Dynamics of Cubic Siegel Polynomials

Abstract

We study the one-dimensional parameter space of cubic polynomials in the complex plane which have a fixed Siegel disk of rotation number θ, where θ is a given irrational number of Brjuno type. The main result of this work is that when θ is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also show that these boundaries vary continuously as one moves in the parameter space. This is most nontrivial near the set of cubics with both critical points on the boundary of their Siegel disk. We prove that this locus is a Jordan curve in the parameter space. Most of the techniques and results can be generalized to polynomials of higher degrees.