Insecurity for lattice translation surfaces of small genus, with applications to polygonal billiards
connecting geodesics | translation surfaces | billiard orbits | polygons | blocking
A configuration in a riemannian manifold is secure if all connecting geodesics can be blocked by a finite subset of the manifold. A space is insecure if it has insecure configurations. If a space is insecure, what is the size of the set of insecure configurations? We study this problem for translation surfaces. We show that for lattice translation surfaces of genus two almost all configurations are insecure. This implies the insecurity of almost all configurations in certain polygons.