Real and complex behaviour of the base change function (was: The "cheta" function)

[jaydfox]

Now I realized that \( f_n \) has a lot of singularities except for \( n=1,2 \) where it is a linear function.

But Bo, these zeroes are a trivial result of the "change of base" concept.

If, starting with the number 1, we perform a change of base operation, from base e to base eta, we get a real number, approximately 6.3344. If we instead start with 0 in base e, we get approximately 5.0179. This is not surprising, as 5.0179 is log_eta(6.3344).

Well, log_eta(5.0179) is approximately 4.3846. Thus, changing base from eta back to base e, we should expect 6.3344 to be 1, 5.0179 to be log_e(1), which is 0, and 4.3846 to be log(log(1)), which is (drum roll please) negative infinity.

What is not as trivial to determine is where the "other" singularities are, if in fact there are any. I assume there are other singularities, but I am so far not having the best of luck in finding them.

By the way, it occurs to me that for converting from base eta to base e (or any other base, I suppose), we must pick a starting point on the real line, which is either above the fixed point at e, or below it. These two approaches will give different results (for if they didn't, then regular iteration above or below would be equivalent, and we know that it is not).

So bear in mind that I'm speaking from the point of view of starting above the fixed point, e.g., at about 5.0179, which would get us back to 0 in base e.

Edit/Update: Sorry, Henryk, I re-read your post and realized that you had in fact specified that you were starting with a real value greater than e for n=3, and larger values as necessary for larger n.