A uniqueness theorem for transitive Anosov flows obtained by gluing hyperbolic plugs

Béguin, François; Yu, Bin

doi:10.2140/agt.2023.23.2673

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A uniqueness theorem for transitive Anosov flows obtained by gluing hyperbolic plugs

François Béguin and Bin Yu

Algebraic & Geometric Topology 23 (2023) 2673–2713

DOI: 10.2140/agt.2023.23.2673

Abstract

In work with C Bonatti, we defined a general procedure to build new examples of Anosov flows in dimension 3. The procedure consists in gluing together some building blocks, called hyperbolic plugs, along their boundary in order to obtain a closed three-manifold endowed with a complete flow. The main theorem of that work states that (under some mild hypotheses) it is possible to choose the gluing maps so the resulting flow is Anosov. Here we show a uniqueness result for Anosov flows obtained by such a procedure. Roughly speaking, we show that the orbital equivalence class of these Anosov flows is insensitive to the precise choice of the gluing maps used in the construction. The proof relies on a coding procedure, which we find interesting for its own sake, and follows a strategy that was introduced by T Barbot in a particular case.

Keywords

Anosov flows, hyperbolic plugs, orbitally equivalent, flows on three-dimensional manifolds

Mathematical Subject Classification

Primary: 37D20

Secondary: 57M99

References

Publication

Received: 13 January 2021

Revised: 3 December 2021

Accepted: 14 January 2022

Published: 7 September 2023

Authors

François Béguin
LAGA, UMR 7539 du CNRS Université Sorbonne Paris Nord Villetaneuse France

Bin Yu
School of Mathematical Sciences Tongji University Shanghai China

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Volume 23, issue 6 (2023)