Just as a background, let me give a short overview about the mathematically founded (as opposed to numerically good looking) approaches:
There is basically two constructions: at one fixed point, and at two fixed points.
Iteration at one fixed point - regular iteration:
The idea is here to make the iterated functions analytic/holomorphic at that fixed point (that is where the word "regular" comes from - in those times one used "regular" as a synonym for what we today call analytic/holomorphic) or at least asymptotically analytic (in the parabolic case). And one can show for the typical cases (hyperbolic/parabolic) that an (asymptotically) analytic solution exists as well as that it is unique by this demand - the method for hyperbolic fixed points is typically called Schröder iteration. This can be approached from the side of (formal) power series, but also there are (proven) limit formulas. The downside is that if the t-iterate is analytic at one fixed point it is not at any other for most t.
Iteration at two fixed points - crescent iteration (this term is my invention, do you have better ones, or will we make it our standard term?):
Kneser's approach is a particular construction of the more general Perturbed Fatou Coordinates. For the latter under certain conditions one can show that, there exists a holomorphic Abel function on a fundamental region/crescent (in the case of Kneser it is constructed via the Riemann mapping theorem, in the case of Perturbed Fatou-Coordinates the measurable Riemann mapping theorem is used.) and it is unique by demanding that it is injective on the fundamental region. Another uniqueness theorem comes from Paulsen, which states that is unique if sequences tending to the first fixed point are mapped by the Abel function to sequences with imaginary part going to infinity, and for the second fixed point to negative infinity (though it is only proven in the context of exponentials, I think it could be a general criterion). The corresponding iteration \(f^t(z) = \alpha^{-1}(t+\alpha(z))\) is not analytic at both fixed points.
While the regular iteration is quite well studied and there are practical formulas for iteration, for the crescent iteration it is a bit more vague (due to the difficulties numerically handling (measurable) Riemann mappings). But typically people identify the Levenstein (Sheldon), Kouznetsov and the Paulsen numerical methods with the crescent iteration.
Both methods are not identical, we know that for \(b>e^{1/e}\) regular iteration at one of the primary fixed points is not real valued on the real axis, and vice versa we know (by computation of Paulsen) that the (b-continued) crescent iteration for bases \(b<e^{1/e}\) is not real valued on the real axis (though only by a tiny imaginary part).
So for making a fractal anything goes, I would say, as long as you add which method you used!
As a side note: I don't think the term tetration-fractal is quite right, as we are not iterating tetration, but exponentials - we don't need any knowledge about tetration, to do this fractal. Similarly with the pentation fractal.
There is basically two constructions: at one fixed point, and at two fixed points.
Iteration at one fixed point - regular iteration:
The idea is here to make the iterated functions analytic/holomorphic at that fixed point (that is where the word "regular" comes from - in those times one used "regular" as a synonym for what we today call analytic/holomorphic) or at least asymptotically analytic (in the parabolic case). And one can show for the typical cases (hyperbolic/parabolic) that an (asymptotically) analytic solution exists as well as that it is unique by this demand - the method for hyperbolic fixed points is typically called Schröder iteration. This can be approached from the side of (formal) power series, but also there are (proven) limit formulas. The downside is that if the t-iterate is analytic at one fixed point it is not at any other for most t.
Iteration at two fixed points - crescent iteration (this term is my invention, do you have better ones, or will we make it our standard term?):
Kneser's approach is a particular construction of the more general Perturbed Fatou Coordinates. For the latter under certain conditions one can show that, there exists a holomorphic Abel function on a fundamental region/crescent (in the case of Kneser it is constructed via the Riemann mapping theorem, in the case of Perturbed Fatou-Coordinates the measurable Riemann mapping theorem is used.) and it is unique by demanding that it is injective on the fundamental region. Another uniqueness theorem comes from Paulsen, which states that is unique if sequences tending to the first fixed point are mapped by the Abel function to sequences with imaginary part going to infinity, and for the second fixed point to negative infinity (though it is only proven in the context of exponentials, I think it could be a general criterion). The corresponding iteration \(f^t(z) = \alpha^{-1}(t+\alpha(z))\) is not analytic at both fixed points.
While the regular iteration is quite well studied and there are practical formulas for iteration, for the crescent iteration it is a bit more vague (due to the difficulties numerically handling (measurable) Riemann mappings). But typically people identify the Levenstein (Sheldon), Kouznetsov and the Paulsen numerical methods with the crescent iteration.
Both methods are not identical, we know that for \(b>e^{1/e}\) regular iteration at one of the primary fixed points is not real valued on the real axis, and vice versa we know (by computation of Paulsen) that the (b-continued) crescent iteration for bases \(b<e^{1/e}\) is not real valued on the real axis (though only by a tiny imaginary part).
So for making a fractal anything goes, I would say, as long as you add which method you used!
As a side note: I don't think the term tetration-fractal is quite right, as we are not iterating tetration, but exponentials - we don't need any knowledge about tetration, to do this fractal. Similarly with the pentation fractal.
