# Is it possible to find aproximation of conformal map from sequences of complex points?

I want to find equation of conformal map (= Fatou function $\Psi : z \to u$ ) which:

• maps some region of complex plane ( attracting petal) to right half of complex plane in u coordinate $Re(u) > 0$
• transforms function $f(z)$ to unit translation $F : u \to u+1$
• unrolls invariant curvs ( orbits ) : maps "circles" to straight lines

Can I find equation which aproximates such map from sequences of points ( complex numbers) ?

The easiest case is $f(z)= z^2 + z$ which has parabolic fixed point at origin ( z=0). Then $\Psi(z) = -1/z$ and $F : u \to u+1+1/(u-1)$, where $2/(u-1)$ is error term ( Adrien Douady, Does a Julia set depend continuously on the polynomial? )

Sequences lay along curves shown inside main chessboard boxe on this image The image is not perfect near boundaries of chessboard box ( there are kinks and curves seems to cross boundary )

On this image one can see the u and z planes for th case f(z)=z^2+z. Src code

• This common point is a fixed point of f(z). The split looks better. – Adam Aug 12 '16 at 20:31
• Yes : "The first step in constructing Fatou coordinate for f0 consists in lifting f0 to a neighborhood of infinity by the coordinate change z→ −1/(qz^q)." arxiv.org/abs/1004.5536. I do not know how to do next steps. I can compute sequences z, f(z), ... easly. Can I ( and how) use it to find more precise aproximation ? – Adam Aug 12 '16 at 20:44
• Can you describe what you have done as an answer ? – Adam Aug 12 '16 at 21:13
• mathoverflow.net/questions/45608/… – Adam Aug 13 '16 at 9:30
• math.stackexchange.com/questions/911818/… – Adam Aug 13 '16 at 9:31