Institute for Mathematical Sciences

Preprint ims90-14

I. L. R. Goldberg
II. L. R. Goldberg and J. Milnor
Fixed Points of Polynomial Maps
I. Rotation Sets
II. Fixed Point Portraits

Abstract: I. We give a combinatorial analysis of rational rotation subsets of the circle. These are invariant subsets that have well-defined rational rotation numbers under the standard self-covering maps of S1. This analysis has applications to the classification of dynamical systems generated by polynomicals in one complex variable.

II. Douady, Hubbard and Branner have introduced the concept of a "limb" in the Mandelbrot set. A quadratic map f(z)=z2+c belongs to the p/q limb if and only if there exist q external rays of its Julia set which land at a common fixed point of f, and which are permuted by f with combinatorial rotation number p/q in Q/Z, p/q ≠ 0). (Compare Figure 1 and Appendix C, as well as Lemma 2.2.) This note will make a similar analysis of higher degree polynomials by introducing the concept of the "fixed point portrait" of a monic polynomial map.

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