for any point $p$ on $\beta^*$ the geodesic ray beginning at $p$ going out the infinite end of $F$ along $\beta^*$ is a critical ray. We shall refer to $\beta^*$ as the canonical Dirichlet geodesic on $F$ and assume it to be oriented out the infinite end, with $\beta^*(0)\in \alpha_0.$

.1in [*Proof of Proposition \[planar\]*]{} Suppose $S$ is a domain in the complex plane with a single infinite end. We refer to the connected components in the complement of $S$ as complementary components. Let $\Delta_{\infty}$ denote the complementary component corresponding to the infinite end of $S$. $S$ is endowed with the unique hyperbolic metric in its conformal equivalence class. We shall define a sequence of simple closed geodesics on $S$ so that each component in the complement of this set of geodesics on $S$ is a triply connected domain, referred to as a pair of pants, (see [@bas2]).

Let $\Delta$ be a complementary component not containing the point at infinity. Define the distance between $\Delta$ and $\Delta_{\infty}$ as the infimum of the (Euclidean) distances between points in $\Delta$ and $\Delta_{\infty}$ and denote this distance by $d(\Delta, \Delta_{\infty}).$ It is possible to index the complementary components not containing the point at 