Why computing Fatou coordinate is so hard? I'm trying to make images of Fatou coordinate for some polynomial maps.  If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. 


*

*Is any example of map for which explicit formula of Fatou coordinate is known ?

*Why computing Fatou coordinate is so hard ?


I am very new to theory, so I am looking for a simple, intuitive answer.
Also I would like to switch from theory to computations and making images.
 A: Will Jagy has an overview of the formal Fatou coordinate (Jean Ecalle at Orsay) for a parabolic point at mathoverflow; I posted some pari-gp code to implement Ecalle's solution below.  Near, but not exactly at a parabolic point, the problem is much more difficult.  Give a function $f(z)$, we call the Fatou Coordinate/Abel function $\alpha(z)$.
$$\alpha(f(z)) = \alpha(z)+1$$
Then the iterated function would be $f^{o z} = \alpha^{-1}(z)$, and I have written a program that calculates $\alpha^{-1}$ for tetration, see the tetration forum, and investigated the properties near the parabolic fixed point (which is a branch point for this family of complex functions).  I have used the same method to calculate $\alpha^{-1}(z)$ for $x^2+c$ near the parabolic branch point, c=0.25, but have not posted it anywhere.  I would also be interested in any other responses.
Some other thoughts.  Consider the case where $f(x)=x^2+0.26$, which has two fixed points, $0.5+/-0.1i$.  The solution I was looking for treats both of these fixed points symmetrically and is based on extending Kneser's solution for tetration, which involves a Riemann mapping, which helps explain why computing such solutions is difficult.  If you only want to calculate $\alpha(z)$ for one of the two fixed points, then the Schroeder function provides a simple well defined solution for $\alpha(z)$.  
Finally, nearby c=0.25, there are also much more complicated parabolic points, where $f^{on}(z)$ is a parabolic point. Near such a point do we compute the Fatou coordinate for $f(z)$, or $f^{on}(z)$?  Will Jagy's link gives a solution for the Fatou coordinate of $f^{on}(z)$.  I also now know to compute the solution for $\alpha^{-1}(z)$ for $f(z)$ using both fixed points; I tried asking a question on math overflow, but I didn't get any relevant responses :(
You could also search parabolic implosion on the web, but I haven't seen any papers showing how to calculate $\alpha(z)$.
EDIT Here is a pari-gp program to implement Jean Ecalle's formal Abel Series, Fatou Coordinate solution for parabolic points with multiplier=1.  This is an asymptotic non-converging series, so there is an optimal number of terms to use, so you may have to iterate $f$ or $f^{-1}$ a few times to get optimally accurate results, so that the coeffient is closer to the fixed point of zero.
abelseries(fz,n) = {
  local(i,z,ns,m,rem);
  kabel=0;
  klog=0;
  m=1;
  while (polcoeff(fz,m+1)==0,m++);
  print("terms with negative coeffients= "m);
  for (i=-m,n,
    if (i==0, klog=acoeff, kabel=kabel+acoeff*x^i);
    rem = Ser(subst(kabel,x,fz) - kabel + klog*log(fz/x) - 1);
    z=polcoeff(rem,i+m);
    z=subst(z,acoeff,x);
    ns=-polcoeff(z,0)/polcoeff(z,1);
    kabel=subst(kabel,acoeff,ns);
    klog=subst(klog,acoeff,ns);
  );
  return([kabel,klog]);
}
/* evaluate kabel and klog after generating abelseries */
eabel(z) = { 
  z=subst(kabel,x,z)+klog*log(z);
  return(z);
}
fz = x+x^2;
abelseries(fz,9); /* initialize kabel and klog for x^2+x */
for (i=-1,9, if (i==0, print(klog"*log(x)"), print(polcoeff(kabel,i)"*x^"i)));
print (eabel(0.2)" "eabel(subst(fz,x,0.2)));
fz = x-2*x^3+x^4;
abelseries(fz,9); /* initialize kabel and klog for for x-2*x^3+x^4 */
print (eabel(0.2)" "eabel(subst(fz,x,0.2)));

A: Because of "The resurgent character of the Fatou coordinates" 
