Ergodic properties of the geodesic flow on noncompact manifolds with nonpositive curvature – GEODE

The aim of this project is the study of ergodic properties of geodesic flows on noncompact manifolds with nonpositive curvature. This study is growing rapidly these last few years. Our project gathers together young french researchers who are already working on the subject, to create a dynamical and efficient group on this subject.

On a noncompact manifold with nonpositive curvature, the classical tools for the study of geodesic flows (harmonic analysis, lattices in semisimple Lie groups, coding through a symbolic dynamic) do not work. Thus, we need to understand finely the behaviour of the geodesic flow, and therefore we have to find robust geometrical arguments that allow to work in this very general framework.

The general philosophy of our project is to try to prove several ergodic properties in the most general possible framework. It should allow us to better understand the relations between geometrical and topological properties of the manifold (curvature, injectivity radius, finitness of the ends, volume, ...), and ergodic properties of the geodesic flow and its stable foliation (horocyclic flow).

The eight members of the project are precisely specialists of such methods, with different mathematical cultures (ergodic theory, dynamical systems, hyperbolic geometry, differential geometry, spectral theory, analysis on manifolds, ...)

The members of our group are geographically very isolated (Amiens, Rennes, Brest, Avignon, Toulouse, Bonn), and our project shoud allow us to structure an active and efficient team in spite of the obstacle of the distance between us.

More precisely, most of us already published alone scientific works in this thematic. The ANR grant will allow us to meet regularly to compare our viewpoints, our ideas, and work together. We also wish to reinforce our relations with national and international teams working on this subject: invitation of international specialists, travels in France and outside. The possibility to teach less than the french teaching charge will allow us to find more time for our researches. This project will therefore be the occasion to complete
several projects of collaborations inside our group, to let emerge new questions, new national and international scientific collaborations.