Generic Absence of Finite Blocking for Interior Points of Birkhoff Billiards

TL;DR

This paper proves that for most smooth, positively curved billiard tables, most pairs of interior points cannot be blocked by a finite set of points, indicating a generic insecurity in such billiard systems.

Contribution

It establishes that for a typical smooth convex billiard table, almost all interior point pairs are insecure, extending understanding of billiard dynamics and blocking properties.

Findings

Generic pairs of interior points are insecure in most billiard tables.

Finite blocking sets do not exist for most interior point pairs.

Insecurity is a generic property in the space of smooth convex billiard tables.

Abstract

Let x and y be points in a billiard table M that is bounded by a curve sigma. We assume that sigma is a simple closed C^r curve with positive curvature, where r is at least 2. A subset B of M\{x,y} is called a blocking set for the pair (x,y) if every billiard path in M from x to y passes through a point in B. If a finite blocking set exists, the pair (x,y) is called secure in M; if not, it is called insecure. We show that for the generic (in the sense of Baire category) curve sigma, the generic pair of interior points is insecure.