On the cohomological equation of magnetic flows

TL;DR

This paper investigates the cohomological equation for magnetic flows on closed manifolds, establishing conditions under which solutions exist, specifically that the right-hand side must be zero and the 1-form must be closed.

Contribution

It extends previous results by proving the cohomological equation has solutions only when the forcing term is zero and the 1-form is closed, without assuming Anosov flow conditions.

Findings

Solutions exist only if h=0 and θ is closed

The result generalizes prior work to non-Anosov flows

Provides conditions for the solvability of the cohomological equation

Abstract

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ \pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.