x↑↑x = -1

[jaydfox]

In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness.

When you say the carleman matrices approach, if you are referring to the "intuitive" approach that Andrew Robbins developed (previously mentioned by others like Peter Walker I think?), then have no worries.

For a real base greater than 1, it has very nice convergence properties. The convergence is slow, but it is quite well-behaved. I've been working on an analysis, based on the intuitive method, as well as my accelerated version of the intuitive method. The convergence is well-behaved for each term of the taylor series, as well as for the root-mean-squared error at any radius up to the convergence radius at the fixed point(s).

For real bases greater than eta, in the principal branch, the closest singularities to the real line are at the primary fixed points, which means you can safely develop the intuitive solution using any development point on the real line, including 0 or 1.

For real bases greater than 1, up to eta, you can develop from any point less than the lower fixed point. Indeed, this is just the regular iteration approach. For example, for base sqrt(2), you can develop at any point on the real line less than 2. I haven't tested, but I assume that developing above the upper fixed point is equivalent to regular iteration at that fixed point. I have my doubts about developing from a point in between.

I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other.