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?


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.

  • $\begingroup$ 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 May 24 '18 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 May 25 '18 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 Jun 1 '18 at 4:07

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