Mutliple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

Abstract: We study time changes of bounded type Heisenberg nilflows $(\phi_t)$ acting on the Heisenberg nilmanifold $M$. We show that for every positive $\tau\in W^s(M)$, $s~>~7/2$, every non-trivial time change $(\phi_t^{\tau})$ enjoys the Ratner property. As a consequence every mixing time change is mixing of all orders. Moreover we show that for every $\tau\in W^s(M)$, $s>9/2$ and every $p,q\in \mathbb{N}$, $p\neq q$, $(\phi_{pt}^\tau)$ and $(\phi_{qt}^\tau)$ are disjoint. As a consequence Sarnak's Conjecture on M\"obius disjointness holds for all such time changes.