Dichotomy for the Hausdorff dimension of the set of nonergodic directions

Abstract

Given an irrational 0<λ<1, we consider billiards in the table P λ formed by a $\tfrac{1}{2}\times1$ rectangle with a horizontal barrier of length $\frac{1-\lambda}{2}$ with one end touching at the midpoint of a vertical side. Let NE (P λ ) be the set of θ such that the flow on P λ in direction θ is not ergodic. We show that the Hausdorff dimension of NE (P λ ) can only take on the values 0 and $\tfrac{1}{2}$ , depending on the summability of the series $\sum_{k}\frac{\log\log q_{k+1}}{q_{k}}$ where {q k } is the sequence of denominators of the continued fraction expansion of λ. More specifically, we prove that the Hausdorff dimension is $\frac{1}{2}$ if this series converges, and 0 otherwise. This extends earlier results of Boshernitzan and Cheung.