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

Vincent Colin (1) , Pierre Dehornoy (2) , Ana Rechtman (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IF - Institut Fourier
3 IRMA - Institut de Recherche Mathématique Avancée

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible 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. Moreover, 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 sphere or on a lens space.

Mots clés: periodic orbits; Reeb vector field; Birkhoff section; broken book decomposition; entropy; open book decomposition

Domaines: Géométrie symplectique [math.SG]; Topologie géométrique [math.GT]; Systèmes dynamiques [math.DS]

Vincent Colin, Pierre Dehornoy, Ana Rechtman. On the existence of supporting broken book decompositions for contact forms in dimension 3. Inventiones Mathematicae, 2023, 231 (3), pp.1489-1539. ⟨10.1007/s00222-022-01160-7⟩. ⟨hal-02426945v4⟩