Dichotomy for the Hausdorff dimension of the set of nonergodic directions

Yitwah Cheung (1) , Pascal Hubert (2) , Howard Masur (3)

1 Department of Mathematics [San Francisco]
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 Department of Mathematics

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.

Mots clés: Hausdorff dimension; billiards; nonergodic directions; Z-expansions; Liouville directions; Loops; slits; summable cross-products; Cantor set construction; foliations

Domaines: Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Théorie des nombres [math.NT]

Yitwah Cheung, Pascal Hubert, Howard Masur. Dichotomy for the Hausdorff dimension of the set of nonergodic directions. Inventiones Mathematicae, 2011, 183 (2), pp.337-383. ⟨10.1007/s00222-010-0279-2⟩. ⟨hal-01299639⟩