Abstract
We consider planar vector fields without zeroes ξ and study the image of the associated Lie derivative operators L ξ acting on the space of smooth functions. We show that the cokernel of L ξ is infinite-dimensional as soon as ξ is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of ξ is “simple enough” (e.g. if ξ is polynomial) then ξ is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of L ξ and to build an embedding of \({\mathbb{R}^{2}}\) into \({\mathbb{R}^{4}}\) which rectifies ξ. Finally, we use this embedding to characterize the functions in the image of L ξ .
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De Leo, R. Solvability of the cohomological equation for regular vector fields on the plane. Ann Glob Anal Geom 39, 231–248 (2011). https://doi.org/10.1007/s10455-010-9231-3
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DOI: https://doi.org/10.1007/s10455-010-9231-3
Keywords
- Cohomological equation
- Foliations of the plane
- Hamiltonian vector fields on the plane
- Linear first-order PDEs