1er étage bâtiment Braconnier, Université Claude Bernard Lyon 1 - La Doua
I will discuss two different approaches to systematically studying invariant sets of Hamiltonian systems. The first approach builds heavily on results due to Viterbo and Vichery. I will discuss how an analogue of Mather's alpha-function arises from homogenized Floer homological Lagrangian spectral invariants and how it gives rise to the existence of an analogue of Mather measures (from Aubry-Mather theory) to general symplectic manifolds. Unlike what happens in the Tonelli case, I will show that the support of these measures can be extremely "wild" in the non-convex case. I will explain how this phenomenon is closely related to diffusion phenomena such as Arnold' diffusion. The second approach builds on work due to Buhovsky-Entov-Polterovich and provides a C^0-analogue of Mather measures for Hamiltonians on "flexible" symplectic manifolds. I will discuss applications to Hamiltonian systems on twisted cotangent bundles and R^2n.