Analytic methods for Dynamical systems and Geometry

Introduction

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.

Focus Areas

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.

Statistical Properties

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

Rigidity Questions

Then, we will address rigidity questions in geometry and dynamics such as:

• Marked length spectrum or boundary/lens rigidity
• Katok's entropy conjecture

Anosov Representations

In a third part, we aim to study Anosov representations and the meromorphic extension of related Poincaré series via microlocal techniques. We expect the tools developed in the first part will help to understand parts two and three.

Detailed Objectives

1. Statistics of weakly hyperbolic flows:

   • Study of transport questions
   • Ergodicity, mixing, polynomial or exponential mixing of partially hyperbolic/non-uniformly hyperbolic systems
   • Study of cohomological equations and prove Livšic-type theorems
   • Study of equilibrium measures (existence, uniqueness, and properties) for compact extensions of Anosov diffeomorphisms/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 Poincaré series
   • If finite, we aim to compute the value of these series at the spectral parameter 0.