Geometric approach to rigidity of horocycles (Q1913550)

The author obtains a generalization of the horocycle rigidity theorem for negatively curved closed oriented surfaces. He shows that the horocycle flows on the unit tangent bundles \(M,M'\) of two such surfaces are orientation-preserving orbit-equivalent under a homeomorphism \(\varphi: M\to M'\) if and only if the geodesic flows on \(M\) and \(M'\) are homothetic. Key observation is that the map \(\varphi\) can be conjugated with the geodesic flows on \(M\) and \(M'\), and a limit of a suitable family of such conjugated maps is a homothety of the geodesic flows. Such a homothety is induced by a homothety of the underlying surfaces.