Abstract

We say that a pair of points x and y is secure if there exists a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena in both the manifold and the billiard table settings. In approaching this, we study whether a product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security which requires that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we construct a compact C(1) surface which contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, and generates billiard tables with similar blocking behavior.