Two days in Symplectic Dynamics
The lectures will take place in room
IA 01/473.
Schedule
Friday, November 26th
Saturday, November 27th
Abstracts

We will present an analog of Santaló point for the area of the boundary of a convex body in a normed space, and explain why they exist in general, and under which conditions they are unique.
We will also provide a characterization of such a point when the normed space is Minkowski.
This is joint work with G. Solanes and K. Tzanev.

On complete, noncompact Riemannian manifolds the possibilities for the
behaviour of a geodesic with respect to infinity leads to a variety of
interesting questions. A geodesic can be bounded or proper or
oscillating. Moreover, and particularly in dimension two, the
injectivity of geodesics is of interest. After a short and incomplete
review of the history of such problems (that starts in the 19th
century), the talk will concentrate on proper, injective geodesics on
complete planes. S.CohnVossen asked in 1936 for the existence of at
least one such geodesic. This was answered affirmatively by the speaker
(1981), when he also asked for the minimal number \(n\) of such geodesics
on a complete plane. The talk will present an example of a complete
plane with only two injective geodesics (joint work with S.Suhr), so \(n\)
is either one or two. We conjecture that actually \(n=2\).

Anna Florio 
Vanishing asymptotic Maslov index for conformally symplectic dynamics
On a symplectic manifold endowed with a Lagrangian bundle and for conformally symplectic dynamics, the asymptotic Maslov index for a positive orbit is for the angles what the Lyapunov exponent is for the norms. In a joint work with MarieClaude Arnaud and Valentine Roos, on a cotangent bundle and for a conformally symplectic dynamics, we show the existence of invariant measure with vanishing asymptotic Maslov index. If moreover the dynamics twists the vertical, then we prove that every Lagrangian submanifold \(L\) Hamiltonianly isotopic to a graph exhibits a point \(x\in L\) of zero index.

The purpose of this talk is to discuss some new sufficient conditions for the existence of rational global surfaces of section (Birkhoff sections) for Reeb flows in dimension three. Our conditions provide perturbative existence statements since they are C^infty generic. They hold, in particular, when the contact form is nondegenerate and the Liouville measure can be approximated by periodic orbits, but we will also explain that these conditions can be checked for certain families of explicit (possibly degenerate) Reeb flows. All results are joint work with Colin, Dehornoy and Rechtman.

The purpose of the talk will be to discuss the relation
between positive and nonnegative Legendrian and contact
isotopies.

Felix Schlenk 
On the topology of strict contactomorphism groups
A strict contactomorphism is a diffeomorphism of a cooriented contact manifold \((M,\alpha)\) that preserves the contact form \(\alpha\). Motivated by a problem on symplectic embeddings, we study the homotopy type of the group of strict contactomorphisms of a closed contact manifold. While this
looks hard in general, computations are possible for generic contact forms, and in special situations as for boundaries of toric domains and for cosphere bundles.
This is work in slow progress joint with Joé Brendel.