Rotation numbers and symbolic dynamics

A degree c rotation set in [0, 1] is an ordered set {t1, . . . , tq} such that there is a
positive integer p such that cti(mod 1) = ti+p(mod q) for i = 1, . . . , q. The rotation number of the set is defined to be p/q. Goldberg has shown that for any rational number p/q ∈ (0, 1) there is a unique quadratic rotation set with rotation number p/q. 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 dynamics. We also deduce a number of additional results from our method, including a characterization of the values of the elements of the rotation sets.