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This is a simple question that I direfully need an answer for. If the response is in the negative, I can work with it. If the response is in the positive, I can also work with it. I just can't seem to find an answer, and I need to direct my proof in one direction or the other.

Consider the exponential functions $\alpha^z$ where $1 < \alpha < e^{1/e}$ and $z \in \mathbb{C}$. These exponential functions notably have a positive real fixed point. These fixed points are geometrically attracting. Are the immediate basins of these fixed points simply connected?

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Yes. All periodic components of the set of normality of any transcendental entire function are simply connected. This is a theorem of Baker, The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 277–283.

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  • $\begingroup$ Thanks! That's a broad result, wouldn't have expected that. I'll have to read how he proved this. The answer to my questions always seems to be Baker... $\endgroup$
    – user78249
    Commented May 24, 2018 at 19:14
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    $\begingroup$ So the natural conclusion is that you should read his papers if you are in this business. $\endgroup$
    – Alexandre Eremenko
    Commented May 25, 2018 at 4:17
  • $\begingroup$ +1 Lol, thanks. I'm kind of forgetful. If I ask another question and the answer is in one of his papers, you can just post the answer "BAKER!" and I'll accept it. $\endgroup$
    – user78249
    Commented Jun 1, 2018 at 4:07

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