On the Rabinowitz Floer homology of twisted cotangent bundles

Consider the cotangent bundle of a Riemannian manifold $(M,g)$ of dimension 2 or more, endowed with a twisted symplectic structure defined by a closed weakly exact 2-form $\sigma$ on $M$ whose lift to the universal cover of $M$ admits a bounded primitive. We compute the Rabinowitz Floer homology of energy hypersurfaces $\Sigma_{k}=H^{-1}(k)$ of mechanical (kinetic energy + potential) Hamiltonians $H$ for the case when the energy value k is greater than the Mane critical value c. Under the stronger condition that k>c_{0}, where c_{0} denotes the strict Mane critical value, Abbondandolo and Schwarz recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k>c, thus covering cases where $\sigma$ is not exact. As a consequence, we deduce that the hypersurface corresponding to the energy level k is never displaceable for any k>c. Moreover, we prove that if dim M > 1, the homology of the free loop space of $M$ is infinite dimensional, and if the metric is chosen generically, a generic Hamiltonian diffeomorphism has infinitely many leaf-wise intersection points in $\Sigma_{k}$.Comment: V4 - final version, accepted for publication in CVPD

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with sublinear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small T>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period T belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.Comment: 24 pages, V2 - minor corrections, final version to appear in JF

Closed orbits of a charge in a weakly exact magnetic field

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $\sigma$ a weakly exact 2-form. Let $\phi_{t}$ denote the magnetic flow determined by $\sigma$, and let $c$ denote the Mane critical value of the pair $(g,\sigma)$. We prove that if $k>c$, then for every non-trivial free homotopy class of loops on $M$ there exists a closed orbit with energy $k$ whose projection to $M$ belongs to that free homotopy class. We also prove that for almost all $k<c$ there exists a closed orbit with energy $k$ whose projection to $M$ is contractible. In particular, when $c=\infty$ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if $\sigma$ is not exact and $M$ has an amenable fundamental group (which implies $c=\infty$) then there exist contractible closed orbits on almost every energy level.Comment: 25 pages. v3 - minor corrections, this version to appear in PJ

Translated points and Rabinowitz Floer homology

We prove that if a contact manifold admits an exact filling then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [San11b]. In addition we prove that if the Rabinowitz Floer homology of the filling is non-zero then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional then every contactmorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover if the contact manifold has dimension greater than or equal to 3, the latter option generically doesn't happen. Finally, we prove that a generic contactomorphism on $\mathbb{R}^{2n+1}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.Comment: 13 pages, v2: numerous corrections, results unchange