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Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensionalmanifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then theconjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosovautomorphism of $\mathbb T^2$ and both flows are constant roof suspensionflows. We deduce several applications. Among them is a new result on rigidityof Anosov diffeorphisms on $\mathbb T^2$ and a new "weighted" marked lengthspectrum rigidity result for surfaces of negative curvature.

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