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 qk+1)/qk is finite, where qk is the the denominator of the kth convergent of the length of the barrier.
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