Title: A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows,
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.