  We prove that all the Tonelli Hamiltonians defined on the cotangent bundle
$T^*\T^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$
integrable, i.e. $T^*\T^n$ is $C^0$ foliated by a family $\Fc$ of invariant
$C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove
that there exists a $G_\delta$ subset $\Gc$ of $\Fc$ such that the dynamics
restricted to every element of $\Gc$ 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.
