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January  2007, 1(1): 37-60. doi: 10.3934/jmd.2007.1.37

On the cohomological equation for nilflows

Livio Flaminio 1, and  Giovanni Forni 2,

1. 

UFR de Mathématiques, Université de Lille 1 (USTL), F59655 Villeneuve d'Asq Cedex, France
2. 

Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Received  March 2006 Published  October 2006

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 $D_n$.
Keywords: Nilflows, Cohomological Equations..
Mathematics Subject Classification: 28Dxx, 43A85, 22E27, 22E40, 58J42. .
Citation: Livio Flaminio, Giovanni Forni. On the cohomological equation for nilflows. Journal of Modern Dynamics, 2007, 1 (1) : 37-60. doi: 10.3934/jmd.2007.1.37
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