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

2010/07/18 by Roberto De Leo, De Leo, Roberto
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #math.DG #math.FA

paper · pdf · doi:10.48550/arxiv.1007.3016

21 pages, 2 figures

arxiv created 2010/07/18 · arxiv updated 2010/07/20

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 R² into R⁴ which rectifies X. Finally we use this embedding to characterize the functions in the image of LX.

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• Solving the cohomological equation for locally hamiltonian flows, part II -- global obstructions