In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, completing the work first initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. A well-known proof of this theorem is due to Bers, who rephrased the problem in terms of extremal quasi-conformal maps between complex surfaces. In joint work with Camille Horbez, we revisit Bers' approach but from the point of view of hyperbolic geometry. This gives a new proof of the classification theorem, as well as new representatives for pseudo-Anosov homeomorphisms as extremal Lipschitz maps between hyperbolic surfaces.