Fractional iteration of x^2+1 at infinity and fractional iteration of exp

[mike3]

then taking your formula for \( f^{-n}(f^n(x) \sqrt{2}) \)

Its a power, not multiplication: f^{-n}(f^n(x)^{\sqrt{2}})

Oh, well, duh, thanks. Now it works better Guess I just didn't notice the superscripting.

Anyway, I think this is not analytic at 0. The iterates of g so formed have a branch point at 0, and also a complementary one at infinity (note that if there is a BP at 0, there must be one at inf, since "circling about inf" is equivalent to circling about 0). The conjugate simply exchanges these two branch points. This would explain how it can approach \( |x| \) as \( t \rightarrow 0 \).