In this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki-Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S 2 × S 3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

Abstract. We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll-Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of nonhyperbolic periodic orbits.

Abstract. Let M be a closed manifold whose based loop space Ω(M ) is "complicated". Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T * M which is fiberwise starshaped with respect to the origin. Choose a function H : T * M → Ê such that Σ is a regular energy surface of H, and let ϕ t be the restriction to Σ of the Hamiltonian flow of H. Theorem 1. The topological entropy of ϕ t is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain-Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T * M is positive.The following corollary extends results of Morse and Gromov on the number of geodesics between two points. Corollary 1. Given q ∈ M , for almost every q ′ ∈ M the number of orbits of the flow ϕ t from Σ q to Σ q ′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed orbits.Corollary 2. Let M be a closed surface different from S 2 , ÊP 2 , the torus and the Klein bottle. Then ϕ t carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

Abstract. We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits.

Abstract. We consider an exact magnetic flow on the tangent bundle of a closed surface. We prove that for almost every energy level κ below the Mañé critical value of the universal cover there are infinitely many periodic orbits with energy κ. Contents