Blocking: New examples and properties of products

TL;DR

This paper explores the concept of security in geodesic and billiard configurations, introducing midpoint security, and demonstrates how product configurations can be secure or insecure, providing new examples and properties.

Contribution

It introduces the concept of midpoint security, proves that products of midpoint secure configurations are midpoint secure, and provides the first example of an insecure product of secure configurations.

Findings

Products of midpoint secure configurations are midpoint secure.

Existence of a compact surface with secure but not midpoint secure configurations.

First example of an insecure product of secure configurations.

Abstract

We say that a pair of points x and y is secure if there exist 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 both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security that imposes 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 give an example of a compact C^1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard table examples.