Rotation numbers and symbolic dynamics

Abstract

A degree c rotation set in (0;1) is an ordered setft1;:::;tqg such that there is a positive integer p such that cti(mod 1) = t i+p(mod q) for i = 1;:::;q. The rotation number of the set is defined to be p . Goldberg has shown that for any rational number p 2 (0;1) there is a unique quadratic rotation set with rotation number p . This result was used by Goldberg and Milnor to study Julia sets of quadratic polynomials (8). In this work, we provide an alternate proof of Goldberg's result which employs symbolic dynam- ics. We also deduce a number of additional results from our method, including a characterization of the values of the elements of the rotation sets. p q . Goldberg's proof relies heavily on geometric arguments. It is noteworthy, how- ever, that Goldberg's result can be stated entirely without reference to complex dynamics, or any sort of geometric interpretation. As such, it is logical to seek an alternate proof which dispenses with this additional machinery. Our goal in this note is to provide a proof of Goldberg's main theorem on quadratic rotation sets using purely symbolic arguments. We also show that such an approach can simplify some of the auxiliary results in Goldberg and Milnor's paper in the degree two case.