On disjointness properties of some parabolic flows

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth Arnol’d flows on T², focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Möbius orthogonality of time-changes of horocycle flows. In fact, we prove Möbius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol’d flows considered.