Orbit equivalences of pseudo-Anosov flows

• 成果类型: Article
• 署名作者: Barthelme, Thomas; Frankel, Steven; Mann, Kathryn
• 署名单位: Queens University - Canada; Washington University (WUSTL); Cornell University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01332-1
• 发表日期: 2025
• 页码: 1119-1192
• 关键词: quasigeodesic flows; 3-manifolds; foliations; systems

摘要

We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold M are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a tree of scalloped regions; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each pi(1 )(M)-orbit of tree gives a complete invariant of orbit equivalence classes of flows. The framework for the proof is a more general result about Anosov-like actions of abstract groups on bifoliated planes, showing that the homeomorphism type of the bifoliation and the conjugacy class of the action can be recovered from knowledge of which elements of the group act with fixed points. As a consequence, we show that Anosov flows are determined up to orbit equivalence by the action on the ideal boundary of their orbit spaces, and more generally that transitive Anosov-like actions on bifoliated planes are determined up to conjugacy by their actions on the plane's ideal boundary: any conjugacy between two such actions on their ideal circles can be extended uniquely to a conjugacy on the interior of the plane.