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For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic foliation by complex curves, the foliation arising from $\dot z=f(z)$. In this case origin is called a singularity.

What is an example of such singular foliation for which the foliation is stable at $0$?

That is, for every open set $W$ containing origin, there is an open set $V\subset W$ which saturation under foliation is contained in $W$? Is there an indirect relation between this question and some methods in "Minimal set problem" about singular holomorphic foliations of $\mathbb{C}P^{2}$?

Note that a minimal set for a singular foliation of $\mathbb{C}P^{2}$ is a compact subset which is invariant(saturated) under foliation and does not contain any singular point. For "minimal set" see here.

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Such an example is impossible. We can always assume $W$ is a polydisc, and part of its boundary $\partial W$ is included in the $3$-space $T=\{(x,y) : |y|=r\}$. Take $p\in T\cap\bar W$. If the foliation were stable then the image of $t\mapsto z(t))$, for small $t$ and with $z(0)=p$, would be included in the adherence of a single connected component of $\mathbb C^2\setminus T$. This contradicts the maximum/minimum modulus principle for the $y$-component of $z(t)$ except if $z=cst$, but in that case the trajectory escapes from $W$ through another component of the boundary. Therefore every leaf of the foliation passing through $\partial W$ escapes from $W$.

You can adapt this construction to the saturation $S$ of $V$ in $W$, to prove the foliation always reaches $\partial W$. If it were not the case, consider the smallest closed polydisc $\bar W_S\subsetneq \bar W$ containing $S$. There is a point $p\in\partial S \cap \partial W_S$, to which the above argument applies: either the trajectory passing through $p$ escapes from $W_S$ (contradiction) or it is included in $\partial W_S$ and therefore leaves the polydisc through another component of $\partial W_S$ (contradiction again).

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  • $\begingroup$ In fact every non-singular trajectory escapes from $W$. $\endgroup$
    – Loïc Teyssier
    Commented May 11, 2015 at 16:07
  • $\begingroup$ Thanks for your answer. I think I can not understand some thing in the last part of your argument or some thing is missing in your answer. An equivalent definition of stability is as follows :for every open set $W$ there is an open set $V\subset W$ such that $V$ is saturated. But why we can choose $V$ in the form of a poly disc? In fact the last part of your statement (which does not depend on real or complex case, I think) is not valid in real case (the center). Moreover By 3-torus do you mean solid two torus?(The boundary is the union of two solid torus) $\endgroup$
    – Ali Taghavi
    Commented May 11, 2015 at 16:52
  • $\begingroup$ Moreover what is the contradiction if in every small neighborhood we have a leaf with constant y-coordinate? Note that we do not assume that the singularity is non degenerate.May be it has vanishing linear term. $\endgroup$
    – Ali Taghavi
    Commented May 11, 2015 at 17:14
  • $\begingroup$ Or vanishing k-Jet for k large. $\endgroup$
    – Ali Taghavi
    Commented May 11, 2015 at 17:26
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    $\begingroup$ @AliTaghavi:I never claimed you could take $V$ as a polydisc, I'm sorry if I was unclear. I modified the post in a way which is clearer I hope. Also, there is no problem in having trajectories with constant $y$-coordinate, they just escape from somewhere else. $\endgroup$
    – Loïc Teyssier
    Commented May 11, 2015 at 18:48

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