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Anosov flows and Birkhoff sections

Abstract : We study the Anosov flows, in dimension 3. These flows have interesting chaotic behaviors, more precisely they have hyperbolic behaviors on the neighborhood of there orbits. To understand these flows, we use some surfaces transverse to the flows, called Birkhoff sections. Thanks to the so called first return map on one Birkhoff section, the dynamic of the flow is partially encoded by the dynamic of a homeomorphism of a surface. This discret dynamic being in dimension 2.In a first part, we explicitly compute le first return maps of a family of Birkhoff sections with a fixed boundary. It allows one to compare the first return maps of the flow on several Birkhoff sections. In a second part, we study the boundaries of the Birkhoff sections and their orientations. A Birkhoff section is interpreted as a transverse cobordism of the flow, from its positive boundary to its negative boundary. Two natural notions appear: the twisted flows (which admit some transverse null-cobordism) and the primitive orbits of these flows (which are the positive boundary of no transverse cobordism). We study these notions, which contain some information on the topology of the flow and of the global manifold.
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Submitted on : Tuesday, January 4, 2022 - 1:55:08 PM
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Théo Marty. Anosov flows and Birkhoff sections. Algebraic Topology [math.AT]. Université Grenoble Alpes [2020-..], 2021. English. ⟨NNT : 2021GRALM028⟩. ⟨tel-03510071⟩

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