Non-ergodic Z-periodic billiards and infinite translation surfaces

• 成果类型: Article
• 署名作者: Fraczek, Krzysztof; Ulcigrai, Corinna
• 署名单位: Nicolaus Copernicus University; University of Bristol
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0482-z
• 发表日期: 2014
• 页码: 241-298
• 关键词: area-preserving flows; geodesic-flows; veech groups; cohomological equation; Hausdorff Dimension; ergodicity; transformations; recurrence; deviation; DYNAMICS

摘要

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on -periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a -periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent -cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincar, maps of the above systems are associated to non-regular -valued cocycles for interval exchange transformations.