Self-similarity of Ziggurat Fringes and Rigidity of Extremal Free Group Actions on the Circle

We use combinatorial and number theoretic techniques to establish several new rigidity and rationality results in the area of nonabelian group actions on the circle. We show that Calegari-Walker {\em ziggurats} -- i.e.\/ the graphs of extremal rotation numbers associated to positive words in free groups -- have projectively self-similar boundary and satisfy a power law for maximal regions of stability, by giving an explicit formula in a certain range. We give bounds on the complexity of the algorithm used to evaluate the formula and give other bounds characterizing the non-linearity of the extremal representations in some specific cases not at the boundary. Additionally, we establish certain sufficiency criteria for rationality of extremal rotation number in the general case of semi-positive and arbitrary words, using tools from one dimensional dynamics and theory of Diophantine approximations.