  Building upon the work of Mitsumatsu and Hozoori, we establish a complete
homotopy correspondence between three-dimensional Anosov flows and certain
pairs of contact forms that we call Anosov Liouville pairs. We show a similar
correspondence between projectively Anosov flows and bi-contact structures,
extending the work of Mitsumatsu and Eliashberg-Thurston. As a consequence,
every Anosov flow on a closed oriented three-manifold $M$ gives rise to a
Liouville structure on $\mathbb{R} \times M$ which is well-defined up to
homotopy, and which only depends on the homotopy class of the Anosov flow. Our
results also provide a new perspective on the classification problem of Anosov
flows in dimension three.
