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

TL;DR

This paper proves that for suspension flows over certain ergodic skew-translations on tori, there exists a dense set of smooth functions inducing mixing flows, extending results to nilflows on quasi-abelian filiform nilmanifolds.

Contribution

It establishes the existence of a dense set of roof functions that induce mixing in suspension flows over skew-translations and nilflows, generalizing previous results to a broader class of nilmanifolds.

Findings

Dense set of smooth functions induce mixing flows

Explicit construction of mixing examples via Fourier coefficients

Generalization of mixing results to quasi-abelian filiform nilflows

Abstract

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^d$, for $d \geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^d)$ of continuous functions, such that every roof function in $\mathscr{R}$ 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.