bo198214 Wrote:How can one find the singularities of \( f^{\circ t}(x) \) and the complex fixed points of \( f(x)=\sqrt{2}^x \) in your drawing?

Actually, the easiest way is to look at the definition of the rslog itself. We know the two primary singularities are at 2 and 4. Therefore, we can find one tier of "images" of these two singularities with:

\( \log_{\sqrt{2}}(p)+\frac{2\pi k i}{\ln(\sqrt{2})},\ p \in \{2, 4\},\ \ k \in \mathbb{Z} \)

We can find the next tier of images by simply repeating this process: every branch of two iterated logarithms:

\( \log_{\sqrt{2}}\left(\log_{\sqrt{2}}(p)+\frac{2\pi j i}{\ln(\sqrt{2})}\left)+\frac{2\pi k i}{\ln(\sqrt{2})},\ p \in \{2, 4\},\ \ j, k \in \mathbb{Z} \)

This can be repeated indefinitely. Indeed, going back to the definition of the rslog, if we look at a solution for a finite n, then we can only repeat the process n times, and only for the images of the fixed point at 2. The fixed point at 4 is only a singularity when n goes to infinity.

And looking at j, k, etc., we can see that setting the innermost branches to 0 allows us to generalize a shorter sequence with the infinite sequence. We can then specify a singularity by indicating if its an image of the upper or lower fixed point, and then specify its infinite sequence of branch choices, starting from the outermost, with the sequence truncated if the branch choices are all 0 beyond a certain iterated logarithm.