# Outer billiards outside regular decagon: periodicity of almost all orbits and existence of an aperiodic point

## TL;DR

This paper proves that almost all orbits in the outer billiard outside a regular octagon are periodic, and it establishes the existence of at least one aperiodic orbit, providing a comprehensive understanding of orbit behavior.

## Contribution

It demonstrates that almost all orbits are periodic and explicitly lists all possible periods, while also proving the existence of an aperiodic orbit outside a regular octagon.

## Key findings

- Almost all orbits are periodic.
- Explicit list of all possible periods.
- Existence of an aperiodic orbit proved.

## Abstract

The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.