Weakly mixing polygonal billiards

• 成果类型: Article
• 署名作者: Chaika, Jon; Forni, Giovanni
• 署名单位: Utah System of Higher Education; University of Utah; University System of Maryland; University of Maryland College Park
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.203.3.1
• 发表日期: 2026-05
• 页码: 695-735
• 关键词: Billiards in polygons; rational polygonal billiards; weak mixing flows; teichmuller flow; Moduli space of Abelian differentials; INTERVAL EXCHANGE TRANSFORMATIONS; periodic-orbits; growth-rate; FLOWS; ergodicity; invariant; SURFACES

摘要

We prove that there exists a G delta dense set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues.