I tried to do analytically one case:

lower fixed point

we get the base by

if then

but :

At this point I do not know what to do with limits -how do You apply n to iterating logarithms, and at the same time to iteration of exponentiation of 1/e inside it? If we could move limit inside, than:

And

for all n, so

which equals

I know it is not complex but real...

Must be a mistake with the limits, but looks nice still...

If someone could explain me on this example how I should have proceeded after the place where I got to limit taking inside iterated logarithm, I promise never to make the same mistake.

If we take the same but:

Then I have to go through the whole procedure again.

In this case, instead of we will have 0, and + before tower as:

this seems to equal 1. So with negative height:

Ivars

lower fixed point

we get the base by

if then

but :

At this point I do not know what to do with limits -how do You apply n to iterating logarithms, and at the same time to iteration of exponentiation of 1/e inside it? If we could move limit inside, than:

And

for all n, so

which equals

I know it is not complex but real...

Must be a mistake with the limits, but looks nice still...

If someone could explain me on this example how I should have proceeded after the place where I got to limit taking inside iterated logarithm, I promise never to make the same mistake.

If we take the same but:

Then I have to go through the whole procedure again.

In this case, instead of we will have 0, and + before tower as:

this seems to equal 1. So with negative height:

Ivars