Tag - Anosov flows

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.

Thomas Massoni: Non-Weinstein Liouville Domains and Three-Dimensional Anosov Flows

25th November, 2022

Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years. Little is known about non-Weinstein Liouville domains, whose first instance is due to McDuff. I will describe two key examples of such domains in dimension four, and then explain how they fit into a general construction based on Anosov flows on three-manifolds. The symplectic invariants of these 'Anosov Liouville domains' constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case.

Thomas Massoni: Three-dimensional Anosov flows and non-Weinstein Liouville domains

17th October, 2022

An Anosov flow Φ on a closed 3-manifold M gives rise to a non-Weinstein Liouville structure on V = [−1,1] × M. Building upon the work of Hozoori, we establish a homotopy correspondence between Anosov flows and certain pairs of contact forms. Moreover, the symplectic invariants of V only depend on the homotopy class of Φ. We focus on a subcategory W0 of the wrapped Fukaya category of V whose objects are in bijection with the simple closed orbits of Φ. In contrast with the Weinstein case, W0 is not homologically smooth, as it is not finitely split-generated in a maximal way. We expect W0 to be a powerful new invariant of Anosov flows.