Periodic orbits in outer billiards

TL;DR

This paper proves that in outer billiards with a piecewise smooth convex boundary, the set of 4-period orbits has no interior points unless four boundary corners form a parallelogram.

Contribution

It establishes a condition under which 4-period orbits are sparse in outer billiards, advancing understanding of orbit structure in these dynamical systems.

Findings

4-period orbits have empty interior in the specified setting

No four corners form a parallelogram in the boundary

Provides conditions for orbit distribution in outer billiards

Abstract

It is shown that the set of 4-period orbits in outer billiard with piecewise smooth convex boundary has an empty interior, provided that no four corners of the boundary form a parallelogram.