Workshop on Recent Developments in Symplectic Dynamics

I will report on a new application of Lagrangian Floer theory in Hamiltonian dynamics, obtained in joint work with Habib Alizadeh. We show that the existence of suitable invariant sets of a Hamiltonian diffeomorphism, which are often Lagrangian, implies that its iterations do not accumulate at the identity in natural metrics of symplectic nature. Our main applications include a solution of Polterovich's well-known rigidity question in Hofer's geometry for surfaces, and a proof of its analogue for autonomous flows on symplectically hyperbolic manifolds. I will also discuss the case of autonomous flows of functions of the moment map on toric manifolds. The arguments apply to the spectral metric and hence often to the $C^0$ metric.