Insecure configurations in lattice translation surfaces, with applications to polygonal billiards

Abstract

A configuration (i.e., a pair of points) in a Riemannian space $X$is secure if all connecting geodesics can be blocked by a finitesubset of $X$. A space is secure if all of its configurations aresecure. Secure spaces seem to be rare.  &nbspIf $X$ is an insecure space, it is natural to ask how big the setof insecure configurations is. We investigate this problem forflat surfaces, in particular for translation surfaces andpolygons, from the viewpoint of measure theory.  &nbspHere is a sample of our results. Let $X$ be a lattice translation surface or a lattice polygon.Then the following dichotomy holds: i) The surface (polygon) $X$ is arithmetic. Then allconfigurations in $X$ are secure;ii) The surface (polygon) $X$ is nonarithmetic. Then almost all configurations in $X$ are insecure.