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

## TL;DR

This paper proves a uniqueness theorem for a class of Anosov flows in 3-manifolds constructed by gluing hyperbolic plugs, showing that their orbital equivalence class is independent of the specific gluing maps used.

## Contribution

It establishes a uniqueness result for Anosov flows obtained via hyperbolic plug gluing, extending previous construction methods with a new coding approach.

## Key findings

- Orbital equivalence class is insensitive to gluing map choices.
- A novel coding procedure for analyzing Anosov flows.
- Extension of previous construction techniques for Anosov flows.

## Abstract

In a previous paper with C. Bonatti ([5]), we have 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 3-manifold endowed with a complete flow. The main theorem of [5] states that (under some mild hypotheses) it is possible to choose the gluing maps so the resulting flow is Anosov. The aim of the present paper is to 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.