On the cohomological equation for nilflows

Livio Flaminio; Giovanni Forni

doi:10.3934/jmd.2007.1.37

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2007, Volume 1, Issue 1: 37-60. Doi: 10.3934/jmd.2007.1.37

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

Received: March 2006

Published: October 2006

Abstract / Introduction Related Papers Cited by

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.

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