THIBAULT LEFEUVRE

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    about

    ERC project ADG — Analytic techniques in Dynamical systems and Geometry

    Practical information

    Contact: Thibault Lefeuvre, tlefeuvre _at_ imj-prg _dot_ fr

    Timeframe: January 2025 – December 2029.

    Project description

    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 Poincaré series via microlocal techniques. We expect the tools developed in the first part will help to understand part two and three.

    • 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 Livsic-type theorems. Finally, we will study equilibrium measures (existence, uniqueness, and properties) for compact extensions of Anosov diffeos / flows.
    • Geometric and dynamical rigidity. 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.
    • Anosov representations. Spectral theory of Anosov actions on infinite volume manifolds, meromorphic extensions of Poincar ́e series. If finite, we aim to compute the value of these series at the spectral parameter 0.

    Project members

    Events

    1. ADG seminar: monthly seminar at IHP, co-organized with M. Cekić
    2. Dynamics and spectral gaps: workshop in Montpellier
    3. Hyperbolic dynamical systems, geometry and spectral theory: workshop in Peyresq

    (Pre)publications

    2025

    1. Guillarmou’s Normal Operator for Magnetic and Thermostat Flows, Sebastián Muñoz-Thon, Sean Richardson, Preprint
    2. The Pestov identity on the frame bundle and associated homogeneous fibrations, Mihajlo Cekić, Thibault Lefeuvre, Andrei Moroianu, Uwe Semmelmann, Preprint
    3. Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric, Karen Butten, Alena Erchenko, Tristan Humbert, Preprint
    4. Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3, Karen Butt, Alena Erchenko, Tristan Humbert, Thibault Lefeuvre, Amie Wilkinson, Preprint
    5. Unstable entropy for Anosov diffeomorphisms on the 3-torus, Tristan Humbert, Preprint
    6. On Kanai’s conjecture for frame flows over negatively curved manifolds, Louis-Brahim Beaufort, Preprint
    7. Microlocal Analysis in Hyperbolic Dynamics and Geometry (book), Thibault Lefeuvre, to appear in the Cours spécialisés de la SMF
    8. Semiclassical defect measure of magnetic Laplacians on hyperbolic surfaces, Laurent Charles, Thibault Lefeuvre, Preprint
    9. Monotonicity of the Liouville entropy along the Ricci flow on surfaces, Karen Butt, Alena Erchenko, Tristan Humbert, Daniel Mitsutani, Preprint
    10. Correspondence between Pestov and Weitzenböck identities, Mihajlo Cekić, Thibault Lefeuvre, Andrei Moroianu and Uwe Semmelmann, to appear in the Mathematical Proceedings of the Cambridge Philosophical Society
    11. Katok’s entropy conjecture near real and complex hyperbolic metrics, Tristan Humbert, to appear in Duke Mathematical Journal
    12. Measure of maximal entropy for minimal Anosov actions, Tristan Humbert, Preprint