Valette, Alain

2011, Heistercamp, Muriel, Schlenk, Félix, Bourgeois, F., Valette, Alain, Gutt, S., Abbondandolo, A., Bertelson, M.

Let M be a smooth closed manifold and T∗M its cotangent bundle endowed with the usual symplectic structure ω = dλ, where λ is the Liouville form. A hypersurface Σ ⊂ T∗M is said to be fiberwise starshaped if for each point q ∈ M the intersection Σ _q := Σ∩T∗_qM of Σ with the fiber at q is the smooth boundary of a domain in T∗M which is starshaped with respect to the origin 0_q ∈ T∗_qM.

In this thesis we give lower bounds on the growth rate of the number of closed Reeb orbits on a fiberwise starshaped hypersurface in terms of the topology of the free loop space of M. We distinguish the two cases that the fundamental group of the base space M has an exponential growth of conjugacy classes or not. If the base space M is simply connected we generalize the theorem of Ballmann and Ziller on the growth of closed geodesics to Reeb flows.