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GDMSeminar: Abstracts of the Talks (2021/7)


Ana Rechtman (Strasbourg)
Broken books and Reeb dynamics in dimension 3

Tuesday, April 13, 2021, 15:00 GMT

Abstract:  Given a contact structure on a closed 3-manifold, Giroux’s correspondance gives an adapted open book decomposition. This implies that there exists a Reeb vector that is tangent to the binding and transverse to the interior of the pages. For this vector field, each page is a Birkhoff section and the dynamics of the flow can be studied from the first return map. This correspondance is unsatisfactory when one wants to study all the Reeb vector fields associated to a contact structure.

In collaboration with V. Colin and P. Dehornoy, we proved that every non-degenerate Reeb vector field on a closed 3-manifold is adapted to a broken book decomposition (a generalisation of open book). This construction gives a system of transverse surfaces with boundary and allows to establish results on the dynamics of the vector field. We proved, for example, that any non-degenerate Reeb vector field has either two or infinitely many periodic orbits.