On geometrically finite degenerations I: boundaries of main hyperbolic components

Abstract

We develop a theory of quasi post-critically finite degenerations of Blaschke products. This gives us tools to study the boundaries of hyperbolic components of rational maps in higher-dimensional moduli spaces. We use it to obtain a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component ${\mathcal{H}}_d$, i.e., the hyperbolic component in the space of monic and centered polynomials that contains $z^d$. We also show that the closure $\overline{{\mathcal{H}}_d}$ is not a topological manifold with boundary for $d\ge 4$ by constructing self-bumps on its boundary.