Mathematical Research Letters

Volume 26 (2019)

Number 3

A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows

Pages: 711 – 728

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n3.a3

Authors

Thomas Barthelmé (Department of Mathematics and Statistics, Queen’s University, Kingston Ontario, Canada)

Andrey Gogolev (Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, New York, U.S.A.; and Dept, of Mathematics, Ohio State University, Columbus, Oh., U.S.A.)

Abstract

We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev [9] about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is $3$-dimensional.

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The second author was partially supported by Simons grant 427063.

Received 3 February 2017

Accepted 8 October 2018

Published 25 October 2019