Generic injectivity of the X-ray transform

In dimensions 3, we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. Building on earlier work by Guillarmou, Knieper and the second author arXiv: 1806. 04218, arXiv: 1909. 08666, this solves locally the marked length spectrum rigidity conjecture in a neighborhood of a generic Anosov metric. This is the first work going beyond the negatively-curved assumption or dimension 2. Our method, initiated in arXiv: 2008. 09191 and fully developed in the present paper, is based on a perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the holonomy inverse problem, as studied in a subsequent article arXiv: 2105. 06376.