Generic dynamical properties of connections on vector bundles

v1: 24 pages. v2: 40 pages, paper is now significantly upgraded. We now prove a generic triviality of twisted CKTs result for the endomorphism bundle. Moreover, we prove that in the Anosov geodesic flow setting, generically parallel transport along geodesics does not preserve non-trivial subbundlesGiven a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $\dim(M) \geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla^{\mathrm{End}(\mathcal{E})}$ on the endomorphism bundle $\mathrm{End}(\mathcal{E})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically $\textit{opaque}$, namely there are no non-trivial subbundles of $\mathcal{E}$ which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called $\textit{operators of uniform divergence type}$, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case)