Geometry and Topology Seminar
A hyperbolic component is said to have bounded escape if there is a sequence of rational maps which is degenerating as conjugacy classes, but for any period $p$, the multipliers of periodic points of period $p$ remain bounded. A hyperbolic component is said to have nested Julia set if the Julia set is a Cantor set of nested continuum.
In this talk, we will study the barycentric extensions of rational maps on hyperbolic $3$ space and its geometric limit as branched coverings on a $\R$-tree.
We will use them to show that a hyperbolic component has bounded escape if and only if it has nested Julia set.
We remark that either phenomenon cannot happen for a finitely generated Kleinian group.