  We establish the existence of new rigidity and rationality phenomena in the
theory of nonabelian group actions on the circle, and introduce tools to
translate questions about the existence of actions with prescribed dynamics
into finite combinatorics. A special case of our theory gives a very short new
proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the
last step in the classification of taut foliations of Seifert fibered spaces.
