Birkhoff billiards are insecure

TL;DR

This paper proves that in any smooth, bounded planar billiard, there are pairs of points for which no finite set of points can block all billiard trajectories, revealing an inherent insecurity in such systems.

Contribution

It establishes the universal insecurity of all smooth, bounded planar billiards, a novel result in billiard dynamics.

Findings

Existence of pairs of points with unblocked trajectories

Universal insecurity of smooth, bounded billiards

No finite blocking set can block all trajectories between certain points

Abstract

We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points $A,B$ such that no finite set of points can block all billiard trajectories from $A$ to $B$.