Dynamical convexity and elliptic periodic orbits for Reeb flows

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively. In this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact forms on $S^3$. Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds.Comment: Version 1: 43 pages. Version 2: revised and improved exposition, corrected misprints, 44 pages. Version 3: final version, 46 pages, 1 figure, to appear in Mathematische Annale

Isometry-invariant geodesics and the fundamental group, II

We show that on a closed Riemannian manifold with fundamental group isomorphic to $\mathbb{Z}$, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.Comment: 23 pages. Version 2: added the proof of Lemma 2.2. To appear in Advances in Mathematic