Rigidity of joinings for some measure-preserving systems

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.

Non-divergence of unipotent flows on quotients of rank-one semisimple groups

Let$G$be a semisimple Lie group of rank one and$\unicode[STIX]{x1D6E4}$be a torsion-free discrete subgroup of$G$. We show that in$G/\unicode[STIX]{x1D6E4}$, given$\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than$\unicode[STIX]{x1D6FF}$for a$1-\unicode[STIX]{x1D716}$proportion of the time, for some$\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup$\unicode[STIX]{x1D6E4}$and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces.Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed$\unicode[STIX]{x1D716}>0$, there exists an injectivity radius$\unicode[STIX]{x1D6FF}$such that, for any unipotent trajectory$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a$1-\unicode[STIX]{x1D716}$proportion of the time in the set with injectivity radius larger than$\unicode[STIX]{x1D6FF}$, for all large$T>0$, or there exists a$\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup$L$of$G$which intersects$g\unicode[STIX]{x1D6E4}g^{-1}$in a small covolume lattice. We also extend these results to when$G$is the product of rank-one semisimple groups and$\unicode[STIX]{x1D6E4}$a discrete subgroup of$G$whose projection onto each non-trivial factor is torsion free.