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Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the following:

For any two points $P,Q \in \mathbb{C}$, can we find a curve $\gamma$ connecting $P$ and $Q$, such that $\Pi^{-1}(f(\gamma))$ lies in a great circle on the unit sphere, and that $\Pi^{-1}(f(\gamma))$ cover the circle at most once as points go from $P$ to $Q$ along the curve $\gamma$?


Any ideas or comments are really appreciated!

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No. A simple example is $f(z)=e^z$, $P=0$, $Q=10\pi i$. For any curve from $P$ to $Q$, the image is a closed curve which winds $5$ times around zero. So it cannot correspond to an arc of the great circle traversed once.

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  • $\begingroup$ Ah, Thank you so much Professor Eremenko! $\endgroup$
    – student
    Jun 3, 2020 at 14:04
  • $\begingroup$ By the way, you mean $P=0$, right? $\endgroup$
    – student
    Jun 3, 2020 at 14:57
  • $\begingroup$ Sorry I asked a stupid question which does not express what I want to truly understand. I just asked a new one, and if you have time and interested in, could you take a look? mathoverflow.net/questions/362105/… $\endgroup$
    – student
    Jun 3, 2020 at 15:25

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