## Institute for Mathematical Sciences

## Preprint ims98-4

** S. Zakeri**
* On Dynamics of Cubic Siegel Polynomials*

Abstract: Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional
slice of the cubic polynomials which have a fixed Siegel disk
of rotation number $\theta$, with $\theta$ being a given
irrational number of Brjuno type. Our main goal is to prove
that when $\theta$ 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 prove that the
locus of all cubics with both critical points on the boundary
of their Siegel disk is a Jordan curve, which is in some sense
parametrized by the angle between the two critical points. A
main tool in the bounded type case is a related space of
degree 5 Blaschke products which serve as models for our
cubics. Along the way, we prove several results about the
connectedness locus of these cubic polynomials.

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