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

Abstract

We study time changes of bounded type Heisenberg nilflows (ϕ t ) acting on the Heisenberg nilmanifold M. We show that for every positive τ∈W s (M), s>7/2, every non-trivial time change (ϕ t τ ) enjoys the Ratner property. As a consequence, every mixing time change is mixing of all orders. Moreover, we show that for every τ∈W s (M), s>9/2 and every p,q∈ℕ, p≠q, (ϕ pt τ ) and (ϕ qt τ ) are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.