Is there an inverse operation of tetration for values between 0 and .3?

Question

So I don't understand everything with tetration, but the graph of $^2x$ or $x^x$ does not have any values between 0 and .3 on the y axis. So if we were to trying inverse tetration on say .2 what is the solution? I realize that negative even integers give answers in this range, but the results don't seem to cover all numbers between 0 and .3.

This is technically a second question, but if there isn't a solution what does that mean for mathematics or is it something similar in nature to that of $i$?

Answer 1

The square super-root is not real-valued for $0<x<e^{-1/e}$. Note that for any $x<0$ that is not an integer, $\Im(x^x)\ne0$.

Answer 2

Technically, for $^x b$, even for $b<e^{1/e}$, The Kneser Tetration becomes ambiguous. For real bases $b>e^{1/e}$, Tetration is well defined, and analytic with singularities at negative integers<=-2. The base $b=e^{1/e}$ is the branch point, where iterating no longer grows arbitrarily large. I investigated what happens to Tetration when we extend it analytically to complex bases, and it turns out that for $b<e^{1/e}$, Tetration is no longer real valued at the real axis. See complex base tetration link

Consider $b=\sqrt{2}$, which has two fixed points, L1=2, and L2=4. Most of the time, when people talk about Tetration for $1<b<e^{1/e}$, they switch to looking at the attracting fixed point, in this case L1=2. Then Tet(z) has the familiar definition, logarithmic singularity at Tet(-2), Tet(-1)=0, Tet(0)=1, and Tet(1)=b, and in the limit as $n\to \infty$, you get the attracting fixed point L1, which is 2 for $b=\sqrt{2}$. But for bases $b>e^{1/e}$, we are using both complex conjugate fixed points to generate Kneser's real valued at the real axis Tetration. And if we move the base in a circle around $e^{1/e}$ slowly using complex bases, from a real base greater than $e^{1/e}$ to one less than $e^{1/e}$, then we get to a function still uses both the attracting and repelling fixed points, but the function is no longer real valued at the real axis. Using the attracting fixed point is not the same function as Kneser's Tetration.