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<p>In recent years deep connections have emerged between contact geometry and Anosov dynamics. Mitsumatsu first noticed that a vector field defining an Anosov flow belongs to the intersection of a pair of transverse contact structures rotating toward each other along the flow-lines. We call these pairs bicontact structures and the flows defined by their intersections <i>projectively Anosov</i> or flows with <i>dominated splitting</i>. Recently Hozoori (J Fixed Point Theory Appl 27(2):37, 2025) gave a characterization of bicontact structures supporting Anosov flow in terms of the Reeb dynamics of the defining contact structures. This link between contact geometry and Anosov dynamics allows us to exploit the technology developed by contact geometers to define new Dehn-type surgeries in hyperbolic dynamics. This survey is an attempt to present in a unified and coherent way the existing construction of Dehn-type surgeries on bicontact structures in the light of the most recent developments of the theory. We highlight the relations with Goodman construction on (pseudo)-Anosov flows and extend it to the realm of projectively Anosov flows. We also describe some novel applications in the context of Anosov flows and reinterpret a classic open question in 3-dimensional Anosov dynamics using the framework of bicontact geometry. We refer to Salmoiraghi (Surgery on Anosov flows using bi-contact geometry. Preprint, 2021) and Salmoiraghi (Goodman surgery and projectively Anosov flows. Preprint, 2021) for the existing applications of bicontact surgery.</p>

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Surgery operations on Anosov flows via contact geometry: a survey

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Federico Salmoiraghi

摘要

In recent years deep connections have emerged between contact geometry and Anosov dynamics. Mitsumatsu first noticed that a vector field defining an Anosov flow belongs to the intersection of a pair of transverse contact structures rotating toward each other along the flow-lines. We call these pairs bicontact structures and the flows defined by their intersections projectively Anosov or flows with dominated splitting. Recently Hozoori (J Fixed Point Theory Appl 27(2):37, 2025) gave a characterization of bicontact structures supporting Anosov flow in terms of the Reeb dynamics of the defining contact structures. This link between contact geometry and Anosov dynamics allows us to exploit the technology developed by contact geometers to define new Dehn-type surgeries in hyperbolic dynamics. This survey is an attempt to present in a unified and coherent way the existing construction of Dehn-type surgeries on bicontact structures in the light of the most recent developments of the theory. We highlight the relations with Goodman construction on (pseudo)-Anosov flows and extend it to the realm of projectively Anosov flows. We also describe some novel applications in the context of Anosov flows and reinterpret a classic open question in 3-dimensional Anosov dynamics using the framework of bicontact geometry. We refer to Salmoiraghi (Surgery on Anosov flows using bi-contact geometry. Preprint, 2021) and Salmoiraghi (Goodman surgery and projectively Anosov flows. Preprint, 2021) for the existing applications of bicontact surgery.