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

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Abstract

We consider suspension flows over uniquely ergodic skew-translations on a (Formula presented.)-dimensional torus (Formula presented.) for (Formula presented.). We prove that there exists a set (Formula presented.) of smooth functions, which is dense in the space (Formula presented.) of continuous functions, such that every roof function in (Formula presented.) 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 [Mixing for time-changes of Heisenberg nilflows. 

Original languageEnglish
Pages (from-to)3407-3436
Number of pages30
JournalErgodic Theory and Dynamical Systems
Volume39
Issue number12
DOIs
Publication statusPublished - Dec 2019
Externally publishedYes

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