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I am a PhD student in several complex variables. I am reading this paper by Orevkov proving that there exists a proper analytic embedding of $\overline{\Bbb C}$ minus a Cantor set into $\Bbb C^2$.

I am really lost and I am writing here to ask some hint/tips on how to proceed reading it. It is not clear why $K$ is a Cantor set, since in order to find out (analytically) its elements one should find roots of $n$-degree polynomials (for all $n\ge1$, and no recursive formula seems to appear). Someone suggested me to argue geometrically, but it is very hard and, anyhow, it doesn't seem to help (for what I tried so far).

Also I cannot see how to prove points 2), 3) and 4) and how does are the exploited in the rest of this short but very dense paper.

Thanks. Of course I don't ask a clear proof of this, but just some guidelines I can follow to going out of this labyrinth.

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    $\begingroup$ I have no access to the paper, but the formulation is ambiguous: does it mean $\mathbf{C}$ minus some Cantor subset, or minus every Cantor subset? while topologically these are all "the same", up to biholomorphy it's far from unique. $\endgroup$
    – YCor
    Mar 18, 2020 at 14:17

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You should be able to find anwers to your question in a well witten book by Franc Forstnerič: Stein Manifolds and Holomorphic Mappings -The Homotopy Principle in Complex Analysis, https://www.springer.com/gp/book/9783319610573

In particular look at sections 9.10, 9.11 and Theorem 9.11.5. The book contains all the background material you need and also contains a rewritten proof of Orevkov's result.

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  • $\begingroup$ Thank You very much for your answer. Nonetheless let me point out a couple of things. 1)Forstneric's book is very very unfriendly, at least for me. Rather than a reading book, it seems he's just collected all the result/papers in several complex variables, to resume the state of art of the subject. 2) I got a solution by myself, without needing all the complex tools in sections 9.10 and 9.11; the difficult part was to interpretate the composition of rational shears. Doing that, making also a sort of graph, one can understand the effect of the composition, leading to the final result. $\endgroup$
    – Joe
    Mar 19, 2020 at 8:25

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