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

✍ Scribed by Roberto De Leo

• Publisher: Springer
• Year: 2010
• Tongue: English
• Weight: 420 KB
• Volume: 39
• Category: Article
• ISSN: 0232-704X
• DOI: 10.1007/s10455-010-9231-3

✦ Synopsis

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 R 2 into R 4 which rectifies ξ . Finally, we use this embedding to characterize the functions in the image of L ξ .