Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as: - the Lyapunov exponents of minimizing measures; -the weak KAM solutions. In particular, we deduce that the support of every minimizing measure all of whose Lyapunov exponents are zero is C1-regular almost everywhere.