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?



Are the immediate basin of these exponential maps 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. 


