On disjointness properties of some parabolic flows

• 成果类型: Article
• 署名作者: Kanigowski, Adam; Lemanczyk, Mariusz; Ulcigrai, Corinna
• 署名单位: University System of Maryland; University of Maryland College Park; University of Zurich; University of Bristol
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00940-y
• 发表日期: 2020
• 页码: 1-111
• 关键词: ratners property; horocycle flows; rotations; joinings

摘要

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth Arnol'd flows on T-2, focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Mobius orthogonality of time-changes of horocycle flows. In fact, we prove Mobius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol'd flows considered.