Local systems and the Hofer-Zehnder capacity

The Hofer-Zehnder capacity of a symplectic manifold is one of the early symplectic invariants: it is a non-negative real number, possibly infinite. Finiteness of this capacity has strong consequences for Hamiltonian dynamics, and it is an old question to decide whether it holds for small compact neighborhoods of closed Lagrangians. I will explain a positive answer to this question for a class of manifolds whose free loop spaces admit nontrivial local systems. The proof is an illustration of the interplay between the symplectic topology of phase spaces and the topology of free loop spaces. Joint work with Peter Albers and Urs Frauenfelder.