Orthogonal powers and M\"obius conjecture for smooth time changes of horocycle flows

Abstract

We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's M\"obius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.