Linearization of singular foliation in the plane

2012-01-30T18:06:32Z

Hello,

I would like to obtain a smooth model around a singular point of a foliation (in my case the model should be the linearization of the foliation). It seems that the answer could be hidden in Grobmann-Hartman, and I apologize for not having read the proof.

Here is my problem:

I have a singular foliation on $\mathbb{R}^2$ given by the smooth $1$-form

$$\beta = a(x,y) dx + b(x,y) dy$$

and I assume that $(0,0)$ is a singular point.

Additionally I suppose that the matrix $$ \begin{pmatrix} \partial_x b & \partial_y b \\ - \partial_x a & -\partial_y a \end{pmatrix} $$ does not have purely imaginary eigenvalues. (In fact, I have $d\beta \ne 0$ so that in the worst case, I could have a 0 eigenvalue, but I assume for now that this is not the case).

Since the kernel of $\beta$ is spanned by the vector field $$ X = b \partial_x - a \partial_y, $$ I can use the Hartman-Grobmann theorem to say that my foliation is homeomorphic to its linearization, but I would like to have a diffeomorphism.

I have seen that there exist counter-examples for vector fields to be smoothly conjugated to its linearization, but on the other hand, I'm not interested in the time parameter of the trajectories and I would only like to map the foliation onto the linearized foliation. Could it be possible that I still get a diffeomorphism in my situation? I would bet that there must be a reference, where this has already been worked out, but googling only confused me (it spoke of resonance conditions and other things).

Thank you very much for any response.

Best Klaus

Linearization of singular foliation in the plane - MathOverflow

Linearization of singular foliation in the plane