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Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results:

In [1] Gross constructed an entire function for which every point in the complex plane is an asymptotic value.

In [2] Heins proved that every Suslin analytic set is an asymptotic set of some entire function.

In [3] Heins proved that for every countable set A in the complex plane there exists an entire function whose set of locally omitted values coincides with A.

(Q1) My question is, are any of these examples locally univalent functions?

Unfortunately I was not able to varify these things by myself since I don't read German [1] and since I was not able to find an article [2] in our library basis or online.

Regarding the third result I think that the answer is positive, since the function is obtained as an associated mapping function of a parabolic surface spread over the sphere with only logarithmic ramification points. Is that correct?

(Q2) Do you know any (other) references with a construction of an entire function having a prescibed singular set - closed subset of a complex plane (I've seen the paper of Bishop, Constructing entire functions by QC-folding, where this is done)

[1] W. Gross: Eine ganze Funktion, fur die jede komplexe Zahl Konvergenzwert ist (German). Math. Ann. 79 (1–2), 201–208 (1918)

[2] M. Heins, The set of asymptotic values of an entire function, Tolfte Skandinaviska Matematikerkongressen (Lund, Sweden, 1953), Proceedings of the Scandinavian Math. Congress,Lund, 1954, pp. 56–60

[3] M. Heins: Asymptotic spots of entire and meromorphic functions. Ann. Math. (2) 66, 430–439 (1957)

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Gross's example and the second example of Heins are locally univalent. (About the second result of Heins, you can check yourself: Annals is available. I just checked). If I remember correctly, Heins's first example is also locally univalent. Proc. of Scandiavian congresses are indeed not available online, but many libraries have them. You can use Interlibrary Loan (ILL), just ask your librarian.

An alternative proof that every Suslin set is a set of asymptotic values (of a meromorphic function) is here:

MR2779072 Cantón, A. Drasin, D. Granados, A. Asymptotic values of meromorphic functions of finite order, Indiana Univ. Math. J. 59 (2010), no. 3, 1057–1095.

Their construction is more complicated than that of Heins because they care about the growth of the function. For a general introduction to singularities of inverse functions you may look at my lectures here:

http://www.math.purdue.edu/~eremenko/dvi/sing1.pdf

and here:

http://www.math.purdue.edu/~eremenko/dvi/mich.pdf

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  • $\begingroup$ Thank you for your answer. $\endgroup$ – user47862 Aug 6 at 7:11

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