Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)

[Ivars]

I checked with the routine in PariTTY most kindly provided by Gottfried to me few trajectories of interesting points on real axis from fractal picture, \( e^{-\pi/2}=I^I \).

If we look for \( h(e^{-\pi/2}) \) without I on top, it is just real number 0.4745409995126511230174679441, while iterating with I on top obviosuly leads to oscillating behaviour since first iteration leads to \( (I^I)^I=I^{-1}= -I. \)

So I am looking for (a^(a^.........(a^I)) where \( a= e^{-\pi/2}=0.2078795763507619085469556198 \)

mybase = exp(-Pi/2)
%27 = 0.2078795763507619085469556198

ffrac_init(1.0*I,mybase)
ffrac_init(1.0*I,mybase)
%28 = 1.000000000000000000000000000*I

for(k=1,20,print(ffrac))
for(k=1,20,print(ffrac))
-2.524354897 E-29 - 1.000000000000000000000000000*I
-2.524354897 E-29 + 1.000000000000000000000000000*I
-7.57306469 E-29 - 1.000000000000000000000000000*I
-7.57306469 E-29 + 1.000000000000000000000000000*I
-2.019483917 E-28 - 1.000000000000000000000000000*I
-2.019483917 E-28 + 1.000000000000000000000000000*I
-5.301145284 E-28 - 1.000000000000000000000000000*I
-5.301145284 E-28 + 1.000000000000000000000000001*I
-1.337908095 E-27 - 1.000000000000000000000000001*I
-1.337908095 E-27 + 1.000000000000000000000000002*I
-3.332148464 E-27 - 1.000000000000000000000000002*I
-3.332148464 E-27 + 1.000000000000000000000000005*I
-8.25464051 E-27 - 1.000000000000000000000000005*I
-8.25464051 E-27 + 1.000000000000000000000000013*I
-2.039678757 E-26 - 1.000000000000000000000000013*I
-2.039678757 E-26 + 1.000000000000000000000000032*I
-5.036088018 E-26 - 1.000000000000000000000000032*I
-5.036088018 E-26 + 1.000000000000000000000000079*I
-1.242739915 E-25 - 1.000000000000000000000000079*I
-1.242739915 E-25 + 1.000000000000000000000000195*I

But perhaps this corresponds to case You mentioned, that iterations with z on top of a does not converge.

Ivars