斜平移和时变准阿贝尔丝状零流上悬浮流的混合
Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows
摘要：
本文考虑$d$维环面$\mathbb{T}^d$上唯一遍历斜平移的悬浮流，其中对$d \geq 2$ .证明了存在一个光滑函数集$\mathscr{R}$，它在连续函数空间$\mathscr{C}(\mathbb{T}^d)$中是稠密的，使得$\mathscr{R}$中的每一个不与常数同调的屋顶函数都诱导一个混合悬浮流.我们还构造了一个稠密的混合的例子，这是明确描述的傅里叶系数。用零流形上的零流的语言，我们的结果意味着，对于拟交换丝状零流形上的每个唯一遍历零流，存在光滑时变的稠密子空间，其中混合发生当且仅当时变不与常数上同调。这推广了Avila，Forni和Ulcigrai（J. Diff. Geom.，2011年）的经典海森堡群。
英文摘要：
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.
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