Outer billiards outside regular octagon: set of full measure and an aperiodic point

TL;DR

This paper thoroughly analyzes outer billiards outside a regular octagon, demonstrating that periodic points occupy full measure and identifying at least one aperiodic point, thus advancing understanding of the system's dynamics.

Contribution

It provides a complete description of periodic points, proves they form a full measure set, and identifies an explicit aperiodic point in the system.

Findings

Periodic points form a set of full measure outside the octagon.

All periodic points and their periods are explicitly described.

An explicit aperiodic point is found.

Abstract

In this paper, outer billiards outside regular octagon are studied in details. We described all periodic points and their periods; also, we proved that the periodic points form a set of full measure outside the octagon and found an aperiodic point.