Workshop on Symplectic Dynamics

June 27-July 01, 2022
Instituto Superior Técnico
Lisboa, Portugal

Titles and Abstracts

Luca Asselle

Title: The dynamics of strong magnetic fields on surfaces: periodic orbits, trapping regions, and rigidity of Zoll systems

Abstract: How does a magnetic field influence the motion of a charged particle on a surface? Are there periodic orbits or trapping regions for the particle? How difficult is it to construct a magnetic field for which all orbits are periodic? In this talk we will see that, if the magnetic field is strong, a normal form going back to the Russian school allows us to use the Poincaré-Birkhoff theorem and KAM theory to tackle these questions. This is joint work with Gabriele Benedetti.

Gabriele Benedetti

Title: A bi-invariant Lorentz-Finsler structure in contact geometry

Abstract: Contact systolic geometry studies the relationship between the minimal period of closed orbits of (autonomous) Reeb flows and the volume of a contact manifold. Extending this study to non-autonomous Reeb flows gives rise to a bi-invariant Lorentz-Finsler structure on the group of contactomorphisms. Although this structure is too flexible to produce non-trivial measurements in general, we will see that it leads to interesting local rigidity phenomena for certain classes of contactomorphisms. This is joint work with Alberto Abbondandolo and Leonid Polterovich.

Frédéric Bourgeois

Title: Generating families and augmentations for Legendrian submanifolds

Abstract: Generating families provide a convenient way to describe wide classes of Legendrian submanifolds in 1-jet spaces, but they can also be used to define a homological invariant via Morse theory. I will  describe a degeneration process that can be used to better understand the differential of the corresponding complex. It turns out that this approach can also lead to a relation between generating families for a Legendrian submanifold and augmentations for its Chekanov-Eliashberg differential graded algebra.

Michael Brandenbursky

Title: $C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

Abstract: Let $\Sigma$ be a surface equipped with an area form. There is a long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of integrable diffeomorphisms. A slightly weaker version of this question asks: "Does every entropy zero Hamiltonian diffeomorphism of a surface lie in the $C^0$-closure of the set of autonomous diffeomorphisms?" In this talk I will answer in negative the later question. In particular, I will show that on a surface $\Sigma$ the set of autonomous Hamiltonian diffeomorphisms is not $C^0$-dense in the set of entropy-zero Hamiltonians. Explicitly constructed examples of such Hamiltonians cannot be approximated by autonomous diffeomorphisms. (Joint with M. Khanevsky).

Urs Frauenfelder

Title: Lagrange multipliers and adiabatic limits

Abstract: This is joint work with Joa Weber. As one learns in calculus there is a one to one correspondence between critical points of the restriction of a functional to a constraint and the Lagrange multiplier functional. The gradient flow lines of the two functional however are different but can be related by an adiabatic limit procedure. In the talk I will present a compactness theorem for this adiabatic limit problem. I also plan to explain an infinite dimensional motivation behind this result concerning Rabinowitz action functional.

Viktor Ginzburg

Title: On the volume of Lagrangian submanifolds

Abstract: We will discuss the continuity property of the surface area of Lagrangian submanifolds, or to be more precise its lower semi-continuity with respect to the gamma-norm, and connections with integral geometry, Floer theory, and the h-principle for isometric embeddings (the Nash theorem). This is a work in progress with Erman Cineli and Basak Gurel.

Basak Gurel

Title: Topological entropy, barcodes and Floer theory

Abstract: Topological entropy is one of the fundamental invariants of a dynamical system, measuring the orbit complexity. In this talk, we discuss a connection between the topological entropy of compactly supported Hamiltonian diffeomorphisms and Floer theory. We introduce a new invariant associated with the Floer complexes of the iterates of such a diffeomorphism, which we call barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in dimension two. The talk is based on joint work with Erman Cineli and Viktor Ginzburg.

Umberto Hryniewicz

Title: Existence and applications of global Surfaces of section

Abstract: A global surface of section (GSS) reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism. In this talk I will present some existence statements within the class of Reeb flows. Firstly, I will focus on the question of finding a GSS spanned by a given collection of periodic orbits, with no genericity assumptions, and state results obtained in collaboration with Salomão and Wysocki. These statements have applications in Celestial Mechanics. Secondly, I will present recent existence statements for rational GSS’s (Birkhoff sections) under genericity assumptions, and explain how to use them to prove that $C^\infty$ generically a Reeb flow on a closed 3-manifold has positive topological entropy. This is the fruit of joint work with Colin, Dehornoy and Rechtman, and is based on broken book decompositions.

Michael Hutchings

Title: Quantitative closing lemmas
Abstract: We consider closing lemmas which assert roughly speaking that a local $C^\infty$ perturbation (of a symplectomorphism or a contact form) leads to the local creation of a periodic orbit. Quantitative closing lemmas assert roughly speaking that an orbit of period $L$ appears within time $O(L^{-1})$. Spectral gaps in Floer theories, or related invariants, can be used to prove such quantitative closing lemmas. In particular we can use periodic Floer homology to prove quantitative closing lemmas for certain area-preserving surface diffeomorphisms, and embedded contact homology to prove quantitative closing lemmas for contact three-manifolds. Based on joint work with Oliver Edtmair.

Jungsoo Kang

Title: Rabinowitz Floer homology for negative line bundles

Abstract: In my previous work with Peter Albers, we proved that the Rabinowitz Floer homology(RFH) for a circle subbundle of a negative line bundle vanishes if the radius of the circle is small. Later Sara Venkatesh showed that, in this setting, the RFH is not always invariant under the change of radius by computing this for some examples. In this talk, I will explain how one can study the RFH for negative line bundles using the filtration introduced by Urs Frauenfelder (also by Viktor Ginzburg and Jeongmin Shon in the context of positive symplectic homology) and revisit the above computations. This is joint work with Peter Albers.

Marco Mazzucchelli

Title: Existence of global surfaces of section for Reeb flows of closed contact 3-manifolds

Abstract: In this talk, based on joint work with Gonzalo Contreras, I will outline a proof of the existence of global surfaces of section for all Reeb flows of closed contact 3-manifolds satisfying the Kupka-Smale condition: non-degeneracy of the closed orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed orbits. This result, in particular, settles the existence of global surfaces of section for the Reeb vector field of a $C^\infty$ generic contact form on any closed 3-manifold, and even for the geodesic vector field of a $C^\infty$ generic Riemannian metric on any closed surface. As an application, I will provide a new characterization of Anosov Reeb flows of closed contact 3-manifolds, which implies the $C^2$ structural stability conjecture for Riemannian geodesic flows of closed surfaces.

Matthias Meiwes

Title: Braid stability and entropy robustness

Abstract: Topological entropy captures the orbit complexity of a dynamical system with the help of a single non-negative number. Detecting robustness of this number under perturbation is a way to understand stability features of a chaotic system. In my talk, I will address the problem of robustness of entropy for Hamiltonian diffeomorphisms in terms of Hofer's metric. Our main focus lies on dimension 2, where there is a strong connection between topological entropy and the existence of specific braid types of periodic orbits. I will discuss a result that states that any braid of non-degenerate one-periodic orbits with pairwise homotopic strands persists under generic Hofer-small perturbations. One application is a local entropy robustness result for surfaces. My talk is based on joint work with Marcelo Alves.

Alexandru Oancea

Title: Secondary continuation map in Floer theory

Abstract: In situations where Floer continuation maps are not isomorphisms, secondary continuation maps obtained by interpolation carry nontrivial topological and dynamical information. I will explain two such examples, with applications to string topology. The talk will be based on joint work with Kai Cieliebak and Nancy Hingston.

Alexander Ritter

Title: Symplectic cohomology of conical symplectic resolutions

Abstract: In this joint work with Filip Zivanovic, we construct symplectic cohomology for a class of symplectic manifolds that admit ${\mathbb C}^*$-actions and which project equivariantly and properly to a convex symplectic manifold. The motivation for studying these is a large class of examples known as Conical Symplectic Resolutions, which includes quiver varieties, resolutions of Slodowy varieties, and hypertoric varieties. These spaces are highly non-exact at infinity, so along the way we develop foundational results to be able to apply Floer theory. Motivated by joint work with Mark McLean on the Cohomological McKay Correspondence, which we also briefly review in the talk, our goal is to describe the ordinary cohomology of the resolution in terms of a Morse-Bott spectral sequence for positive symplectic cohomology. These spectral sequences turn out to be quite computable in many examples. We also obtain a filtration on ordinary cohomology by cup-product ideals, and interestingly the filtration can be dependent on the choice of circle action.

Stefan Suhr

Title: Recent Progress in the Theory of Lyapunov Functions

Abstract: Lyapunov functions are important distinguishers in the theory of dynamical systems as they dissect the underlying space into an “simple” part, i.e. where the dynamics act "gradient like”, and the “interesting” part, i.e. where recurrence appears. The talk will explain a method to construct smooth and complete Lyapunov functions for vector fields (or multi functions). The method is inspired by the closely related problem of time functions in general relativity. If time permits I will give an outlook on generalizations to Lyapunov $1$-forms.

Otto van Koert

Title: A generalization of the Poincare-Birkhoff fixed point theorem

Abstract: In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem and some related problems, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem.