I'd also like to add that if we introduce theta mappings, you can construct more illustrious iterations, but they are sort of offshoots of regular iteration.
For example \(\text{tet}_{1,\log(2)/2}(s)\) constructed through the beta mapping, constructs a function:
\[
f^{\circ s}(z) : \mathbb{C}_{\Re(s)>0} / B \times \mathcal{A}_0/C \to \mathcal{A}_0\\
\]
Where here \(\mathcal{A}_0\) is the immediate basin about the fixed point \(2\) of \(f(z) = \sqrt{2}^z\)--where \(B\) is a branch cut, and \(C\) is another branch cut spawning from the fixed point \(2\). So this wouldn't be holomorphic at \(z= 2\).
This looks a lot like the regular iteration, but it perturbs the regular iteration so that instead of period \(-2\pi i/\log\log(2)\) it now has period \(2 \pi i\). And by consequence adding a lot fractal branching and singularities.
You can construct this with a theta mapping on the original regular iteration.
This is sort of just an off shoot of the regular iteration though, and isn't related to the theta mappings of crescent iteration.
It is also holomorphic on a smaller domain than the regular iteration or the crescent iteration.
Just adding that theta mappings can make uncountably many iterations--not just the crescent iteration. Though it is by far the most important one.
For example \(\text{tet}_{1,\log(2)/2}(s)\) constructed through the beta mapping, constructs a function:
\[
f^{\circ s}(z) : \mathbb{C}_{\Re(s)>0} / B \times \mathcal{A}_0/C \to \mathcal{A}_0\\
\]
Where here \(\mathcal{A}_0\) is the immediate basin about the fixed point \(2\) of \(f(z) = \sqrt{2}^z\)--where \(B\) is a branch cut, and \(C\) is another branch cut spawning from the fixed point \(2\). So this wouldn't be holomorphic at \(z= 2\).
This looks a lot like the regular iteration, but it perturbs the regular iteration so that instead of period \(-2\pi i/\log\log(2)\) it now has period \(2 \pi i\). And by consequence adding a lot fractal branching and singularities.
You can construct this with a theta mapping on the original regular iteration.
This is sort of just an off shoot of the regular iteration though, and isn't related to the theta mappings of crescent iteration.
It is also holomorphic on a smaller domain than the regular iteration or the crescent iteration.
Just adding that theta mappings can make uncountably many iterations--not just the crescent iteration. Though it is by far the most important one.
