Résume | On a negatively-curved Riemannian manifold (and more generally on an Anosov manifold i.e. a Riemannian manifold whose geodesic flow is uniformly hyperbolic), the celebrated Burns-Katok conjecture asserts that the marked length spectrum, namely the length of all closed geodesics marked by the free homotopy of the manifold, should determine the metric (up to isometries). In a similar fashion, on an Anosov manifold, given a vector bundle equipped with a (unitary) connection,one can wonder if the data of the holonomy of the connection along closed geodesics (up to conjugacy) determines the connection (up to gauge-equivalence). This is called the holonomy inverse problem and turns out to be a very rich question, as it brings together different fields of mathematics: microlocal analysis, hyperbolic dynamical systems, theory of Pollicott-Ruelle resonances & Kähler geometry. I will review some recent progress that have been achieved on this question. Joint work with Mihajlo Cekić. |