The problem of gluing two hyperbolic 3-manifolds along a common boundary, and hyperbolizing the result, led Thurston to consider a map from the Teichmuller space of the boundary to itself, translating the hyperbolization problem to a fixed-point problem for this map. The geometry of the map then becomes entwined with the question of understanding the variations of hyperbolic structures for different boundary gluings. Thurston stated informally a theorem on the bounded-image properties of (iterates of) the map, which would improve our understanding of this process. In work in progress with Ken Bromberg and Dick Canary, we are trying to produce a proof of this theorem. I will try to give an overview of this interesting corner of deformation theory.