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. A preliminary draft of this paper, based on conversations with Branner, Douady and Hubbard, was circulated in 1991 but not published. The present version tries to stay close to the original; however, there has been a great deal of progress in the intervening years. (See especially (Faught 92), (Branner and Hubbard 92), (Branner 93), (Roesch 99, 06), and (Kiwi 06).) In particular, a number of conjectures in the original have since been proved; and new ideas have made sharper statements possible. We begin with the period 1 case. Section 2 studies the dynamics of a cubic polynomial map F which has a superattracting fixed point, and whose Julia set J(F ) is connected. The filled Julia set of any such map consists of a central Fatou component bounded bounded by a Jordan curve, together with various limbs sprouting off at internal angles which are explicitly described.