Rotation numbers and symbolic dynamics
David B. Bowman, Ross Flek, Greg Markowsky · Annales Academiae Scientiarum Fennicae Mathematica · 2015
A degree c rotation set in [0, 1] is an ordered set {t 1 , . . ., t q } such that there is a positive integer p such that ct i (mod 1) = t i+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.