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


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