  A central question in dynamics is whether the topology of a system determines
its geometry. This is known as rigidity. Under mild topological conditions
rigidity holds for many classical cases, including: Kleinian groups, circle
diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps
with a break point. More recent developments show that under similar
topological conditions, rigidity does not hold for slightly more general
systems. In this paper we state a conjecture which describes how topological
classes are organized into rigidity classes.
