# Are the immediate basin of these exponential maps simply connected?

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?