# Entropy in Hamiltonian Systems

## Room 008 SeMath, Pontdriesch 14-16 Aachen

## 13-14 February 2020

## Schedule of Talks

13.02.2020, 13:30

**Louis Merlin** (RWTH Aachen):

**On the Volume Entropy of Hilbert Geometries of Low Regularity.**

Hilbert geometries are Finslerian manifolds, arising naturally as convex deformations of the hyperbolic space. In this talk we wish to compare the regularity of the Finlserian structure with the value of the volume entropy. This is based on joint work with Jan Cristina (EPFL).

13.02.2020, 15:00

**Marcelo Alves** (Université Libre de Bruxelles):

*C ^{0}* Entropy Stability for Reeb flows and Geodesic Flows on the 2-dimensional Torus

In this talk I will explain why the topological entropy is bounded from below by positive constants:

- On certain
*C*-open sets in the space of Riemannian metrics on the 2-dimensional torus,^{0} - and on certain
*C*-open sets in the space of contact forms on the 3-dimensional torus endowed with the geodesic contact structure.^{0}

This uses the forcing theories for the topological entropy for geodesic and Reeb flows, combined with ideas from works of Chor-Meiwes and Dahinden. All this is a joint work project with Lucas Dahinden and Matthias Meiwes.

13.02.2020, 16:30

**Discussion.**

14.02.2020, 10:00

**Mathias Meiwes** (RWTH Aachen):

**Topological entropy and Hofer's metric.**

A central object in the study of Hamiltonian diffeomorphisms on a symplectic manifold is Hofer's metric, a bi-invariant metric on the group of Hamiltonian diffeomorphisms, which displays rigidity and flexibility features that are special for those diffeomorphisms. The geometry of this metric and its interplay with the dynamics has been thoroughly studied since its discovery by Hofer and his work in the early 90's. In my talk I address the question of stability of topological entropy with respect to Hofer's metric, and explain a result on the existence of stable lower bounds of it under large perturbations in dimensions 2. Its proof is based among other things on a recent forcing theory of Le Calvez and Tal which will be discussed. This talk is based on joint work with Arnon Chor (Tel Aviv University).

14.02.2020, 11:30

**Lucas Dahinden** (Universität Heidelberg):

**Lightlike Geodesics, Redshift and Topological Entropy**

We can think of the state of the world in a globally hyperbolic Lorentz manifold as a Cauchy hypersurface. I discuss first the Reeb flow on the spherization of the stationary Cauchy hypersurface, and especially how its open string length spectrum changes when viewed from a different Cauchy hypersuface. This is connected to work of Stefan Nemirovski who identified the change of contact form as the Redshift. A second dynamical system is given by a smooth propagation of the Cauchy hypersurface. For this system I can give bounds to the volume growth. This opens the discussion what this means for topological entropy.

14.02.2020, 14:00

**Discussion.**