The "cheta" function

[jaydfox]

Well, it can actually be applied to any base greater than 1.

So did someone check already whether base conversion of regular iteration gives again regular iteration (say at the lower fixed point)?

Well, when I tried a couple years ago, I got different results when using eta and sqrt(2) (using the upper fixed point), so I assume in general that base conversion does not give the same results as regular iteration from the upper fixed point.

Not sure about the lower fixed point, but I would not be surprised if it did match, because regular iteration from the lower fixed point is found by iteratively exponentiating until we reach the fixed point, and the first part of the change of base formula relies on iterative exponentiation.

And at any rate, for base b>eta, it definitely does not converge on the way "up", even if by some miracle it manages to converge on the way back "down"

For bases less than or equal to eta, it will converge on the lower fixed point as we go "up". For bases larger than eta, there aren't any real fixed points, so convergence never happens.

Sorry, I dont know what you mean by "converging on the way up/down".

Ah, sorry. On the way "up" refers to the iterative exponentiation, which for reals will tend to go "up". On the way down refers to the numbers getting smaller with iterated logarithms (at least for the reals).

I think of it as climbing up a mountain of iterated exponentials in one base, then back down a mountain in the other base (undoing the exponentials by taking logarithms). Just a metaphor, and your mileage may vary.