Multiple mixing for a class of conservative surface flows

• 成果类型: Article
• 署名作者: Fayad, Bassam; Kanigowski, Adam
• 署名单位: Sorbonne Universite; Universite Paris Cite; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0596-6
• 发表日期: 2016
• 页码: 555-614
• 关键词: interval exchange transformations; nondegenerate fixed-points; horocycle flows; schmidt games; affine forms; joinings; rotations; PROPERTY; RIGIDITY; systems

摘要

Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, which in turn yields higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.