On the Cubic Polynomial Slice $Per_1(e^{2\pi i \frac{p}{q}})$

Abstract

We prove that every parabolic component in the cubic polynomial slice $Per_1(e^{2\pi i\frac{p}{q}})$ is a Jordan domain. We also show that the central components of its connected locus are copies of the Julia set of the quadratic polynomial $P_{p/q}(z) = e^{2\pi i\frac{p}{q}}z+z^2$