We show that the Birkhoff normal form near a positive definite KAM torus is given by the function $\alpha$ of Mather. This observation is due to Siburg [Si2], [Si1] in dimension 2. It clarifies the link between the Birkhoff invariants and the action spectrum near the torus. Our extension to high dimension is made possible by a simplification of the proof given in [Si2].

Let $L:TM\to ℝ$ be a Tonelli Lagrangian function (with $M$ compact and connected and $dimM\ge 2$). The tiered Aubry set (resp. Mañé set) ${𝒜}^T\left(L\right)$ (resp. ${𝒩}^T\left(L\right)$) is the union of the Aubry sets (resp. Mañé sets) $𝒜(L+\lambda )$ (resp. $𝒩(L+\lambda )$) for $\lambda$ closed 1-form. Then1.the set ${𝒩}^T\left(L\right)$ is closed, connected and if $dim{H}^1\left(M\right)\ge 2$, its intersection with any energy level is connected and chain transitive;2.for $L$ generic in the Mañé sense, the sets $\overline{{𝒜}^T\left(L\right)}$ and $\overline{{𝒩}^T\left(L\right)}$ have no interior;3.if the interior of $\overline{{𝒜}^T\left(L\right)}$ is non empty, it contains a dense subset of periodic points.We then give an example...

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a...

The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. To do so, we develop a set of explicit a-priori estimates for smooth solutions of Hamilton-Jacobi equations, using a combination of methods from viscosity solutions, KAM and Aubry-Mather theories. These estimates are valid in any space dimension, and can be checked numerically to detect gaps between KAM tori and Aubry-Mather sets. We apply these results to detect non-integrable regions in several...