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Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ is in the closure (in $\mathbb{R}^n$) of $\ell$. Sometimes this is known as ''the set of separatrices convergent to $p$''.

Does the set $L$ have any kind of structure? Is it subanalytic, analytic? Or what other conditions should be added to $X$ or $U$ to know that this is the case?

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    $\begingroup$ as a post which is indirectly related to your question please see mathoverflow.net/questions/324207/… $\endgroup$ – Ali Taghavi Apr 3 at 22:47
  • $\begingroup$ Ali Taghavi, thank you but I’m no expecting such things to occur. $\endgroup$ – Paul Apr 3 at 22:51
  • $\begingroup$ If $X$ is defined on $U$ and $p\not\in U$ then how can the flow lines be related to this $p$? Please proofread your question. $\endgroup$ – Alexandre Eremenko Apr 4 at 20:25
  • $\begingroup$ Pick any direction in $R^n$ and take the vector field that is the unit vector in that direction at all points, now take $U$ to be the complement of some closed ball $B$ and restrict $X$ to $U$ (note that the restriction is defined on $U$ only). Then take a point $p$ in the boundary of $B$. There is one flow line whose closure containes $p$. $\endgroup$ – Paul Apr 4 at 21:03
  • $\begingroup$ You might not like my question but it makes perfectly formal sense. $\endgroup$ – Paul Apr 4 at 21:04

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