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

• 成果类型: Article
• 署名作者: Colin, Vincent; Dehornoy, Pierre; Rechtman, Ana
• 署名单位: Nantes Universite; Ecole Centrale de Nantes; Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01160-7
• 发表日期: 2023
• 页码: 1489-1539
• 关键词: reeb vector-fields; pseudoholomorphic curves; topological-entropy; periodic points; HOMOLOGY; Orbits; FLOWS

摘要

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.