end, they are called [*infinite*]{}. Furthermore, a critical or a subcritical ray always goes out some end of $F$ . We shall primarily be interested in the infinite critical and subcritical rays on $F$.

Given a flute surface $F\in \mathcal{  F},$ and an integer $n\geq 0,$ let $\alpha_n$ denote the simple closed geodesic on $F$ in the free homotopy class of the curve $$t\rightarrow (e^{it} , n+\frac{1}{2}), 0\leq
t\leq 2\pi.$$ We shall refer to a geodesic $\alpha_n$ as a [*dividing loop*]{} on $F.$ Note that these are well defined with the possible exception of $\alpha_0$, which exists only if the end corresponding to the ideal boundary $S^1\times \{0\}$ is a hole. In what follows we shall always take this end to be a hole so that $\alpha_0$ does exist.

Let $\beta^* = \{(-1,t)\mid t\in (0,\infty)\}$ and for integers $n\geq 0$ let $\gamma_n =
\{(1,t)\mid n < t< n+1\}$. Suppose in addition that $F$ is an untwisted flute. Then $\beta^*$ and $\gamma_n$ are geodesics since they are fixed by the isometry $r.$ We shall refer to the geodesics $\gamma_n$ as the $\gamma$-curves of $F$. For each $n$ the geodesics $\beta^*$ and $\gamma_n$ are both orthogonal to the dividing loop $\alpha_n$. Also, 