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 [15] a detailed and partially conjectural picture of S n . In the present 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 boundary of each hyperbolic component is a Jordan curve, the points of which are characterized according to the dynamical behaviour of the associated polynomial. The global picture of the connectedness locus is a closed disk together with “limbs” sprouting off it at the cusps of Mandelbrot copies and whose diameter tends to 0 (which corresponds to a qualitative Yoccoz' inequality).