| In this thesis we investigate the topological properties of the action-miniziming sets that appear in the study of Tonelli Lagrangian and Hamiltonian systems.; In the first part, we will focus on understanding the topology of the quotient Aubry set, and in particular its total disconnectedness. This property, in fact, plays a key role in the variational methods developed for constructing orbits with a prescribed behavior or connecting different regions of the state space. We will show how this problem may be related to a Sard-like property for certain subsolutions of Hamilton-Jacobi equation and use this approach to show total disconnectedness under suitable assumptions on the Lagrangian.; In the second part, we will discuss some relations between the dynamics of the system and the underlying symplectic geometry of the space. In particular, we will point out how to deduce from weak KAM theory the symplectic invariance of the Aubry set and the quotient Aubry set, and we will study the action minimizing properties of invariant measures supported on Lagrangian graphs. We will then use these results to deduce uniqueness of invariant Lagrangian graphs in a fixed homology or cohomology class, with particular attention to the case of KAM tori and Herman's Tori. |
