Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows

Abstract

We consider suspension flows over uniquely ergodic skew-translations on a d-dimensional torus T d , for d ≥ 2. We prove that there exists a set R of smooth functions, which is dense in the space C (T d ) of continuous functions, such that every roof function in R which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai (J. Diff. Geom., 2011) for the classical Heisenberg group.In this section, we present the general structure of the proof of Theorem 1.1-(a), stating some intermediate results, which are proved in later sections.Let T be a uniquely ergodic skew-translation as in (1.1). If we denote by E j the image of the linear map (A − Id) j , we have a filtration of R d into rational subspacesUp to a linear isomorphism, we can assume that the basis {e 1 , . . . , e d } of Z d is adapted to the filtration above, in particular {e d0+1 , . . . , e d } is a basis of E k , where d − d 0 = dim E k . Since T is not a rotation, k ≥ 1 and 1 ≤ d 0 ≤ d − 1. We remark that w ∈ E j for j ≥ 1 if and only if there exists v ∈ E j−1 such that v(A − Id) = w, i.e. vA = v + w. In particular,

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

“…Indeed, nilflows are never weakly mixing, because of the presence of a toral factor, corresponding to the projection onto the abelianization of the nilpotent group. 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).…”

“…A shearing phenomenon analogous to the one described above is at the base of several results on quantitative 2-mixing for non-homogeneous parabolic flows, see e.g., [6], [28], [7]. We will use a version of this mechanism in this paper as well, see the proof of Theorems 1.1 and 5.2 in Sections 4 and 5.4.…”

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

“…For Heisenberg nilflows of bounded type the decay of correlations is estimated in [10] to be polynomial, as expected according to the "parabolic paradigm" (see [15], section 8.2.f). The mixing result of [3] was generalized in [24] to a class of nilflows on higher step nilmanifolds, called quasi-Abelian, which includes suspension flows over toral skew-shifts, and then recently to all non-Abelian niflows in [2]. These general results reach no conclusion about the speed of mixing.…”

Slow chaos in surface flows

“…A series of recent works [4,5,74] indicates that, even though classical nilflows are never mixing (see footnote 14), a typical time-change (in a dense class of smooth time-changes) of a mimimal nilflow on any nilmanifold (different from a torus) is mixing.…”

“…24 Furthermore, since this is essentially a geometric mechanism for explaining mixing, this phenomenon persists under perturbation and hence can be used also for time-changes (see [29,61], where we prove quantitative mixing results and show polynomial estimates on the decay of correlations for smooth time-changes of the horocycle flow). A similar mechanism, namely shearing of segments of a suitable foliation (but with the difference that the direction of shearing is not global but depends on the segment considered) was also exploited in [20] to prove mixing in some (exceptional) elliptic flows 25 and in the context of nilflows: while nilflows are never mixing (see footnote 14), in suitable classes of smooth time-changes one can implement this mechanism to prove mixing using shearing, see [4,5,74].…”