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Let Q be a Riemannian manifold such that the Betti numbers of its free loopspace with respect to some coefficient field are unbounded. We show that everycontact form on its unit contangent bundle supporting the natural contactstructure has infinitely many simple Reeb orbits. This is an extension of atheorem by Gromoll and Meyer. We also show that if a compact manifold admits aStein fillable contact structure then there is a possibly different suchstructure which also has infinitely many simple Reeb orbits for everysupporting contact form. We use local Floer homology along with symplectichomology to prove these facts.

Local Floer homology and infinitely many simple Reeb orbits