On disjointness properties of some parabolic flows

TL;DR

This paper advances the understanding of disjointness in parabolic flows by introducing a new criterion based on the switchable Ratner property, and applies it to horocycle and Arnol'd flows, confirming Moebius orthogonality.

Contribution

It develops a general disjointness criterion for smooth parabolic flows using the switchable Ratner property and applies it to specific flows, answering a longstanding question.

Findings

Established disjointness for smooth time changes of horocycle flows.

Proved Moebius orthogonality for all smooth time-changes of horocycle flows.

Extended disjointness results to Arnol'd flows on the torus.

Abstract

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 the torus, focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Moebius orthogonality of time-changes of horocycle flows. In fact, we prove Moebius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol'd flows considered.