In the first part of this article (Sections 2 and 3) the authors discuss strong chain recurrence and strong chain transitivity for flows on metric spaces and theirs characterization in terms of rigidity properties of Lipschitz Lyapunov functions. Let \(\Psi =\{\Psi _t\}_{t\in \mathbb{R}}\) be a continuous flow on the metric space \((X,d)\). A strong \((\epsilon , T)\)-chain from \(x\) to \(y\) is a finite sequence \((x_i, t_i)_{i=1,\dots,n}\), such that if \(t_i \leq T\) for every \(i\), \(x_1=x \) and \(x_{n+1}=y\), then \(\sum_{i=1}^n d(\Psi_{t_i}(x_i), x_{i+1})<\epsilon \). The flow \(\Psi\) is a strongly chain recurrent if for every \(x\in X\), every \(\epsilon >0\) and every \(T\geq 0\) there exists a strong \((\epsilon , T )\)-chain from \(x\) to \(x\). \(\Psi\) is strongly chain transitive if for every \(x,y\in X\), every \(\epsilon >0\) and every \(T\geq 0\) there exists a strong \((\epsilon , T )\)-chain from \(x\) to \(x\).
The authors prove the following theorem: Let \(\Psi\) be a flow on a metric space \((X,d)\) such that \(\Psi _t\) is Lipschitz continuous for every \(t\geq 0\) uniformly for \(t\) in compact subsets of \([0,+\infty )\). Then:
i) \(\Psi\) is strongly chain recurrent if and only if every Lipschitz continuous Lyapunov function is a first integral (Theorem 2.2 from Section 2).
ii) \(\Psi\) is strongly chain transitive if and only if every Lipschitz continuous Lyapunov function is constant (consequence of Propositions 3.2 and 3.3 in Section 3).
In the second part of the article, the authors use the previously established characterizations to give a new proof of a theorem proven by G. P. Paternain et al. [Mosc. Math. J. 3, No. 2, 593–619 (2003; Zbl 1048.53058)].
Let \(T^* M\) be the cotangent bundle of a closed manifold \(M\) with its standard Liouville and sympletic forms and let \(H:T^* M\rightarrow \mathbb{R}\) be fiberwise superlinear and its fiberwise second differential be everywhere positive definite. Then \(H\) is said to be Tonelli Hamiltonian. Let \(\Sigma:= {z\in T^* M| H(z)=c}\) be a non-empty regular level set of \(H\) such that \(\pi(\Sigma) = M\), where \(\pi\) is the canonical projection from the cotangent bundle. \(\Sigma\) is called an optical hypersurface. G. P. Paternain et al. [Mosc. Math. J. 3, No. 2, 593–619 (2003; Zbl 1048.53058)] proved: Let \( \Sigma \) be an optical hypersurface as above. Let \(\Lambda\) be an element of \(\mathcal{L}(T^*M)\) (the set of closed Lagrangian submanifolds \(\Lambda\) of \(T^*M\)) contained in \(\Sigma\). Assume that the restricted Hamiltonian flow \(\Psi^H|_{\mathbb{R}\times \Lambda}:\mathbb{R}\times \Lambda \rightarrow \Lambda\) is strongly chain recurrent. Then, if \(K\) is an element of \(\mathcal{L}(T^*M)\) contained in \(\bar{U_\Sigma}=\{z\in T^* M| H(z)\leq c\} \) and having the same Liouville class as \(\Lambda\), then necessarily \(K=\Lambda\).
The authors give a new proof of this theorem using graph selectors, Lipschitz functions on \(M\) with specific properties whose existence is proved in Theorem 5.2.
In Section 6 it is proved that under the conditions of Paternain et al. [loc. cit.]. If \(\{\Lambda_r\}_{r\in[0,1]}\subset \mathcal{L}(T^*M)\) is an analytic one-parameter family of smooth Lagrangian submanifolds having the same Liouville class of \(\Lambda\), such that \(\Lambda _0=\Lambda\) and \(\Lambda_r \subset U^c_\Sigma \) for all \(r\in[0,1]\), then \(\Lambda_r=\Lambda\) for all \(r\in [0,1].\)