The main cubioid

The connectedness locus in the parameter space of quadratic polynomials is called the Mandelbrot set. A good combinatorial model of this set is due to Thurston. By definition, the principal hyperbolic domain of the Mandelbrot set consists of parameter values, for which the corresponding quadratic polynomials have an attracting fixed point. The closure of the principal hyperbolic domain of the Mandelbrot set is called the main cardioid. Its topology is completely described by Thurston's model. Less is known about the connectedness locus in the parameter space of cubic polynomials. In this paper, we discuss cubic analogues of the main cardioid and establish relationships between them.