In this talk we study the geometry and topology of sets of zeros of power series with certain coefficients and how these properties vary as the coefficients change. Dynamical interest in such sets began with Barnsley and Harrington, who observed that they could also be viewed as a parameter space for certain families of linear iterated function systems. In this talk we discuss various topological, geometric, and dynamical features of such sets. In particular, we study questions of rigidity - to what extent the set of roots uniquely determines the coefficients.