Tonelli Hamiltonians without conjugate points and $C^0$ integrability

Abstract

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\mathbb {T}^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. $T^*\mathbb {T}^n$ is $C^0$ foliated by a family $\mathcal {F}$ of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove that there exists a $G_\delta$ subset $\mathcal {G}$ of $\mathcal {F}$ such that the dynamics restricted to every element of $\mathcal {G}$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.

Appendix

1.1 Aubry sets

If $\lambda$ is a $C^\infty$ closed 1-form of $\mathbb {T}^n$, then the map $T_\lambda \,: T^*\mathbb {T}^n\rightarrow T^*\mathbb {T}^n$ defined by: $T_\lambda (q,p)=(q, p+\lambda (q))$ is a symplectic $C^\infty$ diffeomorphism; therefore, we have: $(\phi ^{H\circ T_\lambda }_t)=(T_\lambda ^{-1}\circ \phi _t\circ T_\lambda )$, i.e. the Hamiltonian flow of $H$ and $H\circ T_\lambda$ are conjugated. Moreover, the Tonelli Hamiltonian function $H\circ T_\lambda$ is associated to the Tonelli Lagrangian function $L-\lambda$, and it is well-known that : $(\varphi _t^L)=(\varphi _t^{L-\lambda })$; the two Euler–Lagrange flows are equal. Let us emphasize that these flows are equal, but the Lagrangian functions, and then the Lagrangian actions differ.

For a Tonelli Lagrangian function ($L$ or $L-\lambda$), Mather introduced in [25] (see [23] too) a particular subset $\mathcal {A}(L-\lambda )$ of $T\mathbb {T}^n$ which he called the “static set” and which is now usually called the “Aubry set”. There exist different but equivalent definitions of this set (see [15, 19, 23]) and it is known that two closed 1-forms that are in the same cohomological class define the same Aubry set:

$$\begin{aligned}{}[\lambda _1]=[\lambda _2]\in H^1(\mathbb {T}^n,\mathbb {R})\Rightarrow \mathcal {A}(L-\lambda _1)=\mathcal {A}(L-\lambda _2). \end{aligned}$$

We can then introduce the following notation: if $c\in H^1(\mathbb {T}^n,\mathbb {R})$ is a cohomological class, we have $\mathcal {A}_c=\mathcal {A}_c(L)=\mathcal {A}(L-\lambda )$ where $\lambda$ is any closed 1-form belonging to $c$. Then $\mathcal {A}_c$ is compact, non empty and invariant under $(\varphi _t^L)$. Moreover, J. Mather proved in [25] that it is a Lipschitz graph above a part of the zero-section (see [19] too).

As we are interested in the Hamiltonian dynamics as well as in the Lagrangian ones, let us define the dual Aubry set:

• if $H$ is the Hamiltonian function associated to the Tonelli Lagrangian function $L$, its dual Aubry set is $\mathcal {A}^*(H)=\mathcal {L}\big (\mathcal {A}(L)\big )$;

• if $c\in H^1(\mathbb {T}^n,\mathbb {R})$ is a cohomological class, then $\mathcal {A}^*_c=\mathcal {A}^*_c(H)=\mathcal {L}\big (\mathcal {A}_c(L)\big )$ is the $c$ -dual Aubry set; let us notice that for any closed 1-form $\lambda$ belonging to $c$, we have: $T_\lambda ( \mathcal {A}^*\big (H\circ T_\lambda )\big )=\mathcal {A}_c^*(H)$.

These sets are invariant by the Hamiltonian flow $(\phi _t^H)$.

Then there exists a real number denoted by $\alpha _H (c)$ such that : $\mathcal {A}^*_c\subset H^{-1} \big (\alpha _H (c)\big )$ (see [13, 24]), i.e. each dual Aubry set is contained in an energy level.

The following property is a well-known characterization of the projected Aubry set: $x_0\in \mathbb {T}^n$ is such that there exists a sequence of absolutely continuous curves $\gamma _k:[0, T_k]\rightarrow \mathbb {T}^n$, with $(T_k) \rightarrow \infty$, such that $\gamma _k(0)=\gamma _k(T_k)=x_0$ and

$$\begin{aligned} \displaystyle {\lim _{k\rightarrow +\infty }\int _0^{T_k}\big (L(\gamma _k, \gamma '_k)-\lambda (\gamma '_k)+\alpha _H(c)\big )=0}, \end{aligned}$$

if and only if $x_0\in \pi (\mathcal {A}_c)$.

The following proposition is proved in [5]:

Proposition

Let $c\in H^1(\mathbb {T}^n, \mathbb {R})$ and $\lambda \in c$, $\varepsilon >0$ and let $L: T\mathbb {T}^n\rightarrow \mathbb {R}$ be a Tonelli Lagrangian function. Then there exists $T_0>0$ such that:

$\forall T\geqslant T_0, \forall (x_0,v_0)\in \mathcal {A}_c, \forall \gamma : [0, T]\rightarrow \mathbb {T}^n$ minimizing for $L-\lambda$ between $x_0$ and $x_0$, i.e.:

$$\begin{aligned}&\forall \eta : [0, T]\rightarrow \mathbb {T}^n, \\&\eta (0)\!=\!\eta (T)\!=\!x_0 \Rightarrow \int _0^T\big (L(\gamma , \gamma ')-\lambda (\gamma ')+\alpha _H(c)\big )\leqslant \int _0^T\big (L(\eta , \eta ')-\lambda (\eta ')+\alpha _H(c)\big ) \end{aligned}$$

then we have: $d\big ((x_0, v_0), (x_0, \gamma '(0))\big )\leqslant \varepsilon$

1.2 Mather sets

The general references for this section are [24] and [26]. Let ${\mathcal {M}} (L)$ be the space of compactly supported Borel probability measures that are invariant by the Euler–Lagrange flow $(\varphi _t^L)$. To every $\mu \in {\mathcal {M}} (L)$ we associate its average action $A_L(\mu )=\int _{T\mathbb {T}^n}Ld\mu$. It is proved in [24] that for every $f\in C^1(\mathbb {T}^n, \mathbb {R})$, we have:

$$\begin{aligned} \int df(q).v d\mu (q,v)=0. \end{aligned}$$

Therefore we can define on $H^1(\mathbb {T}^n, \mathbb {R})$ a linear functional $\ell (\mu )$ by:

$$\begin{aligned} \ell (\mu )([\lambda ])=\int \lambda (q)\cdot vd\mu (q,v) \end{aligned}$$

(here $\lambda$ designates any closed 1-form). Then there exists a unique element $\rho (\mu )\in H_1(\mathbb {T}^n, \mathbb {R})$ such that:

$$\begin{aligned} \forall \lambda ,\quad \int _{T\mathbb {T}^n}\lambda (q)\cdot vd\mu (q,v)=[\lambda ]\cdot \rho (\mu ). \end{aligned}$$

The homology class $\rho (\mu )$ is called the rotation vector of $\mu$. Then the map $\mu \in {\mathcal {M}} (L)\rightarrow \rho (\mu )\in H_1(\mathbb {T}^n, \mathbb {R})$ is onto. Mather’s $\beta$-function $\beta : H_1(\mathbb {T}^n, \mathbb {R})\rightarrow \mathbb {R}$ associates to each homology class $h\in H_1(\mathbb {T}^n, \mathbb {R})$ the minimal value of the average action $A_L$ over the set of measures of ${\mathcal {M}}(L)$ with rotation vector $h$. We have:

$$\begin{aligned} \beta (h)=\min _{\begin{array}{c} \mu \in {\mathcal {M}} (L)\\ \rho (\mu )=h \end{array}}A_L(\mu ). \end{aligned}$$

A measure $\mu \in {\mathcal {M}} (L)$ realizing such a minimum, i.e. such that $A_L(\mu )=\beta \big (\rho (\mu )\big )$ is called a minimizing measure with rotation vector $\rho (\mu )$. The $\beta$ function is convex and superlinear, and its conjugate function (given by Fenchel duality) $\alpha : H^1(\mathbb {T}^n, \mathbb {R})\rightarrow \mathbb {R}$ is defined by:

$$\begin{aligned} \alpha ([\lambda ]) =\max _{h\in H_1(\mathbb {T}^n, \mathbb {R})}\big ([\lambda ]\cdot h-\beta (h)\big )=-\min _{\mu \in {\mathcal {M}}(L)}A_{L-\lambda }(\mu ). \end{aligned}$$

A measure $\mu \in {\mathcal {M}} (L)$ realizing the minimum of $A_{L-\lambda }$ is called a $[\lambda ]$ -minimizing measure. Observe that the function $\alpha$ is exactly the same as the function $\alpha _H$ defined in the section on Aubry sets. It is convex and superlinear.

Being convex, Mather’s $\beta$ function has a subderivative at any point $h\in H_1(\mathbb {T}^n, \mathbb {R})$; i.e. there exists $c\in H^1(\mathbb {T}^n, \mathbb {R})$ such that:

$$\begin{aligned} \forall k\in H_1(\mathbb {T}^n, \mathbb {R}),\quad \beta (h)+c\cdot (k-h)\leqslant \beta (k). \end{aligned}$$

We denote by $\partial \beta (h)$ the set of all the subderivatives of $\beta$ at $h$. By Fenchel duality, we have: $c\in \partial \beta (h)\Leftrightarrow c\cdot h=\alpha (c)+\beta (h)$.

Then we introduce the following notations:

• if $h\in H_1(\mathbb {T}^n, \mathbb {R})$, the Mather set for the rotation vector $h$ is:

  $$\begin{aligned} \displaystyle {\mathcal {M}^h(L)=\bigcup \{\mathrm{supp}(\mu );\quad \mu \,\hbox {is minimizing with rotation vector}\,h\}}; \end{aligned}$$

• if $c\in H^1(\mathbb {T}^n , \mathbb {R})$, the Mather set for the cohomology class $c$ is:

  $$\begin{aligned} \displaystyle {\mathcal {M}_c(L)=\bigcup \{ \mathrm{supp}(\mu );\quad \mu \,\hbox {is}\,c\hbox {-minimizing}\}}; \end{aligned}$$

where $\mathrm{supp}(\mu )$ designates the support of the measure $\mu$.

The sets $\mathcal {M}^h(L)$ and $\mathcal {M}_c(L)$ are invariant by $\varphi _t^L$.

The following equivalences are proved in [26] for any pair $(h, c)\in H_1(M, \mathbb {R})\times H^1(M, \mathbb {R})$:

$$\begin{aligned} \mathcal {M}^h(L)\cap \mathcal {M}_c(L)\not =\varnothing \quad \Longleftrightarrow \quad \mathcal {M}^h(L)\subset \mathcal {M}_c(L)\quad \Longleftrightarrow \quad c\in \partial \beta (h). \end{aligned}$$

The dual Mather set for the cohomology class $c$ is defined by: $\mathcal {M}_c^*(H)=\mathcal {L}\big (\mathcal {M}_c(L)\big )$. If ${\mathcal {M}}^*(H)$ designates the set of compactly supported Borel probability measures of $T^*M$ that are invariant by the Hamiltonian flow $(\phi ^H_t)$, then the map $\mathcal {L}_*: {\mathcal {M}}(L)\rightarrow {\mathcal {M}}^*(H)$ that pushes forward the measures by $\mathcal {L}$ is a bijection. We denote $\mathcal {L}_*(\mu )$ by $\mu ^*$ and say that the measures are dual. We say too that $\mu ^*$ is minimizing if $\mu$ is minimizing in the previous sense.

Moreover, the Mather set $\mathcal {M}_c^*(H)$ is a subset of the Aubry set $\mathcal {A}_c^*(H)$ and every invariant Borel probability measure the support of whose is in $\mathcal {A}_c^*(H)$ is $c$-minimizing.

1.3 Mañé sets

The Mañé set $\mathcal {N}(L)$ of $L$ is the set of $\big (\gamma (0), \gamma '(0)\big )\in T\mathbb {T}^n$ such that for all segment $[a, b]\subset \mathbb {R}$, $\gamma _{|[a, b]}$ is a minimizer for $L$. The dual Mañé set is then $\mathcal {N}^*(H)=\mathcal {L}\big (\mathcal {N}(L)\big )$.

For all $c\in H^1(\mathbb {T}^n, \mathbb {R})$ and $\lambda \in c$, then $\mathcal {N}_c=\mathcal {N}_c(L)=\mathcal {N}(L-\lambda )$ is independent of the choice of $\lambda \in c$ and the $c$ -dual Mañé set is $\mathcal {N}_c^*(H)=\mathcal {L}\big (\mathcal {N}_c(L)\big )=T_\lambda \big (\mathcal {N}^*(H\circ T_\lambda )\big )$. It is invariant under $(\phi _t^H)$, compact and non empty but is not necessarily a graph.

For every cohomological class $c\in H^1(\mathbb {T}^n)$, we have the inclusion : $\mathcal {M}^*_c(H)\subset \mathcal {A}^*_c(H)\subset \mathcal {N}^*_c(H) \subset H^{-1} \big (\alpha _H (c)\big )$ (see [13, 24]), i.e. each dual Mañé set is contained in an energy level.

Moreover, the $\omega$ and $\alpha$-limit sets of every point of the Mañé set $\mathcal {N}^*_c(H)$ are contained in the Aubry set $\mathcal {A}^*_c(H)$.

1.4 The link with the weak KAM theory

The reference for this section is [19]. We just recall some results that are used in the article; a $C^0$ Lagrangian graph is the graph of $a+du:\mathbb {T}^n\rightarrow (\mathbb {R}^n)^*$ where $a\in (\mathbb {R}^n)^*$ and $u\in C^1(\mathbb {T}^n, \mathbb {R})$. Then $a\in H^1(\mathbb {T}^n, \mathbb {R})$ is the cohomology class of the graph. We have:

if $\mathcal {G}$ is a Lagrangian graph with cohomology class $c$ that is invariant by $\Phi _t$, then

$$\begin{aligned} \mathcal {A}^*_c\subset \mathcal {G}\subset \mathcal {N}^*_c. \end{aligned}$$

Moreover, if $\mathcal {A}^*_c$ (resp. $\mathcal {N}^*_c$) is a graph above the whole $\mathbb {T}^n$, then we have $\mathcal {A}^*_c=\mathcal {N}^*_c$ and it is a $C^0$ Lagrangian graph.

It is proved in [18] that every $C^0$ Lagrangian graph that is invariant by a Tonelli Hamiltonian is a Lipschitz graph.

1.5 Green bundles

Recall (see [2, 14] for details) that if $s \in \mathbb {R}\longmapsto (x,p) = \phi _s^H(x_0,p_0) \in T^*\mathbb {T}^n$ is an orbit of the Hamiltonian flow that is free of conjugate points, one may define two bundles $G_-$ and $G_+$ (called the Green bundles) by

$$\begin{aligned} G_+(x,p) = \lim _{t \longrightarrow +\infty } D \phi _t^H \big ( \phi _{-t}^H(x,p) \big ) \cdot V^*\big ( \phi ^H_{-t}(x,p) \big ) \end{aligned}$$

and

$$\begin{aligned} G_-(x,p) = \lim _{t \longrightarrow +\infty } D \phi _{-t}^H \big ( \phi _{t}^H(x,p) \big ) \cdot V^*\big ( \phi ^H_{t}(x,p) \big ). \end{aligned}$$

Then $G_-$ is the negative Green bundle and $G_+$ is the positive one.

Every $G_\pm (x,p)$ is a Lagrangian subspace of $T_{(x,p)}T^*\mathbb {T}^n$ that is transverse to the vertical space $V^*(x,p)$, and this bundle is invariant by the Hamiltonian flow: $D \phi _t^H G_\pm (x,p) = G_\pm \big ( \phi ^H_t(x,p) \big )$ for all $t \in \mathbb {R}$.

We have of the following criteria (see [2, 14]): if $w \in T_{(x,p)}(T^*\mathbb {T}^n)$, then

$$\begin{aligned} w \notin G_+(x,p)&\Longrightarrow&\lim _{t \rightarrow + \infty } \vert \vert D \left( \pi \circ \phi ^H_{-t}\right) (x,p) \cdot w \vert \vert = + \infty ,\\ w \notin G_-(x,p)&\Longrightarrow&\lim _{t \rightarrow + \infty } \vert \vert D \left( \pi \circ \phi ^H_{t}\right) (x,p) \cdot w \vert \vert = + \infty , \end{aligned}$$

where $\vert \vert \cdot \vert \vert$ denotes the Euclidean norm.

Moreover, $G_+$ is upper semi-continuous and $G_-$ is lower semi-continuous, and we have at every point: $G_-\leqslant G_+$ (for the usual order relation on the Lagrangian subspaces that are transverse to the vertical, given by the order on symmetric matrices, see [2] for details). Hence $\{ G_-=G_+\}$ is a $G_\delta$ subset of $T^*\mathbb {T}^n$.

It is proved in [2] that if $G$ is any invariant Lagrangian subspace that is transverse to the vertical space (for example the tangent to some invariant Lipschitz Lagrangian graph), then we have: $G_-\leqslant G\leqslant G_+$.

There is a strong link between Oseledet’s bundle and Green bundles, as explained in [6]:

Theorem

Let $H~: T^*\mathbb {T}^n\rightarrow \mathbb {R}$ be a Tonelli Hamiltonian and let $\mu$ be an ergodic minimizing probability measure. Then the two following assertions are equivalent:

• at $\mu$ almost every point, $\dim \big (G_-(x)\cap G_+(x)\big )=p$;

• $\mu$ has exactly $2p$ zero Lyapunov exponents, $n-p$ positive ones and $n-p$ negative ones.

Moreover, if the Oseledet’s splitting along the support of $\mu$ is denoted by $E^s\oplus E^c\oplus E^u$, then we have: $G_-=E^s\oplus (G_-\cap G_+)$ and $G_+=E^u\oplus (G_-\cap G_+)$.

1.6 Generalized tangent vectors and Green bundles

There exist different notions of tangent vectors for a subset of a manifold that is not necessarily a submanifold. A geometric one is due to Bouligand [10]. The Bouligand contingent cone to a set $A\subset T^*M$ at $a\in A$ is defined in a chart as being the set of the limits

$$\begin{aligned} v=\lim _{k\rightarrow +\infty } \frac{1}{t_k}(x_k-a) \end{aligned}$$

where $(x_k)$ is a sequence of points of $A$ that converges to $a$ and $t_k$ a sequence of real numbers. The contingent cone to $A$ at $a$ is denoted by $T^G_aA$ and one of its elements is a generalized tangent vector.

If $\Gamma \subset T^*M$ is the Lipschitz graph of $\lambda$, we have at every point of differentiability $x$ of $\lambda$

$$\begin{aligned} T_{(x, \lambda (x))}\Gamma =\{ (v, D\lambda (x)v); v\in T_xM\} = T^G_{(x, \lambda (x))}\Gamma ; \end{aligned}$$

hence this notion of generalized tangent vector extends the notion of tangent vector.

Let us recall two results that are proved in [2] (propositions 4.6 and 4.3).

Proposition

Let $\Gamma$ be the Lipschitz Lagrangian graph of $\lambda$. Let $x\in M$ be a point such that $T^G_{(x, \lambda (x))}\Gamma$ is a subspace that has the same dimension as $M$. Then $\lambda$ differentiable at $x$ and $T^G_{(x, \lambda (x))}\Gamma =T_{(x, \lambda (x))}\Gamma$.

Proposition

Let $\Gamma$ be the Lipschitz Lagrangian graph of $\lambda$ that is invariant by the Hamiltonian flow of the Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb {R}$. Let $x$ be a point of differentiability of $\lambda$. Then we have:

$$\begin{aligned} G_-(x, \lambda (x))\le T_{(x, \lambda (x))}\Gamma \le G_+(x, \lambda (x)). \end{aligned}$$