On geometrically finite degenerations I: boundaries of main hyperbolic components

Abstract: In this paper, we develop a theory on the degenerations of Blaschke products $\mathcal{B}_d$ to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component $\mathcal{H}_d$ containing $z^d$. We show the closure $\overline{\mathcal{H}_d}$ is not a topological manifold with boundary for $d\geq 4$ by constructing self-bumps on its boundary.