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