Generically Mane set supports uniquely ergodic measure for residual   cohomology class
  In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set $\mathcal{G}\subset H^1(M,\mathbb{R})$ such that \[ \widetilde{\mathcal{M}}(c)=\widetilde{\mathcal{A}}(c)=\widetilde{\mathcal{N}}(c),\quad \forall c\in\mathcal{G} \] with $\widetilde{\mathcal{M}}(c)$ supports on a uniquely ergodic measure. 
Jianlu Zhang
math.DS