Journal of Modern Dynamics (JMD)

On the cohomological equation for nilflows

Pages: 37 - 60, Issue 1, January 2007      doi:10.3934/jmd.2007.1.37

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Livio Flaminio - UFR de Mathématiques, Université de Lille 1 (USTL), F59655 Villeneuve d'Asq Cedex, France (email)
Giovanni Forni - Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada (email)

Abstract: 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.

Received: March 2006;      Available Online: October 2006.