Inventiones mathematicae

, Volume 183, Issue 2, pp 337–383

Dichotomy for the Hausdorff dimension of the set of nonergodic directions



Given an irrational 0<λ<1, we consider billiards in the table Pλ 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 (Pλ) be the set of θ such that the flow on Pλ in direction θ is not ergodic. We show that the Hausdorff dimension of NE (Pλ) 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 {qk} 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.LATP, case cour AFaculté de Saint JérômeMarseille Cedex 20France
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations