Dichotomy for the Hausdorff dimension of the set of nonergodic directions

TL;DR

This paper investigates the Hausdorff dimension of the set of nonergodic directions in a specific billiard table, establishing a dichotomy based on the barrier length's number-theoretic properties.

Contribution

It proves that the Hausdorff dimension of nonergodic directions is either 0 or 1/2, depending on a precise Diophantine condition on the barrier length.

Findings

Hausdorff dimension of nonergodic directions is either 0 or 1/2.

Dimension equals 1/2 if and only if the barrier length satisfies P'erez Marco's condition.

The result links geometric billiard dynamics with number-theoretic properties of the barrier length.

Abstract

We consider billiards in a (1/2)-by-1 rectangle with a barrier midway along a vertical side. Let NE be the set of directions theta such that the flow in direction theta is not ergodic. We show that the Hausdorff dimension of the set NE is either 0 or 1/2, with the latter occurring if and only if the length of the barrier satisfies the condition of P'erez Marco, i.e. the sum of (loglog q_{k+1})/q_k is finite, where q_k is the the denominator of the kth convergent of the length of the barrier.