Infinite tetration and superroot of infinitesimal

[andydude]

But is not applying of infinite log(log(log( .........h(z)) from the right leading back to z?

No, no, no! What h represents is a fixed point of an exponential function. So if \( a = h(z) \) then \( a = z^a \) and if you are taking the base-z logarithm of both sides then you can see that \( \log_z(a) = a \) which means that a is not only a fixed point of the base-z exponential function, but it is also a fixed point of the base-z logarithm function as well. And if it is a fixed point of the logarithm function, then there is no amount of iteration that will give any other output, so \( \log_z^{\infty}(a) = a \), so infinitely iterated logarithms are not the inverse of infinitely iterated exponentials, only \( z = a^{1/a} \) is the inverse of infinitely iterated exponentials.

Andrew Robbins