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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
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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