Building Anosov flows on 3–manifolds

Béguin, François; Bonatti, Christian; Yu, Bin

doi:10.2140/gt.2017.21.1837

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Building Anosov flows on $3$–manifolds

François Béguin, Christian Bonatti and Bin Yu

Geometry & Topology 21 (2017) 1837–1930

DOI: 10.2140/gt.2017.21.1837

Abstract

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:

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

Keywords

Anosov flows, $3$–manifolds, hyperbolic plugs

Mathematical Subject Classification 2010

Primary: 37D20

Secondary: 57M99

References

Publication

Received: 7 March 2016

Accepted: 28 June 2016

Published: 10 May 2017

Proposed: Danny Calegari

Seconded: Benson Farb, Leonid Polterovich

Authors

François Béguin
LAGA, UMR 7539 du CNRS Université Paris 13 Nord 99 ave. J B Clement 93430 Villetaneuse France

Christian Bonatti
Institut de Mathematiques de Bourgogne Universite de Bourgogne 9 ave. A Savary 21000 Dijon France

Bin Yu
School of Mathematical Sciences Tongji University Shanghai, 200092 China

Volume 21, issue 3 (2017)