I've asked this question on mathoverflow, but I thought it may be helpful to restate here since I guess it might be a not so hard question to solve.
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) belonging to complete geodesics with at most $k$ self intersections are nowhere dense and of Hausdorff dimension $1$.
In section two of this article, they introduce non-exceptional geodesics relative to a fundamental region $R$, which are the complete geodesics that don't pass through the vertices of the fundamental region $R$ or don't have self intersections on the vertices. They claim that we can take $3$ fundamental regions so that any simple complete geodesic is non-exceptional relative to at least one of them.
I was trying to justify this part for myself, and I wasn't successful. Could it be possible for someone to help me prove this part.