通过接触几何研究三维流形上Anosov流的有限性
On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry
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中文总结 AI 辅助
该论文通过接触几何证明闭双曲三维流形上Anosov流模同痕轨道等价类数量有限，还给出其数量显式界，并控制非双曲三维流形上Anosov流的动力学。
中文摘要 AI 辅助
继Eliashberg-Thurston和Mitsumatsu的工作后，可将一对横截且反向的接触结构与任意Anosov流关联。我们证明这些接触结构的同痕类完全决定了该流的同痕轨道等价类，这意味着闭双曲三维流形上模同痕轨道等价的Anosov流数量是有限的。该方法还能以通用紧接触结构的数量给出Anosov流轨道等价类数量的显式界。此外，我们证明了伪Anosov流的有限性，其中奇异轨道的补是不可约环面的；在非双曲情形下，我们对JSJ分解的双曲片上Anosov流的动力学进行了控制。
英文摘要
Following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that the number of Anosov flows modulo isotopy orbit equivalence on a closed hyperbolic $3$-manifold is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non-hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.
发表机构
- Leibniz University Hannover（汉诺威莱布尼茨大学）
- Nantes Université（南特大学）
机构由 AI 辅助整理，请以论文原文为准。
