Dissecting Andrew's slog solution

[bo198214]

The main reason for iterating 0 rather than 1 is because the radius of convergence of Andrew's slog limits me to this range. (The radius of convergence is limited by the singularity at the fixed point.)

Oh this is interesting. It is known that a regular iteration at a hyperbolic fixed point has no singularity at this fixed point. But the regular iterations are complex valued (not real for real argument, forget what I wrote about it elsewhere) and hence not equal to any real solution.
So any real solution must have singularities at the non-real fixed points (which are always hyperbolic).