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 τ t ) 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 τ pt ) and (φ τ qt ) are disjoint. As a consequence Sarnak Conjecture on Möbius disjointness holds for all such time changes.

Polynomial 3-mixing for smooth time-changes of horocycle flows

“…Nevertheless, non trivial time-changes, within a natural class of "polynomial" functions on the nilmanifold, destroy the toral factor and are strongly mixing, as was shown by Avila, Forni, Ulcigrai, and the second author in [1], extending previous results in [2] and in [28]. For time-changes of bounded type Heisenberg nilflows, one obtains an even stronger dichotomy, [11]: either the time-change is trivial (in which case the toral factor-persists), or the time-changed flows is mildly mixing (it has no non-trivial rigid factors).…”

Spectral Theory of Dynamical Systems

“…In the case the flow is of bounded type, the authors prove polynomial speed of decay of correlations. Moreover, in Forni and Kanigowski (2020), the authors show that for time-changes of bounded type Heisenberg nilflows, every non-trivial time-change enjoys the R-property and as a consequence is mildly mixing. Moreover, in the above setting, it also follows that every mixing time-change is mixing of all orders.…”

Slow chaos in surface flows

“…The first examples outside the homogeneous world were given by Frączek and Lemańczyk in [33][34][35] (in the setting of special flows). The two authors could also show in [33] that a variant of Ratner's property hold for some surface flows, more precisely in a class of flows on genus one tori known as von Neumann flows 28 (for non generic flows, corresponding to a measure zero set of frequencies). However, the flows in [33] are not (globally) smooth.…”

“…The new criterion for disjointness introduced in [40] has already proved useful in different contexts, see for example the recent works [16,28] where it is applied to study disjointess phenomena respectively for Heisenberg nilflows in [28], for von Neumann flows in genus one in [16]. Finally, the disjointness criterion is used in [40] also to show that a typical Arnold flow is disjoint from any smooth time change of the horocycle flow (and in particular from the classical horocycle flow itself), thus showing that these two classes of parabolic flows are truly distinct.…”

“…One can more in general consider a shift p, which belongs to a fixed compact set P so that now ϕ t+ p (x) and the time-shifted orbit ϕ t+ p (x ) are close. The original Ratner property, where P = {+1, −1}, is now sometimes called 2-point Ratner property, while the generalization to P finite first and to P compact later were defined by by Frączek and Lemańczyk in[33,34] and called respectively finite Ratner and weak Ratner properties 28. Von Neumann flows are called in this way since they first appeared in von Neumann's work[89].…”