## 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 |
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Pages (from-to) | 227-234 |

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