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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 imageSequences lay along curves shown 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 enter image description here

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  • $\begingroup$ This common point is a fixed point of f(z). The split looks better. $\endgroup$
    – Adam
    Aug 12, 2016 at 20:31
  • $\begingroup$ 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 ? $\endgroup$
    – Adam
    Aug 12, 2016 at 20:44
  • $\begingroup$ Can you describe what you have done as an answer ? $\endgroup$
    – Adam
    Aug 12, 2016 at 21:13
  • $\begingroup$ mathoverflow.net/questions/45608/… $\endgroup$
    – Adam
    Aug 13, 2016 at 9:30
  • $\begingroup$ math.stackexchange.com/questions/911818/… $\endgroup$
    – Adam
    Aug 13, 2016 at 9:31

1 Answer 1

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There are three methods for numerically approximating conformal maps from arbitrary domains to the unit disc:

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