Inventiones mathematicae

, Volume 183, Issue 2, pp 337-383

First online:

Dichotomy for the Hausdorff dimension of the set of nonergodic directions

  • Yitwah CheungAffiliated withDepartment of Mathematics, San Francisco State University Email author 
  • , Pascal HubertAffiliated withLATP, case cour A, Faculté de Saint Jérôme
  • , Howard MasurAffiliated withDepartment of Mathematics, University of Chicago

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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.