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# On the cohomological equation for nilflows

• Let $X$ be a vector field on a compact connected manifold $M$. An important question in dynamical systems is to know when a function $g: M\to \mathbb{R}$ is a coboundary for the flow generated by $X$, i.e., when there exists a function $f: M\to \mathbb{R}$ such that $Xf=g$. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions $D_n$ such that any sufficiently smooth function $g$ is a coboundary iff it belongs to the kernel of all the distributions $D_n$.
Mathematics Subject Classification: 28Dxx, 43A85, 22E27, 22E40, 58J42.

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