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 $d\ge 4$ by constructing self-bumps on its boundary.

68 pages, 34 figures. Final version, to appear in J. Eur. Math. Soc. (JEMS)

References in corpus (1)

• Hyperbolic components and cubic polynomials

Cited by in corpus (1)

• Combination Theorems in Groups, Geometry and Dynamics