Cubic Polynomial Maps with Periodic Critical Orbit, Part I

The parameter space for cubic polynomial maps has complex dimension 2. Its non-hyperbolic subset is a complicated fractal locus which is difficult to visualize or study. One helpful way of exploring this space is by means of complex 1-dimensional slices. This note will pursue such an exploration by studying maps belonging to the complex curve Sp consisting of all cubic maps with a superattracting orbit of period p . Here p can be any positive integer.