Analytic methods for Dynamical systems and Geometry

The aim of this project is to study a broad class of dynamical systems by using tools from the fields of harmonic analysis and PDEs (semiclassical, microlocal analysis), and to apply these new results to a variety of problems of geometric origin.In a first part, we will mainly focus on systems exhibiting a weak hyperbolic behaviour (partially, non-uniformly hyperbolic systems) for which analytic techniques are far less understood compared to the uniformly hyperbolic setting.

We plan to study statistical properties of such systems, and the regularity of solutions to transport / cohomological equations.

Then, we will address rigidity questions in geometry and dynamics such as marked length spectrum or boundary / lens rigidity, Katok's entropy conjecture.

In a third part, we aim to study Anosov representations and meromorphic extension of related Poincar series via microlocal techniques.

We expect the tools developed in the first part will help to understand part two and three. 1) Statistics of weakly hyperbolic flows, study of transport questions. Ergodicity, mixing, polynomial or exponential mixing of partially hyperbolic / non-uniformly hyperbolic systems. We also plan to study cohomological equations and prove Livv sic-type theorems.

Finally, we will study equilibrium measures (existence, uniqueness, and properties) for compact extensions of Anosov diffeos / flows.2) Geometric and dynamical rigidity for flows / actions.

Marked or unmarked length spectrum rigidity conjecture for (non-)uniformly hyperbolic geodesic flows, lens and boundary rigidity, Katok's entropy rigidity conjecture, rigidity of Anosov actions (Katok-Spatzier's conjecture), Kanai's regularity conjecture. 3) Anosov representations. Spectral theory of Anosov actions on infinite volume manifolds, meromorphic extensions of Poincar series.

If finite, we aim to compute the value of these series at the spectral parameter 0.