Dichotomy for the Hausdorff dimension of the set of nonergodic directions

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.

45 pages, 2 figures. Added motivational material requested by referee, in particular, a detailed sketch of the proof of main theorem was added to introduction. The notion of Z-expansion/Liouville direction was added as in order to abstract the proof of the Hausdorff dimension zero result.