Closedness of the Range for Vector Fields on the Torus

We consider vector fields of the formL=∂t+a(x)∂xon the 2-torus and study the closedness of the range ofLin spaces of functions and distributions. Whenavanishes somewhere but not everywhere we show that the range is closed if and only if all zeros ofaare of finite order. In the most interesting case one obtains a finiteness condition on the speed of accumulation of orbits, similar to what happens in the known tube case.