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

This article focus on the connected locus of the cubic polynomial slice $Pe{r}_1(\lambda )$ 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 $Pe{r}_1(0)$. We prove that ${\mathcal{K}}_{\lambda }$ is almost a double covering of the filled-in Julia set of the quadratic polynomial $P_{\lambda }(z)=\lambda z+z^2$.