Siegel and Cremer building blocks for polynomial Julia sets
by
John C. Mayer
University of Alabama at Birmingham

Suppose that J is the connected Julia set of a polynomial P of degree d ³ 2. For simplicity, let 0 be a fixed irrationally indifferent point of P with derivative exp(2pi q). If P is linearizable at 0 we are in the Siegel case and there is a maximal disk D of linearizability with boundary S. If P is not linearizable at 0, we are in the Cremer case, and set S={0}. We make a topological assumption about J: assume J is hereditarily decomposable (this can be weakened).

On the circle of prime ends (external rays) of J, consider the map s_d:S¹®S¹ defined by s_d(t)=t mod 1. We investigate the connection between an invariant Cantor set C in the circle of prime ends with a well-defined irrational rotation number q under s_d|_C and an invariant nowhere dense (in J) continuum B É S which we call the Siegel (respectively, Cremer) building block of J associated with the irrationally indifferent fixed point 0. (B is defined by prime end impressions.) The issue is complicated by the fact that for degree d > 2, there are uncountably many Cantor sets in S¹ with rotation number q.

Date received: March 1, 2005