On dynamical parameter space of cubic polynomials with a parabolic fixed point

Abstract

This article focuses on the connectedness locus of the cubic polynomial slice ${\mathrm{Per}}_1\left(\lambda \right)$ with a parabolic fixed point of multiplier $\lambda =e^{2\pi ip/q}$. We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum $K_{\lambda }$ called the central part in the connectedness locus is introduced. This is the natural analogue to the closure of the main hyperbolic component of ${\mathrm{Per}}_1\left(0\right)$. We prove that $K_{\lambda }$ is almost a double covering of the filled-in Julia set of the quadratic polynomial $p\left(z\right)=\lambda z+z^2$.