From: "Christian G. Bower" <bowerc@usa.net>
iter_exp( z ) = lim n -> oo exp^n( z0 + z0^(z-n) )
iter_exp( z + 1 ) = exp( iter_exp( z ) )
Which is nice, but there are no z's where iter_exp(z) = 0 or 1 or e or e^e...
I don't think there are z's that map to 0, but remember exp is not 1-1 in the complex plane. Find a place where iter_exp(z) = 2 pi i and that yields a place where iter_exp(z) = 1
I find one around 1.511 - .217i
Giving us 1 at around 2.511 - .217i
Yes, I was premature...1 and so e, e^e, etc., all in a row. I haven't plotted the curve that connects those, nor the curves where z is closer to the real line, past about e^e. The wobbles in the imaginary part there exceed +/- pi, and soon after that the paths must be really nuts, and then hopelessly nuts, etc. Also the fixed points can be reached in similar ways. I've avoided branches and cuts my whole life until now. I think I can map the inverse function which is 1-M, so it won't cause overlaps, and maybe get an interesting fractal, but I still need to wrap my head around the branches (or vice-versa) first. --Steve