Straight line on the Poincare disk hitting points almost everywhere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:23:41Z http://mathoverflow.net/feeds/question/34578 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34578/straight-line-on-the-poincare-disk-hitting-points-almost-everywhere Straight line on the Poincare disk hitting points almost everywhere gomaff 2010-08-05T02:57:08Z 2010-08-05T04:29:19Z <p>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. </p> <p>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?</p> <p>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?</p> <p>Thank you.</p>