Merged fixpoints of 2 iterates ? Asymptotic ? [2019]

[tommy1729]

Consider the function f(z) = exp(z) + z.

f(z) has no finite fixpoints.

We do iterates ( fractional near the real line ) by moving the fixpoint at oo to a finite place. 
Of other ways.

I Came to consider the following

f^[2](z) = exp(exp(z)) exp(z) + exp(z) + z.

The finite fixpoints are then 

Exp(z) ( exp(exp(z)) +1 )  + z = z

=>

Exp(z) ( exp(exp(z)) + 1) = 0

=> 

Z = ln( ln(-1) ) 

One solution is

 ln( pi ) + pi / 2 i = A

And another the complex conjugate of A : B.

Now i wonder if we take the half- iterates of f^[2] from those 2 fixpoints ( A and B ) based on koenigs solutions ,

Call them 

FA(z) and FB(z).

And then take the average of them : 

C(z) = ( FA(z) + FB(z))/2  
[ i call this merged fixpoints method ]


Then is C(x) , where x is real , close to f(x) ??

And how does C(z) behave on the complex plane ?
----

Similar questions for the merged fixpoints of g(z,t) = (z^2 + t) + z with t ≥ 1.

Notice g has only 2 fixpoints.


Plots would be nice too.


Regards

Tommy1729

[sheldonison]

Consider the function f(z) = exp(z) + z.

f(z) has no finite fixpoints.

We do iterates ( fractional near the real line ) by moving the fixpoint at oo to a finite place. 
...
Now i wonder if we take the half- iterates of f^[2] from those 2 fixpoints ( A and B ) based on koenigs solutions 

Iterating \( z\mapsto \exp(z)+z \) is congruent to iterating
\( y \mapsto y{e^y};\;\;\;y=e^z \)

Then we use the method of Ecalle to generate the Abel function for the parabolic fixed point of \( y=y{e^y} \) at y=0, which corresponds to the fixed point at \( z=-\infty \).  The two periodic fixed points correspond to \( y=\pm\pi i \)

Then iterating y is the inverse of iterating the lambert-w function; so that is well defined too.  The problem of using the pair of two-periodic fixed points to generate a real valued soution requires something much much more complicated than just Koenigs; probably the Op has in mind something more like Kneser's method of using a complex conjugate pair of repelling fixed points?   

Similar questions for the merged fixpoints of g(z,t) = (z^2 + t) + z 

I have implemented iterating \( g(z)=z^2+z+t;\;\;z\mapsto g(z);\;\;g^{[\circ z]} \) where \( \Re(t)>0 \), using both complex conjugate fixed points in a method analogous to Kneser's solution for tetration.  In complex dynamics, this Abel function is the perturbed fatou coordinate.  This implementation is actually one of the dozen or so things included in my program fatou.gp by using the parameter x2mode=1.

Code:

\r fatou.gp
x2mode=1;
loop(0.5);  /* this is my value for k; equal to the Op's "t"; works for k<=0.68; crashes for k>0.69 */
abel(5);    /*  3.33194722643578               */
invabel(4); /* 14.7851562500000; invabel(0)=0  */