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

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.