  In dimensions $\geq 3$, 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 $\textit{holonomy inverse problem}$, as
studied in a subsequent article [arXiv:2105.06376].

Analysis of PDEs;Differential Geometry;Dynamical Systems;Spectral Theory