About a week ago I asked this question. I have edited the question several times and have started a bounty. So far it has received more views than any of my other questions, but it has only gotten a few comments and has no answers. I simply cannnot understand this.
What can I do to improve the post? Is my question unclear in its intent? Is it too detailed? Or is it just very difficult and only interesting to a few people?
For me it is very difficult to answer. I worked on that what I understood in the first read, also because I'm interested in this myself. Then I became aware, that I was misreading the question in some details (for instance, your formula for $t$ looked very familiar and I just used my procedures for experimenting, and found later, that your version $t= W(-\log(z))$ and my version as well as the common formulae in the literature $t=-W(-\log(z))$ such that $z = \exp(t \cdot \exp (-t))$ and $t$ is the log of the fixpoint) differ simply by one sign - to find this reason for some strange errors has not been easy and needed much time) and all my computations and even carefully made pictures were actually unrelated from the focus of your question. I began with some arbitrary starting value of the iterations, where I realized only after some work, that always the wished orbit should include the complex numbers 0,1,z,.. - which is a very specific orbit. It is of course an interesting special case, but I remember that some articles I read recently just adressed the same problem working on the very orbit - so I was doubting whether I (or your question) might be overlooking some trivial or known thing. It would for instance have been helpful if that specifity of orbits had been pointed out explicitely - but well, that can well be considered as a problem of attention/precision of mine, which indeed is not so good as in the younger age ago.
Now I understand, that your question puts up a conjecture, which applies to arguments nearly from the whole complex plane except the case 3b, which is likely much too broad to get proven by amateurs.
So what I'm telling here is not, that something with your question is wrong, but for whatever reason it has been (and is still) difficult for me to get the feeling to have really understood what about the real question is here.
Practically, I'm beginning to reconsider my model-computations to reproduce the graphs (which is some work using Pari/GP and Excel) but it seems, that even if I use now the correct parameters the results are again completely aside of your intended question. So... just wait a bit more for my next contribution, maybe today in the evening or tomorrow...
