Solvability of the cohomological equation for regular vector fields on the plane

TL;DR

This paper investigates the solvability of the cohomological equation for regular planar vector fields, revealing conditions for the image of the Lie derivative operator and constructing explicit embeddings to analyze solutions.

Contribution

It characterizes the image and cokernel of the Lie derivative operator for planar vector fields and provides explicit embeddings to understand the solutions of the cohomological equation.

Findings

Cokernel of LX is infinite-dimensional unless X is topologically conjugate to a constant vector field.

If trajectories are topologically simple, X is transversal to a Hamiltonian foliation.

Constructs an explicit embedding of R^2 into R^4 to analyze the image of LX.

Abstract

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial) then X is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of LX and to build an embedding of R^2 into R^4 which rectifies X. Finally we use this embedding to characterize the functions in the image of LX.