Hausdorff dimension of three-period orbits in Birkhoff billiards

TL;DR

This paper establishes an upper bound of one on the Hausdorff dimension of three-period orbits in classical billiards and characterizes the geometric structure of the set when this bound is attained.

Contribution

It proves that the Hausdorff dimension of three-period orbits is at most one and describes the tangent line structure if the dimension equals one.

Findings

Hausdorff dimension of three-period orbits is at most one

If the dimension is one, the set has a tangent line at almost every point

Provides geometric insight into the structure of these orbits

Abstract

We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.