regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational)

[Gottfried]

Therefore how do we generate \( \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) \in (2, 4) \)?
Do we create a middle super function?

Beginning at x=1 we use the lower fixpoint (at 2) and the iterates are always between -infty and 2. if we begin at x=3 the iterates are always between 2 and 4. However, we can connect the two areas. If we begin at x=1 and iterate with the complex height h=0 + 2*Pi*i/log(log(2)) then we get exactly one value in the 2..4 interval. That is that we just switch the sign of the value of the schröder-function from positive to negative (one half round in the complex plane). That makes it also possible to define a "norm"-height for that values in the 2..4-interval. We set the real part of the height = 0 where x=1 was mapped to.

Gottfried