It is well known that the study of Liouville geometry, in the case of gradient-like Liouville dynamics, can be reduced to a Morse theoretical description in terms of symplectic handle decompositions. Such examples are called Weinstein. On the other hand, the construction and properties of non-Weinstein Liouville structures are far less understood. The first examples of non-Weinstein Liouville manifolds were constructed by McDuff (1991) and Geiges (1995), which were later on generalized by Mitsumatsu (1995), hinting towards further interactions with hyperbolic dynamics. More specifically, Mitsumatsu proved that given an arbitrary closed 3-manifold equipped with a uniformly hyperbolic (i.e. Anosov) flow, one can construct a non-Weinstein Liouville structure on its 4-dimensional thickenning. The purpose of this talk is to show that Mitsumatsu�s construction in fact provides a framework for a contact/symplectic geometric theory of Anosov flows. Furthermore, we discuss how deeper phenomena from the regularity and stability theory of Anosov flows is inherited in the resulting Liouville structure, portraying strong dynamical and geometric rigidity. In particular, we show that our geometric model gives a characterization of non-vanishing Liouville 4-manifolds with C^1-persistent 3-dimensional skeleton in terms of Anosov dynamics.