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The globally positive diffeomorphisms of the 2 n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry-Mather theory was developed by Garibaldi and Thieullen that provides the existence of some minimizing measures. Using the two Green bundles $${G_-}$$ and $${G_+}$$ that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h: T→ R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h: T→ R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C 1 (Theorem 3). Then we consider the case of the C 0 integrable symplectic twist maps and we prove that for such a map, there exists a dense G δ subset of the set of its invariant curves such that every curve of this G δ subset is C 1 (Theorem 4). [ABSTRACT FROM AUTHOR]