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On the existence of supporting broken book decompositions for contact forms in dimension 3

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Abstract

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the tight sphere or on a tight lens space. Moreover we get that if M is a closed oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy.

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Acknowledgements

We thank Oliver Edtmair, Umberto Hryniewicz, Michael Hutchings, Rohil Prasad and the anonymous referees for useful exchanges and suggestions. V. Colin thanks the ANR Quantact for its support, P. Dehornoy thanks the ANR projects IdEx UGA and Gromeov for their supports, and A. Rechtman thanks the IdEx Unistra, Investments for the future program of the French Government. This project started during the Matrix event “Dynamics, Foliations and Geometry in dimension 3” held at Monash University in 2018. We thank these institutions for their support.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, LMJL, CNRS, Nantes Université, 44000, Nantes, France

    Vincent Colin

  2. CNRS, Institut Fourier, Univ. Grenoble Alpes, 38000, Grenoble, France

    Pierre Dehornoy

  3. Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 Rue René Descartes, 67084, Strasbourg, France

    Ana Rechtman

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  1. Vincent Colin
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Correspondence to Vincent Colin.

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Colin, V., Dehornoy, P. & Rechtman, A. On the existence of supporting broken book decompositions for contact forms in dimension 3. Invent. math. 231, 1489–1539 (2023). https://doi.org/10.1007/s00222-022-01160-7

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