For over a century, geodesic flows of hyperbolic manifolds have been one of the main examples of hyperbolic flows. The deep knowledge of the underlying geometry, the homogeneous structure of the model hyperbolic space, the reversibility property of the dynamics allowed a lot of breakthroughs, and led to a very good understanding of the ergodic properties of the associated flow, in particular entropy, (speed of) mixing, equi-distribution and counting of periodic orbits, thermodynamical formalism, etc.

In the last decade, there was a new focus on these questions, by various very active communities of researchers mostly interested in non-positive curvature, non-compact manifolds, and higher-rank representations of (possibly infinitely generated) discrete groups. At the crossroads of many questions, Hilbert geometries and their geodesic flows are a connecting bridge between hyperbolic dynamics in negative curvature and higher-rank geometry and dynamics. The spirit of these recent types of research is the same: to benefit from the good understanding of the underlying geometry in order to get new dynamical results.

The goal of this workshop is to gather specialists from these different areas to benefit from our respective expertise in close but distinct topics.

Among many possibilities, we are particularly interested in the Finsler geodesic flows associated with weakly regular Hilbert geometries and the structure of their Gibbs measures, some being non reversible, especially the SRB measure.

Those Finsler metrics are not constrained as in the Riemannian case by the Gauss- Bonnet formula. The Busemann Finsler area of compact surfaces, equipped with their Hilbert metric, is flexible and can be deformed to infinity, paving the way to new phenomena.

We will try to understand the degeneracies of these geometric structures also from the thermodynamic formalism point of view, with a focus on the behaviour of Gibbs measures associated with higher-rank representations of discrete groups and looking at questions about the existence of phase transitions for Lyapounov exponents, entropy or pressure.

Aother related topic would be to discuss the status of some of the most famous conjectures still pending, namely, at the crossroads of topology and group theory, the Cannon conjecture which asserts that a hyperbolic group whose boundary at infinity is homoeomorphic to $S^2$ should be virtually (up to a finite index) a lattice in PSL (2, C). (Perelman’s works are quite involved and having a simpler proof in that case would already be a great progress). We could also focus on the entropy rigidity conjecture for Riemannian geodesic flows of closed manifolds, which still holds in dimension higher than 2. It asserts that, if the Bowen-Margulis measure and the SRB measure coincide, then the metric should be locally symmetric.

We also wish to invite young researchers in the early stage of their career. We’re actively seeking to increase the diversity of our attendees and speakers by a variety of perspectives, and our goal is to create an inclusive, respectful conference environment.