  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.
