Published online by Cambridge University Press: 30 April 2021
We introduce two properties: strong R-property and
$C(q)$
-property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the
$C(q)$
-property. Moreover, we show that if
$u_t$
is a unipotent flow on
$G/\Gamma $
with
$\Gamma $
irreducible, then
$u_t$
satisfies the
$C(q)$
-property provided that
$u_t$
is not of the form
$h_t\times \operatorname {id}$
, where
$h_t$
is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.