Dichotomy for the Hausdorff dimension of the set of nonergodic directions

• 成果类型: Article
• 署名作者: Cheung, Yitwah; Hubert, Pascal; Masur, Howard
• 署名单位: California State University System; San Francisco State University; Aix-Marseille Universite; University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0279-2
• 发表日期: 2011
• 页码: 337-383
• 关键词: foliations

摘要

Given an irrational 0 < lambda < 1, we consider billiards in the table P-lambda formed by a 1/2 x 1 rectangle with a horizontal barrier of length 1-lambda/2 with one end touching at the midpoint of a vertical side. Let NE(P-lambda) be the set of theta such that the flow on P-lambda in direction theta is not ergodic. We show that the Hausdorff dimension of NE(P-lambda) can only take on the values 0 and 1/2, depending on the summability of the series Sigma(k) log log(qk+1)/q(k) where {q(k)} is the sequence of denominators of the continued fraction expansion of lambda. More specifically, we prove that the Hausdorff dimension is 1/2 if this series converges, and 0 otherwise. This extends earlier results of Boshernitzan and Cheung.