Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a set of measure zero. In other words, the elements of A are scattered throughout the octagon almost everywhere. Suppose a straight ray were drawn from the origin to the edge of $\mathbb{D}$, i.e. going to infinity.

Keeping in mind that the tiled octagons are identified, so the elements of A remain in the same locations for each octagon, is it possible to have such a ray that is not periodic on the octagons AND never intersects A?

If not, can any geodesic be drawn on these tiled octagons (again, starting from the origin and ending at the edge) in order to not be periodic and avoid intersecting A?

Thank you.