Tag - Reeb flows

Viktor Ginzburg: Invariant Sets and Hyperbolic Periodic Orbits

10th May, 2024

The presence of hyperbolic periodic orbits or invariant sets often has an affect on the global behaviour of a dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards Franks’s theorem for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Gurel and asserting that no periodic orbit of a Hamiltonian pseudo-rotation is locally maximal.

Yasha Savelyev: Gromov-WItten Invariants of Riemann-Finsler Manifolds

12th February, 2024

I will give a construction of certain ℚ-valued deformation invariants of (in particular) complete non-positively curved Riemannian manifolds. These are obtained as certain elliptic Gromov-Witten curve counts. As one immediate application we give the (possibly) first generalization to non-compact fibrations, of Preissman's now classical theorem on non-existence of negative sectional curvature metrics on compact products. One additional goal of the talk is to use the above theory to motivate a very elementary but deep open problem in Riemannian geometry/dynamics concerning existence of Reebable and geodesible sky catastrophes. I will give a partial answer to this problem for surfaces.

Matthias Meiwes: C0 Stability of Topological Entropy for 3-Dimensional Reeb Flows

24th November, 2023

The C0 distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms. In this talk, I will explain some recent progress on the stability properties of the topological entropy with respect to this distance obtained in collaboration with M. Alves, L. Dahinden, and A. Pirnapasov. Our main result states that the topological entropy for closed contact 3-manifolds is lower semi-continuous in the C0 distance for C∞-generic contact froms. Applying our methods to geodesic flows of surfaces, we obtain that the points of lower-semicontinuity of the topological entropy include non-degenerate metrics. In particular, given a geodesic flow of such a metric with positive topological entropy, the topological entropy does not vanish for sufficiently C0-small perturbations of the metric.

Marco Mazzucchelli: Locally Maximal Closed Orbits of Reeb Flows

6th November, 2023

A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gurel, I will present a 'forced existence' result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension:

- If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits. 

- If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption.

These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gurel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.

Marco Mazzucchelli: Surfaces of Section, Anosov Reeb Flows, and the C2-Stability Conjecture for Geodesic Flows

3rd March, 2023

In this talk, based on joint work with Gonzalo Contreras, I will briefly sketch the proof of the existence of global surfaces of section for the Reeb flows of closed 3-manifolds satisfying a condition à la Kupka-Smale: non-degeneracy of the closed Reeb orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed Reeb orbits. I will then present an application of this theorem to hyperbolic Reeb dynamics: a Reeb flow on a closed 3-manifold is Anosov if and only if the closure of the subspace of closed Reeb orbits is hyperbolic and the Kupka-Smale transversality condition holds. This result implies the validity of the C2 stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C2 structurally stable within the class of Riemannian geodesic flows must be Anosov.

Jo Nelson: Floer Theories and Reeb Dynamics for Contact Manifolds

13th February, 2023

Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a nondegeneracy condition called maximal non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will then explain how to construct and compute Floer-theoretic contact invariants. These are a sort of infinite-dimensional version of Morse theory wherein the chain complexes are generated by closed Reeb orbits and the differential counts certain J-holomorphic curves. This talk will feature numerous graphics and anecdotes.

Christian Lange: Orbifolds and Systolic Inequalities

13th January, 2023

In this talk, I will first discuss some instances in which orbifolds occur in geometry and dynamics, in particular, in the context of billiards and systolic inequalities. Then I will present topological conditions for an orbifold to be a manifold together with applications to foliations and to Besse geodesic and Reeb flows (joint work with Manuel Amann, Marc Kegel and Marco Radeschi). Here a flow is called Besse if all its orbits are periodic. Such flows are related to systolic inequalities. Namely, I will explain a characterization of contact forms on 3-manifolds whose Reeb flow is Besse as local maximizers of certain 'higher' systolic ratios, and mention other related systolic-like inequalities (joint work with Alberto Abbondandolo, Marco Mazzucchelli and Tobias Soethe).

Marcelo Alves: Hofer’s Geometry and Braid Stability

16th December, 2022

Hofer's metric dH is a remarkable bi-invariant metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. In my talk, I will explain a result, obtained jointly with Matthias Meiwes, which says that the braid type of a set of periodic orbits of a Hamiltonian diffeomorphism on a closed surface is stable under perturbations that are sufficiently small with respect to Hofer's metric. As a consequence of this we obtained that the topological entropy, seen as a function on the space of Hamiltonian diffeomorphisms of a closed surface, is lower semi-continuous with respect to the Hofer metric dH.

If time permits, I will explain related questions for Reeb flows on 3-manifolds and Hamiltonian diffeomorphisms on higher-dimensional symplectic manifolds, and recent progress on these problems obtained by myself, Meiwes, Abror Pirnapasov and Lucas Dahinden.

Julian Chaidez: The Ruelle invariant and convexity in higher dimensions

24th June, 2022

I will explain how to construct the Ruelle invariant of a symplectic cocycle over an arbitrary measure preserving flow. I will provide examples and computations in the case of Hamiltonian flows and Reeb flows (in particular, for toric domains). As an application of this invariant, I will construct toric examples of dynamically convex domains that are not symplectomorphic to convex ones in any dimension.

Jonathan Zung: Reeb flows transverse to foliations

25th June, 2021

Eliashberg and Thurston showed that taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.