  We consider Milnor's "tower algorithm" in the space of piecewise monotone
maps, an iterative algorithm on the space of metrics which unifies, on the one
hand, Thurston's iterative scheme which converges to holomorphic models, and,
on the other hand, the theory of piecewise linear models coming from kneading
theory. We prove that the algorithm converges for unimodal maps of high
entropy, and provide examples where it does not converge for unimodal maps of
low entropy and multimodal maps of higher degree.
