On Dynamical Parameter Space of Cubic Polynomials with a Parabolic Fixed Point

Abstract

This article focus on the connected locus of the cubic polynomial slice $Per_1(\lambda)$ with a parabolic fixed point of multiplier $\lambda=e^{2\pi i\frac{p}{q}}$. We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum $\mathcal{K}_\lambda$ called the central part in the connected locus is defined. This is the natural analogue to the closure of the main hyperbolic component of $Per_1(0)$. We prove that $\mathcal{K}_\lambda$ is almost a double covering of the filled-in Julia set of the quadratic polynomial $P_{\lambda}(z) = \lambda z+z^2$.