AI 中文总结
该研究发展三维双接触塞理论，构造Anosov流等结构稳定流，回答了Béguin-Bonatti-Yu的问题，拓展了相关构造并揭示了双接触塞的Liouville结构性质。
AI 中文摘要
我们在三维空间中发展了双接触塞理论，以构造Anosov流或更一般的结构稳定无奇点流。对于具有可填充Morse-Smale边界层状结构的横向可定向双曲塞，我们证明：每个强横向粘合映射都可通过强横向映射同痕变换，以确定强适配接触形式的符号。所得流是双曲的，当商空间为闭时即为Anosov流。在所述可定向性假设下，这回答了Béguin-Bonatti-Yu提出的问题，该结果还可扩展至粘合无奇点部分双曲流。通过反射实现的Anosov完备化将每个此类双曲塞嵌入其定向二重覆盖上的Anosov流中。进一步应用包括：无鞍周期轨道的结构通用横向定向投影Anosov流的分类、具有指定鞍点数量的Morse-Smale实例、Bonatti-Bowden-Potrie构造的推广——将（吸引型）双曲塞嵌入具有相同拓扑熵的（部分双曲）投影Anosov流，以及在具有受控不变环面动力学的二重覆盖和环面丛上的构造。最后，每个共定向部分双曲双接触塞在其加厚区域上都具有Liouville结构，其Liouville轨道投影为正重参数化的流线；排斥型塞产生四维Liouville域，通常具有混沌骨架。
英文摘要
We develop a theory of bi-contact plugs in dimension three to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a transversely orientable hyperbolic plug with orientable filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted contact form up to sign. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu under the stated orientability assumptions. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic transversely oriented projectively Anosov flows without saddle periodic orbits, Morse-Smale examples with prescribed saddle count, a generalization of the construction of Bonatti-Bowden-Potrie to give embeddings of (attracting) hyperbolic plugs into (partially hyperbolic) projectively Anosov flows with the same topological entropy, and constructions on doubles and torus bundles with controlled invariant torus dynamics. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.
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