In the first part of the talk we will discuss some aspects of a celebrated theorem due to Bangert and Hingston which says the following: on any closed manifold Q, which is not a circle and has fundamental group Z, the number of geometrically distinct closed geodesics grows like the prime numbers. We will give a rough sketch of the proof and put an emphasis on the use of Lusternik–Schnirelmann theory therein. In the second part of the talk we will explain how Bangert and Hingston's theorem can be restated in terms of Hamiltonian dynamics and discuss the natural generalization from geodesics to Reeb orbits. Under an additional circle action assumption and the use of Floer theory, we proceed to give a proof of a Bangert and Hingston type result for closed Reeb orbits on starshaped hypersurfaces.
