iterating exp(z) + z/(1 + exp(z))

[tommy1729]

I come back to an old idea.

The issue with iterating is exp is that it's inverse is not real for negative real imput.

So I considered iterating exp(x) + x which has no finite fixpoint.

There is a thread about that.

BUT a better asymptotic to a linear function for negative x and exp for positive x is the simple 

exp(z) + z/(1 + exp(z))

It also has a closed form inverse and two fixpoints who can also be given in closed form.
That closed forms requires lambertW.

More methods will work here too.

But my main question is how its super/abel/semi-iterate etc behave and look on the complex plane.
How different it is from iterations of exp , dexp or 2sinh.

And - more advanced - how the super of the super differs from pentation.

I prefer the abel function to be completely monotone beyond a certain real value y.
( as i do for slog btw , still no solution as far as i know ??? )

Iterating this functions feels natural too me !

Regards

Tommy1729