The holonomy inverse problem

Let (M, g) be a smooth Anosov Riemannian manifold and C^ the set of its primitive closed geodesics. Given a Hermitian vector bundle E equipped with a unitary connection ^E, we define T^ (E, ^E) as the sequence of traces of holonomies of ^E along elements of C^. This descends to a homomorphism on the additive moduli space A of connections up to gauge T^: (A, ) ^ (C^), which we call the primitive trace map. It is the restriction of the well-known Wilson loop operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map T^ is locally injective near generic points of A when (M) 3. We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of A to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.