Closedness of the range for vector fields on the torus (Q1288866)

The authors consider vector fields on the torus \(T^2\cong R^2/2\pi\mathbb{Z}^2\) of the form \(L= \partial_t+ a(x)\partial_x\), where \(a\) is a smooth, real-valued function on the unit circle \(S^1\) and study the closedness of the range of the operators \(L: C^\infty(T^2)\to C^\infty(T^2)\) and \(L: D'(T^2)\to D'(T^2)\). Assuming that \(a(x)\not\equiv 0\), the authors show that the range of \(L\) is closed if and only if all zeros of \(a\) are of finite order. In the most interesting case one obtains a finiteness condition on the speed of accumulation of orbits.