Here's a graph of \(\text{tet}_{\sqrt{2}}\) with period \(2 \pi i/\lambda = 2 \pi i/0.38\). This is just before it turns into regular iteration. The weak julia set is the bits of black/white, which will turn into nothing as we let \(\lambda \to 0.3665.... = - \log \log 2\). These are again, fractal lines and the such, a.e. this tetration converges (When we think of a lebesgue area measure).
For reference, this is what the regular iteration looks like:
I'd also like to add that for \(b = e\) and say, period \(2 \pi i\), that this will be SMOOTH on \((-2,\infty)\) and be a smooth bijection to \(\mathbb{R}\). In the complex plane we get a weird equivalent of "smooth". In the sense that we make an arbitrarily accurate approximation, but the taylor expansion is nowhere analytic.
When you let \(\lambda \to 0\) though, as you let \(2 \pi i/\lambda \to i \infty\), you get a much calmer feature in the taylor expansion. And by proxy, the Taylor series is no longer zero radius of convergence. This is sort of the reallly reallllllly hard idea, I have no idea how to justify. All I know is that the Taylor coefficients calm down a lot, and it starts to look like the Crescent iteration (taylor coefficient wise).
For reference, this is what the regular iteration looks like:
I'd also like to add that for \(b = e\) and say, period \(2 \pi i\), that this will be SMOOTH on \((-2,\infty)\) and be a smooth bijection to \(\mathbb{R}\). In the complex plane we get a weird equivalent of "smooth". In the sense that we make an arbitrarily accurate approximation, but the taylor expansion is nowhere analytic.
When you let \(\lambda \to 0\) though, as you let \(2 \pi i/\lambda \to i \infty\), you get a much calmer feature in the taylor expansion. And by proxy, the Taylor series is no longer zero radius of convergence. This is sort of the reallly reallllllly hard idea, I have no idea how to justify. All I know is that the Taylor coefficients calm down a lot, and it starts to look like the Crescent iteration (taylor coefficient wise).
