[MSE] Shape of orbit of iterations with base b on Shell-Thron-region

[sheldonison]

There you also cover the dynamics for the rational case as well.  Since the Schroeder function doesn't exist when the multiplier at the fixed point is 1, or a rational root of 1, then one can use Ecalle's solution for the Abel function.  

Hi
I want to use fatou.gp.ecalle() evaluate Shell-Thron-region rational base, but my series input isn't work. can you give me some code for examples? 

And...I want to evaluate derivative for sexp/slog/pent/sroot, Can I just use deriv(sexptaylor()), deriv(slogtaylor()), deriv(kecalle), deriv(pentaylor()), deriv(gm)? 
I see the ihex_deriv() you use are not simple.

l=1/exp(1); /* the fixed point */
b=exp(-exp(1)); /* the base */
f=b^b^(l+x)-l; /* this is the f(f(x); a function with a derivative=1 and a fixed point of ~zero */
f=strip0fx(f); /* strip off the x^0 term which was only approximately zero; it is required to be exactly zero */

ecalle(f,16); /* initialize formal series for ecalle assymptotic; 16 positive terms; two neg terms; one log term */
z1=0.1; z2=b^b^(z1+l)-l; /* test example; z1=0.1; z2=f(f(0.1)); */
ecalleu(z1) /* [abel_function,abel_function_derivative]; [44.892671576806680805639501168174, */
ecalleu(z2) /* [abel_function,abel_function_derivative]; [45.892671576806680693723673529703, */

the closer z1 and z2 are to zero, the more accurate the asymptotic series; in this case the abel function is accurate to 16 decimal digits.  Notice that f has fixed point of zero, derivative of 1, and no x^2 term

f = x*  1.0000000000000000000000000000000
+x^ 2* -4.324402048806904993 E-39   /* ecalle smart enough to ignore x^2 */
+x^ 3* -1.2315093498217750378717379100958
+x^ 4*  0.83689737179948615587202206894090
+x^ 5*  0.90996916721907065130183768671435

This leads to an ecalle assymptotic form for f with two neg terms; a_2/x^2 + a_1/x; a log term, and we calculated 16 terms of the assymptotic above.
- Sheldon