Real and complex behaviour of the base change function (was: The "cheta" function)

[sheldonison]

Walker showed a similar convergence for \( f_n=\operatorname{dexp}^{[-n]}\circ \exp^{[n]} \), where dexp(x) = exp(x)-1.
He showed that the limit is infinitely differentiable on the real axis.
That means that he also wasnt clear about the complex behaviour otherwise he would have shown that the limit is holomorphic as a consequence of local uniform convergence.
But he could prove that local uniform (or compact) convergence only on the real axis, which does not suffice to imply holomorphy (because it could be that during the convergence non-real singularities get dense towads points on the real axis). I will persue this topic in the next days and have still some unexplored ideas at my hands.

First off, the graph below, and the earlier now incorrect results I posted to this thread correspond to the equation Henryk says Walker analyzed, except with the exponentials being "exp(x) = exp(x)-1", and the logarithms being base e. The "exp(x)=exp(x)-1" corresponds to base eta with the initial operands divided by e. Here is small slice of the complex plane, that seems to have a reasonable chance of converging as k increases.

\( f(x) =
\lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (
4.7 + \Im
) \right) \)

I don't have much time right now, but here is a graph for f(x), where real(x)=4.7 and imag(x) varies from 0 to i*0.5, where I show the graphs for k=5, k=6, and k=7. I haven't analyzed whether the windings work for larger values of K, but I hope they might. Also, I zoomed in on the transition near i=0.1, and the k=7 graph is off by one winding; this isn't visible at this scale.



Next, I was interested in f(4.7+0.2i) and I thought it might converge to the fixed point of e. Not so! For k=7, it converges to -0.516080387 + i*0.262012723. The method of convergence is to track down those pesky windings. so as to guarantee that all of the steps in the iterated logarithms are continuous, it winds up, and eventually the imaginary portion goes negative, and it stops winding, and more or less freezes at approximately i=0.1, there is a small amount of change (not visible in the graph), when i continues growing past 0.4

For this one segment of the complex plane, F appears to converge, and for the value in question may allow for a continuous extension of Jay's base change to the complex plane, but there is much more to do. In other words,
\( F(4.7+0.2i)=exp(F(\eta^(4.7+0.2i)) \)

- Shel