An explanation for this?

[bo198214]

What is this proven algorithm

Michael Yampolski referred me to one of his articles [1].
I think you need a really good background in holomorphic dynamics to understand the article. Before that you should understand the construction of Shishikura [2] of the Fatou coordinates/Abel function where he uses the measurable Riemann theorem. Because Gaidashev gives a *constructive* measurable Riemann theorem, i.e. one where there is an algorithm to calculate it, and uses it to calculate the Fatou coordinates, if I understood that properly.

The Kneser construction is very specific, it constructs the (i.e. the unique) Fatou coordinates for exp (which I showed in [3]) with the (plain) Riemann theorem. Then Jaydfox used an algorithm for the Riemann mapping theorem to compare Kneser's with Andrew's solution.

Just wanted to show the similarity and that it seems we can not circumvent the (measurable) Riemann mapping theorem, which gives no closed form coefficients; which was regarding your question:

, and does it provide any clues to finding the explicit (or close enough, e.g. if in terms of, say, infinite sums) coefficients of the Taylor series

[1] Gaidashev, D., & Yampolsky, M. (2007). Cylinder renormalization of Siegel disks. Exp. Math., 16(2), 215–226.

[2] Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000).

[3] Trappmann, H., & Kouznetsov, D. (2010). Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Math.