Tag - Reeb flows

Agustin Moreno: On the spatial restricted three-body problem

18th June, 2021

In his search for closed orbits in the planar restricted three-body problem, Poincaré’s approach roughly reduces to:

   1.  Finding a global surface of section;
   2.  Proving a fixed-point theorem for the resulting return map.

This is the setting for the celebrated Poincaré-Birkhoff theorem. In this talk, I will discuss a generalization of this programme to the spatial problem.

For the first step, we obtain the existence of global hypersurfaces of section for which the return maps are Hamiltonian, valid for energies below the first critical value and all mass ratios. For the second, we prove a higher-dimensional version of the Poincaré-Birkhoff theorem, which gives infinitely many orbits of arbitrary large period, provided a suitable twist condition is satisfied. Time permitting, we also discuss a construction that associates a Reeb dynamics on a moduli space of holomorphic curves (a copy of the three-sphere), to the given dynamics, and its properties.

Vincent Colin: Reeb dynamics in dimension 3 and broken book decompositions

11th September, 2020

In joint work with Pierre Dehornoy and Ana Rechtman, we prove that on a closed 3-manifold, every non-degenerate Reeb vector field is supported by a broken book decomposition. From this property, we deduce that in dimension 3 every non-degenerate Reeb vector field has either 2 or infinitely periodic orbits and that on a closed 3-manifold that is not graphed, every non-degenerate Reeb vector field has positive topological entropy.

Abror Pirnapasov: Reeb orbits that force topological entropy

5th June, 2020

A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy. We also generalize to the category of Reeb flows a beautiful result due to Denvir and Mackay, which says that if a Riemannian metric on the two-dimensional torus has a contractible closed geodesic then its geodesic flow has positive topological entropy.

Jo Nelson: Reflections on Cylindrical Contact Homology

15th May, 2020

This talk beings with a light introduction, including some historical anecdotes to motivate the development of this Floer theoretic machinery for contact manifolds some 25 years ago. I will discuss joint work with Hutchings which constructs nonequivariant and a family Floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over ℤ and capture interesting torsion information. I will explain the need for an obstruction bundle gluing correction term in the expression of the differential in the presence of contractible Reeb orbits, which is essential even in the simple example of an ellipsoid. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

Marco Mazzucchelli: Spectral characterizations of Besse and Zoll Reeb flows

8th May, 2020

In this talk, I will address a geometric inverse problem from contact geometry: is it possible to recognize whether all orbits of a given Reeb flow are closed from the knowledge of the action spectrum? Borrowing the terminology from Riemannian geometry, Reeb flows all of whose orbits are closed are sometimes called Besse, and Besse Reeb flows all of whose orbits have the same minimal period are sometimes called Zoll. In the talk I will summarize recent results on this inverse problem in a few settings: geodesic flows (joint work with Stefan Suhr), closed contact 3-manifolds (joint work with Daniel Cristofaro-Gardiner), convex contact spheres and, more generally, restricted contact type hypersurfaces of symplectic vector spaces (joint work with Viktor Ginzburg and Basak Gürel). I will also mention a few conjectures and open problems.

Kei Irie: C∞ closing lemma for three-dimensional Reeb flows via embedded contact homology

16th February, 2017

Cr closing lemma is an important statement in the theory of dynamical systems, which implies that for a Cr generic system the union of periodic orbits is dense in the nonwondering domain. C1 closing lemma is proved in many classes of dynamical systems, however Cr closing lemma with r > 1 is proved only for few cases. In this talk, I'll prove C∞ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH). In particular, the key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M. Asaoka) will be also presented.

Kai Zehmisch: Disc filling and connected sum

27th February, 2015

In my talk I will report on recent work with Hansjörg Geiges about a strong connection between the topology of a contact manifold and the existence of contractible periodic Reeb orbits. Namely, if the contact manifold appears as non-trivial contact connected sum and has non-trivial fundamental group or torsion-free homology, then the existence is ensured. This generalizes a result of Helmut Hofer in dimension three.

Mark McLean: Minimal Discrepancy of Isolated Singularities and Reeb Orbits

4th April, 2014

Let A be an affine variety inside a complex N-dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization.

Jo Nelson: Cylindrical contact homology as a well-defined homology?

7th February, 2014

In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop a theory without the use of virtual chains or polyfolds in 3 dimensions. It turns out that this can be accomplished by placing strong conditions on the growth rates of the indices of Reeb orbits. In addition we sketch a new approach allowing us to compute cylindrical contact for a large class of examples which admit contact forms that are admissible under the stronger conditions required. This approach is applicable to prequantization spaces and the links of simple singularities.