The parameter space poly2 for the family of dynamical systems (Pc)c2C, where Pc : C ! C are the quadratic polynomials given by Pc(z) = z2 + c, is well understood. It is divided into the Mandelbrot set M, corresponding to maps with connected Julia sets, and its complement, corresponding to maps with totally disconnected Julia sets.

The space poly2 is naturally embedded as the “central slice” in the larger moduli space of all quadratic rational maps, calledM2, where two maps belong to the same equivalence class if they are conjugate by some M¨obius transformation. The moduli spaceM2 is isomorphic to C2 and there is a subset, isomorphic to D×C, of so-called polynomial–like maps, wherein each slice is well understood as a quasi–conformal copy of poly2. Whereas hyperbolic dynamical systems are conjecturally dense in the polynomial–like locus, its boundary consists of slices where the dynamical systems are all non–hyperbolic.

In the talk I will give a more elaborate description of the above mentioned properties, and present results about the structure of those slices in the boundary of the polynomial–like locus, that consist of maps with a parabolic fixed point.