Ziggurats and rotation numbers

TL;DR

This paper introduces new rigidity and rationality phenomena in nonabelian group actions on the circle, providing tools to connect dynamics questions with finite combinatorics and offering a new proof of a key classification theorem.

Contribution

It establishes novel phenomena in group actions on the circle and develops tools linking dynamics to finite combinatorics, including a new proof of Naimi's theorem.

Findings

New rigidity phenomena in nonabelian group actions

Tools translating dynamics questions into combinatorics

A concise proof of Naimi's theorem

Abstract

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.