The tiered Aubry set for autonomous Lagrangian functions

If L is a Tonelli Lagrangian defined on the tangent bundle of a compact and connected manifold whose dimension is at least 2, we associate to L the tiered Aubry set and the tiered Mane set (defined in the article). We prove that the tiered Mane set is closed, connected, chain transitive and that if L is generic in the Mane sense, the tiered Mane set has no interior. Then, we give an example of such an explicit generic Tonelli Lagrangian function and an example proving that when M is the torus, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.Comment: 28 pages; to appear in Ann. Inst. Fourier number 58 (2008

Lyapunov exponents for conservative twisting dynamics: a survey

Finding special orbits (as periodic orbits) of dynamical systems by variational methods and especially by minimization methods is an old method (just think to the geodesic flow). More recently, new results concerning the existence of minimizing sets and minimizing measures were proved in the setting of conservative twisting dynamics. These twisting dynamics include geodesic flows as well as the dynamics close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite . Two aspects of this theory are called the Aubry-Mather theory and the weak KAM theory. They were built by Aubry \& Mather in the '80s in the 2-dimensional case and by Mather, Ma{\~n}{\'e} and Fathi in the '90s in higher dimension. We will explain what are the conservative twisting dynamics and summarize the existence results of minimizing measures. Then we will explain more recent results concerning the link between different notions for minimizing measures for twisting dynamics: their Lyapunov exponents; their Oseledet's splitting; the shape of their support. The main question in which we are interested is: given some minimizing measure of a conservative twisting dynamics, is there a link between the geometric shape of its support and its Lyapunov exponents? Or : can we deduce the Lyapunov exponents of the measure from the shape of the support of this measure? Some proofs but not all of them will be provided. Some questions are raised in the last section.Comment: 28 page

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