# Flow lines of a real analytic vector field convergent to a point

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?

• as a post which is indirectly related to your question please see mathoverflow.net/questions/324207/… – Ali Taghavi Apr 3 at 22:47
• Ali Taghavi, thank you but I’m no expecting such things to occur. – Paul Apr 3 at 22:51
• 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. – Alexandre Eremenko Apr 4 at 20:25
• 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 $p$. – Paul Apr 4 at 21:03
• You might not like my question but it makes perfectly formal sense. – Paul Apr 4 at 21:04