Hyperbolic components of polynomials with a fixed critical point of maximal order

Abstract

For the study of the 2-dimensional space of cubic polynomials, J. Milnor considers the complex 1-dimensional slice S_n of the cubic polynomials which have a super-attracting orbit of period n. He gives in [M4] a detailed conjectural picture of S_n. In this article, we prove these conjectures for S_1 and generalize these results in higher degrees. In particular, this gives a description of the closures of the hyperbolic components and of the Mandelbrot copies sitting in the connectedness locus. We prove that the closure of hyperbolic components is a Jordan curve, the points of which are characterized according to their dynamical behaviour. The global picture of the connectedness locus is a closed disk together with ``limbs'' sprouting off at the cusps of Mandelbrot copies and whose diameter tends to 0 (which corresponds to the Yoccoz inequality in the quadratic case). [[M4] J. Milnor - On cubic polynomials with periodic critical point, preprint (1991).]