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
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:

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.
Keywords
Anosov flows, $3$–manifolds, hyperbolic plugs
Primary: 37D20
Secondary: 57M99