DCDS
Linearization of cohomology-free vector fields
Livio Flaminio Miguel Paternain
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1031-1039 doi: 10.3934/dcds.2011.29.1031
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
keywords: Cohomological Equations Greenfield-Wallach and Katok conjectures.
JMD
On the cohomological equation for nilflows
Livio Flaminio Giovanni Forni
Journal of Modern Dynamics 2007, 1(1): 37-60 doi: 10.3934/jmd.2007.1.37
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.

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