I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A vague statement is this: 


An inner function is a continuous deformation of a holomorphic function.


I am looking for a simple version rather than the most general one.
Any accessible reference is welcome (I understand Romanian).


Thanks in advance.



Stoilow Theorem
First of all, it seems that every author has his own favourite definition of "inner mapping". The best thing would probably be to adopt [the one used by Martin Jurchescu](https://books.google.com/books?id=8v7sCgAAQBAJ&pg=PA420): a continuous, open, zero-dimensional mapping.


Stoilow's theorem says, then, that if $\phi : R \to S^2$ is a map from a Hausdorff, connected topological surface $R$ to the Riemann sphere $S^2$, then the following are equivalent:


* $\phi$ is an interior mapping
* $\phi$ behaves locally like the mapping $\{ |z| \le 1 \} \ni z \mapsto z^n \in \{ |z| \le 1 \}$
* there exist a unique Weyl-Rado Riemann surface on $R$ such that $\phi$ becomes a holomorphic mapping.


To make things a bit confusing, [the article about "inner mapping" on the Springer encyclopedia of mathematics](https://www.encyclopediaofmath.org/index.php/Interior_mapping) gives a slightly different statement of the theorem.


For more details, check "Analysis and Topology: A Volume Dedicated to the Memory of S Stoilow" (World Scientific, 1998), pages 420-421.


