  A convenient measure of a map or flow's chaotic action is the topological
entropy. In many cases, the entropy has a homological origin: it is forced by
the topology of the space. For example, in simple toral maps, the topological
entropy is exactly equal to the growth induced by the map on the fundamental
group of the torus. However, in many situations the numerically-computed
topological entropy is greater than the bound implied by this action. We
associate this gap between the bound and the true entropy with 'secondary
folding': material lines undergo folding which is not homologically forced. We
examine this phenomenon both for physical rod-stirring devices and toral linked
twist maps, and show rigorously that for the latter secondary folds occur.
