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Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory") is seldom Hausdorff. Such foliated structures have been intensively studied, and a complete $C^r$-classification is due to Haefliger and Reeb in the case where $X$ is regular on a simply connected region (thanks to the same topological niceness of the plane used in the proof of Bendixon-Poincaré's theorem). Almost any reasonable one-dimensional, simply-connected non-Hausdorff manifold can be realized as the orbit space of a foliation.

The two main sources of non-separability of orbits are:

  1. Saddle singularities: the stable and unstable (half-)manifolds cannot be separated.
  2. Limit cycles: the limit cycle cannot be separated from the accumulating trajectories.

I believe that orbits space coming from real-analytic foliations should have a "nicer" structure. I expect also that the work of Kaplan, Haefliger, Reeb dating back from the 40--50's should have been generalized to the analytic setting. Is that so? Is there any special structure / characterization on the (non-Hausdorff) analytic orbits space of a real-analytic planar foliation that I should be aware of (and where can I find it)?

A special case of particular interest is where the the vector field $X$ is the realification of a holomorphic vector field on $\mathbb C\simeq \mathbb R^2$. Now there are no limit cycles. The topology of the phase-portrait looks simpler and the orbit space also. Is there any known characterization of the analytic one-dimensional (non-Hausdorff) manifolds that can arise in this very special case?

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You wrote "I believe that orbits space coming from real-analytic foliations should have a "nicer" structure".

I think that this nicer structure arises when we consider a more technical "Leaf space" so called "Groupoid foliation" not mereley the orbit space. The orbit space has a very poor topology. For example it can not distingishe two topological different foliation of the torus with different slopes $"\sqrt 2"$ and $"\pi"$ see "The" kronecker foliation or "a" kronecker foliation?

Note that the holomorphicity can not do any thing special if we resist on the plain orbit space and do not consider the groupoid foliation. Because the simplest holomorphic map in the world, the constant map $z'=c$ produce a complicated kronecker foliation. But the k theory of $C^*$ algebra of the corresponding groupoid foliation, is the true useful tool for studing the invisible features of the foliation.

So with consideration of groupoid foliation $G(M,F)$, it is always a Haussdorf space if the foliation is real analytic. In general $G(M,F)$ is $k+n$ (not necessarily Hausdorf) manifold where $F$ is a $k$ -dimensional foliation of an n manifold

Your question reminds me of an MO question of mine as follows. I have neither deleted that question nor I can find it in my question list.

Is there a non analytic foliation whose groupoid foliation is diffeomorphic to an $S^3 $ with two north pole, a compact 3 dimensional analogy of a line with two origin?

https://ncatlab.org/nlab/show/line+with+two+origins

We identify two disjoint $S^3$ at all its points exept at north pole:

In the disjoint union $S^3\times\{0\} \coprod S^3\times \{1\}$ we identify $(x,0)$ with $(x,1)$ for all $x\in S^3\setminus \{N\}$ so we obtain a 3-sphere with two north pole.

Any way if you give a precise reference to papers of Haefliger et al one can search for thses papers to realize which kind of orbit space they are working with.

P.S: for the holomorphic foliation $Z'=e^Z$ we have infinite number of points(in the leaf space according to your terminology) with the following property: There are infinite paires such that each pair consite of two points which can not seperate from each other.

But is there an entire holomorphic function(non vanishing) for which the corresponding foliation does not admit infinit number of paires with the above property, but any way the leaf space is non Haussdorf? On the other extrem is there a non vanishing entire function whose corresponding leaf space(according to your terminology) which contains infinite number of points which mutually are non seperable?

Note: You wrote: "Limit cycles : the limit cycle cannot be separated from the accumulating trajectories"

But surprisingly in the groupoid foliation a limit cycle can seperates from other trajectories!. The only case it can not seperate is that "it is center from exterior and it is limit cycle from the interior or it is exteror limit cycle and interior center. Of course this can not happen in real analytic case.

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