Ziggurats and rotation numbers

I will discuss new rigidity and rationality phenomena

(related to the phenomenon of Arnold tongues) in the theory of

nonabelian group actions on the circle. I will introduce tools that

can translate questions about the existence of actions with prescribed

dynamics, into finite combinatorial questions that can be answered

effectively. There are connections with the theory of Diophantine

approximation, and with the bounded cohomology of free groups. A

special case of this 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. This is joint work with Alden Walker.