is conformally equivalent to a connected domain in ${{\mathbb{C}}}$ with a single infinite end.

Let $S$ be a hyperbolic surface and $\sigma :[0,\infty)\rightarrow
S$ a geodesic ray. Here and henceforth, all geodesics are parameterized by arc length. Define the function $\Delta_{\sigma}(t)= t - d_{S}(\sigma(0),\sigma(t)) $, where $d_{S}$ denotes distance as defined by the hyperbolic metric on $S$. The ray $\sigma$ is then said to be [*horocyclic, critical, [*or*]{} subcritical*]{} if $\Delta_{\sigma}$ is respectively, unbounded, zero, or nonzero but bounded. A critical ray may be said to travel directly out an end of $S,$ and a subcritical ray may be said to travel almost directly out an end. It is known (see, [@haas; @Waterman]) that critical rays are simple and subcritical rays are simple beyond some point.

Let ${\sf E} = \{{\sf E}_i\}$ be an end of a flute $F$. It is known from [@haas] that for any $p\in F$ there is a critical ray with initial point $p$ that goes out the end [E]{}. When the end ${\sf E}$ is a finite end, critical and subcritical rays that go out the end ${\sf E}$ are called [*finite*]{} critical or subcritical rays; otherwise, if the end ${\sf E}$ is an infinite 