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

This article focuses on the connectedness locus of the cubic polynomial slice (Formula presented.) with a parabolic fixed point of multiplier (Formula presented.). We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum (Formula presented.) called the central part in the connectedness locus is introduced. This is the natural analogue to the closure of the main hyperbolic component of (Formula presented.). We prove that (Formula presented.) is almost a double covering of the filled-in Julia set of the quadratic polynomial (Formula presented.).