Volume 21, issue 3 (2017)

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Building Anosov flows on $3$–manifolds

François Béguin, Christian Bonatti and Bin Yu

Geometry & Topology 21 (2017) 1837–1930

We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example:

  1. We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field.
  2. For any n, we build a closed three-dimensional manifold M supporting at least n pairwise different Anosov vector fields.
  3. We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors.
  4. We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori.

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Anosov flows, $3$–manifolds, hyperbolic plugs
Mathematical Subject Classification 2010
Primary: 37D20
Secondary: 57M99
Received: 7 March 2016
Accepted: 28 June 2016
Published: 10 May 2017
Proposed: Danny Calegari
Seconded: Benson Farb, Leonid Polterovich
François Béguin
LAGA, UMR 7539 du CNRS
Université Paris 13 Nord
99 ave. J B Clement
93430 Villetaneuse
Christian Bonatti
Institut de Mathematiques de Bourgogne
Universite de Bourgogne
9 ave. A Savary
21000 Dijon
Bin Yu
School of Mathematical Sciences
Tongji University
Shanghai, 200092