44 research outputs found
Insecurity for compact surfaces of positive genus
A pair of points in a riemannian manifold is secure if the geodesics
between the points can be blocked by a finite number of point obstacles;
otherwise the pair of points is insecure. A manifold is secure if all pairs of
points in are secure. A manifold is insecure if there exists an insecure
point pair, and totally insecure if all point pairs are insecure.
Compact, flat manifolds are secure. A standing conjecture says that these are
the only secure, compact riemannian manifolds. We prove this for surfaces of
genus greater than zero. We also prove that a closed surface of genus greater
than one with any riemannian metric and a closed surface of genus one with
generic metric are totally insecure.Comment: 37 pages, 11 figure
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure