Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 227-234

ROTATION NUMBERS AND SYMBOLIC DYNAMICS

David Bowman, Ross Flek and Greg Markowsky

University of Adelaide, School of Mathematical Sciences
Adelaide SA 5005, Australia; david.bowman 'at' adelaide.edu.au

The New School, Eugene Lang College For Liberal Arts, Interdisciplinary Science Department
New York, NY 10011, U.S.A.; flekr 'at' newschool.edu

Monash University, School of Mathematical Sciences
Victoria 3800, Australia; gmarkowsky 'at' gmail.com

Abstract. A degree c rotation set in [0,1] is an ordered set {t₁,...,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.

2010 Mathematics Subject Classification: Primary 37E45, 37E10, 37E15.

Key words: Symbolic dynamics, rotation sets, complex dynamics, doubling map, combinatorial dynamics.

Reference to this article: D. Bowman, R. Flek and G. Markowsky: Rotation numbers and symbolic dynamics. Ann. Acad. Sci. Fenn. Math. 40 (2015), 227-234.

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doi:10.5186/aasfm.2015.4015