Sobre o funcional de ação mínima de Mather: propriedades genéricas e diferenciabilidade

Abstract

In this thesis we present some generic properties and its consequences to the dynamics of the Aubry set that appear in the Mathers theory about Tonelli Lagrangians and Hamiltonian systems. We also study conditions for differentiability of the Mathers minimal action functional and some implications of its regularity to the dynamics of the system. In the first part, we prove that for the set of exact magnetic Lagrangians, the property There exist finitely many static classes for every cohomology class is generic. We also obtain some dynamical consequences of this property. In the second part, we present a dynamical condition in order to obtain differ-entiability of Mathers -function. More preciselly, we obtain differentiability of on all homology classes corresponding to rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian absorbing graph, invariant by Tonelli Hamil-tonians. We also show the relationship between local differentiability of and local integrability of the Hamiltonian flow. In the last part, we show an example of Mathers -function on the homology group of two torus T2, by using the results obtained about its differentiability.