The ultrafunction really seems to be something of a mathematical gold mine (or at least a copper mine), and as I was thinking about it earlier, I’ve had some more ideas about it.
Recall that the ultrafunction is defined thus:
u(f(a),b) = f(f(f(…f(a)…) (repeated b times)
Are there any functions f_i(a) such that u(f_i(a),n) = f_i(a)? I shall refer to these as ultraidentity functions. Here are a few, defined on the set of real numbers:
f(x) = x
f(x) = c
There are also some semiultraidentity functions, which I’ll refer to as n-SI functions. The simplest example of a 2-SI function is:
f(x) = 1/x
Since 1/(1/x) = x, and then f(x) = 1/x, and so on ad infinitum.
f(x) = 1/(x^2), however, is not an ultraidentity function or an SI function, and in fact, taking the ultrafunction of 1/(x^2) repeatedly yields:
1/(1/x^2)^2 = x^4, 1/(x^4)^2 = 1/(x^8), etc.