Related papers: Closed Reeb orbits on the sphere and symplecticall…

We construct a contact form on R^{2n+1}, n at least 2, equal to the standard contact form outside a compact set and defining the standard contact structure on all of R^{2n+1}, which has trapped Reeb orbits, including a torus invariant under…

We show the existence of a contractible periodic Reeb orbit for any contact structure supported by an open book whose binding can be realised as a hypersurface of restricted contact type in a subcritical Stein manifold. A key ingredient in…

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one…

We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section…

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a…

We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field…

We study the $J-$holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one periodic Reeb orbits in any closed contact manifold with any contact form by using the well-known Gromov's…

We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, we…

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic orbits,…

A contact form on the tight $3$-sphere $(S^3,\xi_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,\xi_0)$ admitting a…

In this paper we prove two isomorphisms in the local symplectic homology of a simple, which is to say non iterated, isolated Reeb orbit. The isomorphisms are in $S^1$-equivariant and nonequivariant symplectic homology, relating the local…

We prove that there exists at least one close orbit in a given contact hypersurface in some symplectic manifolds.

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition…

We prove the existence of infinitely many periodic orbits of symplectomorphisms isotopic to the identity if they admit at least one hyperbolic periodic orbit and satisfy some condition on the flux. Our result is proved for a certain class…

Let $\varphi$ be any flow on $T^n$ obtained as the suspension of a diffeomorphism of $T^{n-1}$ and let $\mathcal A$ be any compact invariant set of $\varphi$. We realize $(\mathcal A, \varphi|_{\mathcal A})$ up to reparametrization as an…

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…

In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of…

The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the…

Let Q be a Riemannian manifold such that the Betti numbers of its free loop space with respect to some coefficient field are unbounded. We show that every contact form on its unit contangent bundle supporting the natural contact structure…