On extension to "other" iteration roots

[JmsNxn]

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

I realized that that infimum appears meaningless.

There's a split in the sense that \(b^z\) no longer has an attracting fixed point on the real line; there is not a discontinuity in the Fixed points. The fixed points follow a continuous curve; I know this. There's no splitting of fixed points. I said they are not "primary"; and once you have \(b < \eta^-\), we no longer have an attracting fixed point on the real line. Okay, so to maintain real valued; we need our "primary" fixed point to be the up/down fixedpoints in the complex plane. And in doing such, we won't have the orbit about \(0\) in our image; so it won't have \(0,1,b,...\) in its image.

What I meant by "split" I meant: not a split in the continuity of the fixed points; there's a split in our choice of my words to describe what primary is. By which we don't have \(0,1,b\) in the orbit, but it is still a real valued super function.

I think I just misspoke one too many times and I keep digging a hole here. I'm happy to take the L, but when you switch from \(b > \eta^-\) to \(b < \eta^-\) there is a "split" which I used in my own internal language. I apologize bo, if I can appear thick.