Projects per year
Abstract
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.
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.
Original language  English 

Pages (fromto)  227234 
Number of pages  8 
Journal  Annales Academiae Scientiarum Fennicae Mathematica 
Volume  40 
DOIs  
Publication status  Published  2015 
Keywords
 Symbolic dynamics
 rotation sets
 complex dynamics
 doubling map
 combinatorial dynamics
Projects
 2 Finished

Planar Brownian motion and complex analysis
Australian Research Council (ARC)
2/01/14 → 11/01/17
Project: Research

Finite Markov chains in statistical mechanics and combinatorics
Garoni, T., Collevecchio, A. & Markowsky, G.
Australian Research Council (ARC)
2/01/14 → 31/12/17
Project: Research