Does tetration take the right half plane to itself?

[sheldonison]

This question is rather straightforward. Let . We know that there exists a unique bounded tetration function that is holomorphic for . But is ? Does the bounded tetration function take the right half plane to itself?

So are you asking if it is a 1-1 mapping from the right half of the plane to itself?  Certainly there's a singularity at z=-2.  
Lets think about base sqrt(2), with two fixed points, 2 and 4. As gets arbitrarily large  

At the imaginary axis, sexp(0)=1.  At half the period, sexp(period/2) the value would have grown to something like 2.468.  In between, imag(z) gets to about +/- 0.7208.

So at the imaginary axis, it will roughly "circle" or loop around the fixed point of 2 an infinite number of times, as follows:  This very roughly circular path is what the right half of the plane is mapped to!   Of course, it would be more complicated if I repeated this experiment at the singularity at z=-2 and again mapped one imaginary period, but I don't see the plane mapped to itself at z=-2 either.  The imaginary period is  17.143