GLOBAL PROPERTIES OF TIGHT REEB FLOWS WITH APPLICATIONS TO FINSLER GEODESIC FLOWS ON S2

Abstract. We show that if a Finsler metric on S2 with reversibility r has flag curvatures K satisfying ( r r+1)2 < K ≤ 1, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok [21], proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels C2-close to S3