Related papers: Invariant Sets and Hyperbolic Closed Reeb Orbits

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…

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a…

As a refinement of the Weinstein conjecture, it is a natural question whether a Reeb orbit of particular types exists. D. Cristofaro-Gardiner, M. Hutchings and D. Pomerleano showed that every nondegenerate closed contact three manifold with…

We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to…

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…

When the Poincar\'{e} map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby…

We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all…

We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of…

We prove the non--triviality of the Reeb flow for the (2n+1)--dimensional standard contact spheres inside the fundamental group of their contactomorphism group, n greater than 3. The argument uses the existence of homotopically non--trivial…

We use the inverse pressure concept to estimate the stable dimension for hyperbolic non-invertible maps which are conformal in the stable fibers. The non-invertible case is different than the diffeomorphism case. In particular we show that…

In contact geometry, a systolic inequality is a uniform upper bound on the shortest period of a closed Reeb orbit, in terms of the contact volume. We prove a general systolic inequality valid on Seifert bundles with non-zero Euler number…

We give a dynamical characterisation of odd-dimensional balls within the class of all contact manifolds whose boundary is a standard even-dimensional sphere. The characterisation is in terms of the non-existence of short periodic Reeb…

A classical problem in smooth dynamical systems is known as smooth realization problem. It asks if given a compact manifold $M$, one can construct a volume preserving diffeomorphism with prescribed ergodic properties. We study the decay of…

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 give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S*M,\xi) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of…

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian…

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of…

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true…

We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have…

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key…