Quando uma variedade Lagrangiana, invariante por um fluxo Hamiltoniano, é uma seção?

Abstract

This thesis consists in a search by suficient conditions for the graph property of a Lagrangian manifold W, invariant by a Tonelli Hamiltonian ow. This kind of study started with Birkhoof in the 1920' and later after 1980' with Russian and French school and also with the work of Carneiroand Ruggiero in Brazil. In section 2.1, we study the case where W has no conjugate point. More precisely, we show that when in the energy level there exists a neighborhood U of W, such that the positive semi-orbit of a point (...) has no point conjugate to alpha then W is a graph. In section 2.2, we consider the case where W is contained in a Mañé set. It is a well known fact that a necessary condition to a Lagrangian invariant manifold to be a graph is that it is contained in some Mañé set. A natural question is if that is also a su_cient condition. On this direction, we show that if W is a subset of a Mañé set and any two wandering orbits have either the same alpha limit set or the same omega limit set, then W is a graph. In section 2.4, we restricted our study to the geodesic ow on the torus (...). In this case, we show that when the geodesics with initial conditions in W minimize the distance in the covering space (...) and besides that any twonon recurrent orbits have either the same alpha limit set or the same omega limit set, then W is a graph.