  The aim of this paper is twofold. On the one hand, we discuss the notions of
strong chain recurrence and strong chain transitivity for flows on metric
spaces, together with their characterizations in terms of rigidity properties
of Lipschitz Lyapunov functions. This part extends to flows some recent results
for homeomorphisms of Fathi and Pageault. On the other hand, we use these
characterisations to revisit the proof of a theorem of Paternain, Polterovich
and Siburg concerning the inner rigidity of a Lagrangian submanifold $\Lambda$
contained in an optical hypersurface of a cotangent bundle, under the
assumption that the dynamics on $\Lambda$ is strongly chain recurrent. We also
prove an outer rigidity result for such a Lagrangian submanifold $\Lambda$,
under the stronger assumption that the dynamics on $\Lambda$ is strongly chain
transitive.
