Consider the modular surface $X:=\mathbb{H^2}/PSL_2(\mathbb{Z})$. Fix a width-of-cusp parameter $w, 0<w<<1$. Let $B_w$ be the cusp neighborhood of width $w$. (So $w=1$ corresponds to the quotient of $\{\Im(z)>1\}$.)
Let $\gamma$ be a geodesic joining the cusp to itself. It leaves $B_w$, winds around, may re-enter $B_w$ several times, and eventually re-enters $B_w$ for good. We may lift $\gamma$ to a vertical geodesic joining $\infty$ to a point $p/q$ in $[0,1)$.
Let $(l_1, l_2, \ldots, l_n)$ be the sequence of hyperbolic lengths of maximal subsegments of $\gamma$ lying outside $B_w$. Let $[0; a_1, a_2, \ldots, a_m]$ be the continued fraction expansion of $p/q$.
What, if any, is the relationship between these two sequences? I am aware of a relationship between the continued fraction expansion and ``cusp excursions'' expressed in terms of intersections with Farey edges having a common endpoint [R. Moeckel, Ergod. Thy. and Dynam. Sys. (1982), 2, 69-83.] But here I seek information on the geometry {\bf outside} the cusps.
