Dynamics of Cubic Rational Maps Under Certain Constraints on Critical Points

There is a neat dichotomy for the Julia sets of quadratic rational maps; that is, they are either connected or a Cantor set. In contrast to the quadratic case, the Julia sets of rational maps of of degree ≥ 3 have more variations. In this project, we study the Julia sets of cubic rational maps under some constraints. We first extend the Julia set dichotomy to the cubic rational maps with all critical points escaping to an attracting fixed point. Then we consider two more classes of cubic rational maps: one class consists of the cubic rational maps with two attracting fixed points and the other class is comprised of the cubic rational maps with two critical points on a 2-cycle. We obtain the following results: 1. There exists a map in the first class having a Herman ring of period 1, but no map in this class has a Herman ring of period ≥ 2. 2. Those maps in the first class without Herman rings have only two types of Julia set: either connected or a semi-Cantor set. 3. Any map in the second class cannot have Herman rings in its Fatou set. 4. The Julia sets of maps in the second class come in three varieties: connected, a semi- Cantor set, or disconnected but not a semi-Cantor set