let $\rho _1$, $\rho _2$ be two extremal points in $F$ which are linearly independent and let $\mu _i\in {\mathcal M}_{\rho _i}(L)$, $i=1, 2$. Then there is a cohomology class $c=[\omega ]$ such that

$$\begin{aligned} F = \rho ({\mathcal M}^\omega (L)) =\Bigl \{\,h\in H_1(M,\mathbb {R})\;|\; \alpha _L(c) + \beta _L (h) = \langle h, c\rangle \;\Bigr \}. \end{aligned}$$

Since the ergodic components of $\mu _1$ and $\mu _2$ are also in ${\mathcal M}^\omega (L)$, their homologies are also in $F$. Since $\rho _1, \rho _2$ are extremal points of $F$ and $\rho$ is linear, the homologies of the ergodic components of $\mu _1$ and $\mu _2$ are respectively $\rho _1$ and $\rho _2$. In conclusion, we can assume that $\mu _1$, $\mu _2$ are ergodic. We need to show that the projection of the orbits in the support of $\mu _1$ and $\mu _2$ intersect. Since $\mu _1, \mu _2\in M^\omega (L)$ are minimizing measures for the Lagrangian $L-\omega$, the intersection will contradict the Mather’s graph property, proving the result.

In the 2-torus $M$, any two integral homology classes in $H_1(M,\mathbb {Z})$ which are linearly independent intersect. Let $I:H_1(M,\mathbb {Z})\times H_1(M,\mathbb {Z})\rightarrow \mathbb {R}$ be the intersection form which extends by bilinearity to real homologies. Then, if $r_2$ is not a multiple of $r_1$ in $H_1(M,\mathbb {R})$, we have $I[r_1,r_2]\ne 0$.

We denote by $\pi :TM\rightarrow M$ the canonical projection. For each $i=1,2$, let $(x_i,v_i)\in TM$ be a generic point for $\mu _i$, $\Sigma _i$ a small transversal segment to $v_i$ in $M$ containing $x_i$, $T$ a large return time to $\Sigma _i$ of the projected flow of $(x_i,v_i)$ so that $\pi \bigl (\phi ^L_T(x_i,v_i)\bigr )\in \Sigma _i$, and $\Gamma _i(T)$ a small segment in $\Sigma _i$ joining $x_i$ to $x_i(T)=\pi \bigl (\phi ^L_T(x_i,v_i)\bigr )$. For each $i=1,2$, we define $C_i(T)$ to be the closed curve $C_i(T):= \pi \bigl (\phi ^L_{[0,T]}(x_i,v_i)\bigr )*\Gamma _i(T)$ obtained by concatenating $\pi \bigl (\phi ^L_{[0,T]}(x_i,v_i)\bigr )$ with $\Gamma _i(T)$. Note that, without loss of generality, we may assume that $\Sigma _1 \cap \Sigma _2= \emptyset$. Choose now two sequences of return times $\{T^1_k\}_k,\{T_l^2\}_l$ such that $\lim _{k \rightarrow \infty } T^1_k = \lim _{l \rightarrow \infty } T^2_l = +\infty$ and $\lim _{k \rightarrow \infty } \text {diam }\Gamma _i(T^1_k)= \lim _{l\rightarrow \infty } \text {diam }\Gamma _i(T^1_l)= 0$. Since the points $(x_i,v_i)$ are generic points for $\mu _i$, Birkhoff’s Theorem ensures that

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{1}{T^1_k}\left[ C_1\left( T^1_k\right) \right] =\rho _1 \quad \text{ and } \quad \lim _{k\rightarrow \infty } \frac{1}{T^2_l}\left[ C_2\left( T^2_l\right) \right] =\rho _2 \quad \text{ in } H_1(M,\mathbb {R}). \end{aligned}$$

Then by bilinearity of the intersection form, we have

$$\begin{aligned} 0\ne I[\rho _1,\rho _2] = \lim _{k,l\rightarrow \infty }\frac{1}{T^1_k\, T^2_l} I\left[ C_1\left( T^1_k\right) ,C_2\left( T^2_l\right) \right] . \end{aligned}$$

In order to obtain the contradiction we have to show that there is at least one intersection in $I\big [C_1(T^1_k),C_2(T^2_l)\big ]$ which is not due to the small closing segments $\Gamma _1(T^i_k), \Gamma _2(T^i_l)$.

Note that if $\mu _1$ (resp. $\mu _2$) is supported on a periodic orbit then we can take as $T^1_k$ (resp. $T^2_l$) a multiple of the period and there is no joining segment $\Gamma _1(T^1_k)$ (resp. $\Gamma _2(T^2_l)$). This proves that the intersection occurs when both $\mu _1$, $\mu _2$ are supported on periodic orbits, giving the desired contradiction.

For the general case, let $\psi$ be the induced Hamiltonian flow on the projected Aubry set $\mathcal A(c)$ in $M$, that is

$$\begin{aligned} \psi _t(x):=\pi ^*\bigl (\phi _t^H(x,du(x)) \bigr ), \end{aligned}$$

where $u:M\rightarrow \mathbb {R}$ is a critical solution (see Proposition 2.5). We fix $\tau >0$ small enough so that $\psi _{(0,\tau ]}\big ({\mathcal A}(c) \cap \Sigma _1\big )\cap \Sigma _1 = \emptyset$, and define $B_1(T^1_k):= \psi _{[0,\tau ]}(\mathcal A(c)\cap \Gamma _1(T^1_k))$. Let $\chi _{B_1(T^1_k)}$ be the characteristic function of $B_1(T^1_k)$. Since $\chi _{B_1(T^1_k)}\le 1$ and the part of $C_2(T^2_l)$ which may intersect $\Gamma _1(T^1_k)$ is contained in $\mathcal {A}(c)$ (recall that, by construction, $\Sigma _1\cap \Sigma _2 = \emptyset$), we have

$$\begin{aligned} \# \left[ C_2\left( T^2_l\right) \cap \Gamma _1\left( T^1_k\right) \right] \le \frac{1}{\tau }\int \limits _0^{T^2_l} \left( \chi _{B_1\left( T^1_k\right) }\circ \pi \right) \left( \varphi ^L_s(x_2,v_2)\right) \; ds \le \frac{T^2_l}{\tau }\quad \forall \,k,l. \end{aligned}$$

Therefore

$$\begin{aligned} \lim \sup _{l\rightarrow \infty }\frac{1}{T^2_l} \# \left[ C_2\left( T^2_l\right) \cap \Gamma _1\left( T^1_k\right) \right] \le \frac{1}{\tau }\quad \forall \,k, \end{aligned}$$

which implies

$$\begin{aligned} \lim _{k,l \rightarrow \infty } \frac{1}{T^1_k\,T^2_l} \left| I\,\left[ \Gamma _1\left( T^1_k\right) , C_2\left( T^2_l\right) \right] \,\right| \le \lim _{k\rightarrow \infty } \frac{1}{T^1_k}\; \frac{1}{\tau }=0. \end{aligned}$$

Similarly

$$\begin{aligned} \lim _{k,l \rightarrow \infty } \frac{1}{T^1_k\,T^2_l} \left| I\,\left[ \Gamma _2\left( T^2_k\right) , C_1\left( T^1_l\right) \right] \,\right| = 0, \end{aligned}$$

which proves that projection of the orbits in the support of $\mu _1$ and $\mu _2$ intersect, a contradiction. $\square$

Let $h\in H_1(M,\mathbb {R})$ be an irrational homology class, and let $c=[\omega ]\in H^1(M,\mathbb {R})$ be a cohomology class such that $\alpha _L(c)+\beta _L(h) = \langle c,h\rangle$. Since $\beta _L$ has no flat, the set $\rho ({\mathcal M}^\omega (L))$ is a singleton [see (4.1)]. Let $\Lambda$ be a minimal set in $\tilde{\mathcal A}(c)$, and $U\in C^\infty (M,\mathbb {R})$ a $C^2$-small smooth non-negative function on $M$ such that $U^{-1}(\{0\}) =\pi ^*( \Lambda )$. Then the Aubry set for the Lagrangian $L-\omega +U$ is the minimal set $\Lambda$. Moreover $\Lambda$ is not a closed orbit because (the image through the Legendre transform of) any ergodic measure in $\Lambda$ is in $\mathcal M^\omega (L)$, thus has homology $h$, which is irrational.

The Euler–Lagrange flow of $L$ is the geodesic flow of the metric $g$, which is uniformly hyperbolic (outside $TD'$). Then the Lagrangian $L-\omega$ has the same flow as $L$. Moreover since the projected Aubry set does not cross $D'$ (Lemma 4.1) and $U$ is $C^2$-small, the invariant set $\Lambda$ remains hyperbolic with respect to $L-\omega +U$. Hence $\Lambda$ is a non-trivial minimal hyperbolic Aubry set.

4.3 The case of surfaces of higher genus

A similar construction can be made in a surface of higher genus as follows. Let $M_1$ be a $2$-torus, let a hyperbolic metric on $M_1 \setminus \{P_1\}$ with $P_1 \in M_1$, $\chi _1$, $\chi _1'$ two curves surrounding the point $P_1$ as above, and let $M_2 \setminus \{P_2\}$ be a punctured surface of genus $g$, $g\ge 1$ equipped with a hyperbolic metric. Construct two curves $\chi _2$ and $\chi _2'$ bounding the puncture as in the example above, cut $M_2$ through $\chi _2'$ and join it smoothly to $M_1$ along $\chi _1'$. Join also smoothly the Riemannian metrics on $M_1$ and $M_2$ in the tube between $\chi _2'$ and $\chi _1'$. Then define $M := M_1\# M_2$ and consider the smooth geodesic Lagrangian given by (4.2). By construction, $H_1(M,\mathbb {R})=H_1(M_1,\mathbb {R})\oplus H_1(M_2,\mathbb {R})$ and the minimizing measures with homologies in $H_1(M_1,\mathbb {R})$ or in $H_1(M_2,\mathbb {R})$ do not cross $\chi _1'=\chi _2'$. Also the $\beta$ function satisfies $\beta _{L}= \beta _{L_1} \oplus \beta _{L_2}$, where $L_1$ and $L_2$ denote the geodesic Lagrangians obtained as above on $M_1$ and $M_2$.

Take an irrational homology class $h\in H_1(M,\mathbb {R})$ and let $c=[\omega ]\in H^1(M,\mathbb {R})$ be a cohomology class such that

$$\begin{aligned} \beta _L(h\oplus 0)+\alpha _L(c) = \langle c, (h\oplus 0)\rangle , \end{aligned}$$

where $(h\oplus 0)\in H_1(M,\mathbb {R})=H_1(M_1,\mathbb {R})\oplus H_1(M_2,\mathbb {R})$. Let $\Lambda \subset TM$ be the minimal set obtained in Sect. 4.2, and let $U\in C^\infty (M,\mathbb {R})$ be a $C^2$-small smooth non-negative function on $M$ such that $U^{-1}(\{0\}) = \Lambda$. Then $\Lambda$ is a non-trivial minimal hyperbolic Aubry set for the Lagrangian $L-\omega +U$ on $TM$.