Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity

Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensional manifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then the conjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosov automorphism of ${\mathbb{T}}^2$ and both flows are constant roof suspension flows. We deduce several applications. Among them is a new result on rigidity of Anosov diffeorphisms on ${\mathbb{T}}^2$ and a new “weighted” marked length spectrum rigidity result for surfaces of negative curvature.