On extension to "other" iteration roots

[bo198214]

Let's say that \(z_0 = \inf_{z \in \mathbb{C}} \{|\Im(z_0)|\,|\, b^{z_0} = z_0\}\). Then within the shell thron region for \(b > e^{-e}\) the infimum is \(z_0\) and it is real valued. When you let \(b < e^{-e}\), the value \(z_0\) is split into fixed point pairs \(z_0^{\pm}\) where \(|\Im(z_0)|> 0\).

We really have a problem here. I thought from the picture I gave it was clear that there is no split at eta minor - it is a totally continuous behaviour regarding the fixed points. The \(\inf \{|\Im(z_0)|\,|\, b^{z_0} = z_0, z_0 \in \mathbb{C}\}\) is 0 because of the fixed point on the real line (in and out of the STR). I really have no clue what split you are referring to or what essentially changes in the fixed points when passing eta minor. Inside and outside we have one real fixed point and many conjugated fixed point pairs.