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I would like to point out the following very nice reference (unfortunately I cannot plug into MathSciNet right now and only have a preprint version on my computer, but you should be able to track it down if you're interested)

• K. Pilgrim, «Polynomial vector fields, dessins d'enfants, and circle packings»

where he establishes the relationship between usual Belyi polynomials on the Riemann sphere, and real flows of complex polynomial of the Riemann sphere $\dot{z}(t)=P(z(t))$. The topological classification of these flows has been done by A. Douady and P. Sentenac (a very nice paper unfortunalety yet unplished), and it boils down to building a graph whose vertices are the singular points of the system, and whose edges link two singular points lying in the limit set of a same trajectory. The graph in question is actually a tree and its combinatorial data is related to the good-bracketing problem. The invariant of the flow is the class of such a graph up to obvious combinatorial equivalence. The result of Pilgrim states that there is a very simple correspondance between $P$ and the Belyi polynomial $B$ associated to the dessin d'enfants embodied in the combinatorial invariant. You obtain $P$ by simply droping the exponents of the irreducible factors of $B$, and vice-versa.

Pilgrims's paper is also related to two questions of Douady:

1. Is it possible to compute explicitely the combinatorial invariant being given a flow?

2. Conversely, being given a combinatorial data is it possible to produce an explicit example of a flow with the corresponding invariant in the same combinatorial class?

To the best of my knowledge both questions remain open for degrees bigger than 4, as they are equivalent to the very same problem regarding dessins d'enfants, which is a difficult question if I understood things properly.

That being said, I would conjecture that the question context of the OP OP's question is related (say, on $\mathbb{C}$ ) to the topological classification of entire flows $\dot{z}\left(t\right)=f\left(z\left(t\right)\right)$ with $f\in\mathcal{O}(\mathbb{C})$. This classification should also give a topological classification of flows in a neighborhood of an essential singularity. Such a classification is not done (again, to the best of my knowledge). A limiting procedure allows yet to build a locally finite graph which should embody the topological data. Then one would have to adapt the construction of Pilgrim to bind these objects and the objects defined by the OP. This is of course of a very speculative nature. I'm also aware of the fact that it does not answer the OP question directly, but this was too long a comment (and I hope it is nonetheless relevant regarding the question).

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I would like to point out the following very nice reference (unfortunately I cannot plug into MathSciNet right now and only have a preprint version on my computer, but you should be able to track it down if you're interested)

• K. Pilgrim, «Polynomial vector fields, dessins d'enfants, and circle packings»

where he establishes the relationship between usual Belyi polynomials on the Riemann sphere, and real flows of complex polynomial of the Riemann sphere $\dot{z}(t)=P(z(t))$. The topological classification of these flows has been done by A. Douady and P. Sentenac (a very nice paper unfortunalety yet unplished), and it boils down to building a graph whose vertices are the singular points of the system, and whose edges link two singular points lying in the limit set of a same trajectory. The graph in question is actually a tree and its combinatorial data is related to the good-bracketing problem. The invariant of the flow is the class of such a graph up to obvious combinatorial equivalence. The result of Pilgrim states that there is a very simple correspondance between $P$ and the Belyi polynomial $B$ associated to the dessin d'enfants embodied in the combinatorial invariant. You obtain $P$ by simply droping the exponents of the irreducible factors of $B$, and vice-versa.

Pilgrims's paper is also related to two questions of Douady:

1. Is it possible to compute explicitely the combinatorial invariant being given a flow?

2. Conversely, being given a combinatorial data is it possible to produce an explicit example of a flow with the corresponding invariant in the same combinatorial class?

To the best of my knowledge both questions remain open for degrees bigger than 4, as they are equivalent to the very same problem reagrding regarding dessins d'enfants, which is a difficult question if I understood things properly.

That being said, I would conjecture that the question of the OP is related (say, on $\mathbb{C}$ ) to the topological classification of entire flows $\dot{z}\left(t\right)=f\left(z\left(t\right)\right)$ with $f\in\mathcal{O}(\mathbb{C})$. This classification should also give a topological classification of flows in a neighborhood of an essential singularity. Such a classification is not done (again, to the best of my knowledge). A limiting procedure allows yet to build a locally finite graph which should embody the topological data. Then one would have to adapt the construction of Pilgrim to create a bound between bind these objects and the objects defined by the OP. This is of course of a very speculative nature. I'm also aware of the fact that it does not answer the OP question directly, but this was too long a comment (and I hope it is nonetheless relevant regarding the question).

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I would like to point out the following very nice reference (unfortunately I cannot plug into MathSciNet right now and only have a preprint version on my computer, but you should be able to track it down if you're interested)

• K. Pilgrim, «Polynomial vector fields, dessins d'enfants, and circle packings»

where he establishes the relationship between usual Belyi polynomials on the Riemann sphere, and real flows of complex polynomial of the Riemann sphere $\dot{z}(t)=P(z(t))$. The topological classification of these flows has been done by A. Douady and P. Sentenac (a very nice paper unfortunalety yet unplished), and it boils down to building a graph whose vertices are the singular points of the system, and whose edges link two singular points lying in the limit set of a same trajectory. The graph in question is actually a tree and its combinatorial data is related to the good-bracketing problem. The invariant of the flow is the class of such a graph up to obvious combinatorial equivalence. The result of Pilgrim states that there is a very simple correspondance between $P$ and the Belyi polynomial $B$ associated to the dessin d'enfants embodied in the combinatorial invariant. You obtain $P$ by simply droping the exponents of the irreducible factors of $B$, and vice-versa.

Pilgrims's paper is also related to two questions of Douady:

1. Is it possible to compute explicitely the combinatorial invariant being given a flow?

2. Conversely, being given a combinatorial data is it possible to produce an explicit example of a flow with the corresponding invariant in the same combinatorial class?

To the best of my knowledge both questions remain open for degrees bigger than 4, as they are equivalent to the very same problem reagrding dessins d'enfants, which is a difficult question if I understood things properly.

That being said, I would conjecture that the question of the OP is related (say, on $\mathbb{C}$ ) to the topological classification of entire flows $\dot{z}\left(t\right)=f\left(z\left(t\right)\right)$ with $f\in\mathcal{O}(\mathbb{C})$. This classification should also give a topological classification of flows in a neighborhood of an essential singularity. Such a classification is not done (again, to the best of my knowledge), though I try to work on it from time to time. A limiting procedure surely allows yet to build a locally finite graph which should embody the topological data. Then one would have to adapt the construction of Pilgrim to create a bound between these objects and the objects defined by the OP. This is of course of a very speculative nature. I'm also aware of the fact that it does not answer the OP question directly, but it may be this was too long a leadcomment (and I hope it is nonetheless relevant regarding the question).

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