We prove a rigidity result for group actions on the line whose elements have what we
call “hyperbolic-like” dynamics. Using this, we give a rigidity theorem for
–covered Anosov
flows on
–manifolds,
characterizing orbit equivalent flows in terms of the elements of the fundamental
group represented by periodic orbits. As consequences of this, we give an
efficient criterion to determine the isotopy classes of self-orbit equivalences of
–covered
Anosov flows, and prove finiteness of contact Anosov flows on any given
manifold.
In the appendix, with Jonathan Bowden, we prove that orbit equivalences of
contact Anosov flows correspond exactly to isomorphisms of the associated
contact structures. This gives a powerful tool to translate results on Anosov
flows to contact geometry and vice versa. We illustrate its use by giving two
new results in contact geometry: the existence of manifolds with arbitrarily
many distinct Anosov contact structures, answering a question of Foulon,
Hasselblatt and Vaugon, and a virtual description of the group of contact
transformations of a Anosov contact structure, generalizing a result of Giroux and
Massot.