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

## TL;DR

This paper investigates the nature of self orbit equivalences in transitive Anosov flows on 3-manifolds, revealing conditions under which they preserve or alter orbits and implications for topological rigidity of certain fiber bundles.

## Contribution

It removes an unnatural assumption in Farrell and Gogolev's result, showing that self orbit equivalences are either orbit-preserving or highly constrained in fiberwise Anosov flows.

## Key findings

- Self orbit equivalences either preserve all orbits or are of a specific type in certain Anosov flows.
- The result simplifies understanding of topological rigidity in 3-dimensional fiber bundles with fiberwise Anosov flows.
- The paper extends previous rigidity results by removing restrictive assumptions.

## 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 about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is $3$-dimensional.