# Scarcity of periodic orbits in outer billiards

## TL;DR

This paper provides a simple proof that in certain convex outer billiards, the set of period 4 orbits has empty interior unless boundary corners form a parallelogram, and extends results to periods 5 and 6.

## Contribution

It offers a simplified proof of previous results on period 4 orbits and extends analysis to periods 5 and 6 in outer billiards with convex boundaries.

## Key findings

- Period 4 orbits have empty interior unless corners form a parallelogram.
- Results on the structure of period 5 and 6 orbits.
- Simplified proof technique for orbit periodicity properties.

## Abstract

We give a simple proof of our previous result with V. Zharnitsky that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a parallelogram. We also obtain results on period 5 and 6 orbits.